Daily Papers

The advancement of high-performance computing has enabled the generation of large direct numerical simulation (DNS) datasets of turbulent flows, driving the need for efficient compression/decompression techniques that reduce storage demands while maintaining fidelity. Traditional methods, such as the discrete wavelet transform, cannot achieve compression ratios of 8 or higher for complex turbulent flows without introducing significant encoding/decoding errors. On the other hand, a super-resolution-based generative adversarial network (SR-GAN), called ZipGAN, can accurately reconstruct fine-scale features, preserving velocity gradients and structural details, even at a compression ratio of 512, thanks to the more efficient representation of the data in compact latent space. Additional benefits are ascribed to adversarial training. The high GAN training time is significantly reduced with a progressive transfer learning approach and, once trained, they can be applied independently of the Reynolds number. It is demonstrated that ZipGAN can enhance dataset temporal resolution without additional simulation overhead by generating high-quality intermediate fields from compressed snapshots. The ZipGAN discriminator can reliably evaluate the quality of decoded fields, ensuring fidelity even in the absence of original DNS fields. Hence, ZipGAN compression/decompression method presents a highly efficient and scalable alternative for large-scale DNS storage and transfer, offering substantial advantages over the DWT methods in terms of compression efficiency, reconstruction fidelity, and temporal resolution enhancement.

Transformer neural operators have recently become an effective approach for surrogate modeling of systems governed by partial differential equations (PDEs). In this paper, we introduce a modified implicit factorized transformer (IFactFormer-m) model which replaces the original chained factorized attention with parallel factorized attention. The IFactFormer-m model successfully performs long-term predictions for turbulent channel flow, whereas the original IFactFormer (IFactFormer-o), Fourier neural operator (FNO), and implicit Fourier neural operator (IFNO) exhibit a poor performance. Turbulent channel flows are simulated by direct numerical simulation using fine grids at friction Reynolds numbers Re_{tau}approx 180,395,590, and filtered to coarse grids for training neural operator. The neural operator takes the current flow field as input and predicts the flow field at the next time step, and long-term prediction is achieved in the posterior through an autoregressive approach. The results show that IFactFormer-m, compared to other neural operators and the traditional large eddy simulation (LES) methods including dynamic Smagorinsky model (DSM) and the wall-adapted local eddy-viscosity (WALE) model, reduces prediction errors in the short term, and achieves stable and accurate long-term prediction of various statistical properties and flow structures, including the energy spectrum, mean streamwise velocity, root mean square (rms) values of fluctuating velocities, Reynolds shear stress, and spatial structures of instantaneous velocity. Moreover, the trained IFactFormer-m is much faster than traditional LES methods. By analyzing the attention kernels, we elucidate the reasons why IFactFormer-m converges faster and achieves a stable and accurate long-term prediction compared to IFactFormer-o. Code and data are available at: https://github.com/huiyu-2002/IFactFormer-m.

Advances in time-resolved Particle Tracking Velocimetry (4D-PTV) techniques have been consistently revealed more accurate Lagrangian particle motions. A novel track initialisation technique as a complementary part of 4D-PTV, based on local temporal and spatial coherency of neighbour trajectories, is proposed. The proposed Lagrangian Coherent Track Initialisation (LCTI) applies physics-based Finite Time Lyapunov Exponent (FTLE) to build four frame coherent tracks. We locally determine the boundaries (i.e., ridges) of Lagrangian Coherent Structures (LCS) among neighbour trajectories by using FTLE to distinguish clusters of coherent motions. To evaluate the proposed technique, we created an open-access synthetic Lagrangian and Eulerian dataset of the wake downstream of a smooth cylinder at a Reynolds number equal to 3900 obtained from 3D Direct Numerical Simulation (DNS). The dataset is available to the public. Performance of the proposed method based on three characteristic parameters, temporal scale, particle concentration (i.e., density), and noise ratio, showed robust behaviour in finding true tracks compared to the recent initialisation algorithms. Sensitivity of LCTI to the number of untracked and wrong tracks are also discussed. We address the capability of using the proposed method as a function of a 4D-PTV scheme in the Lagrangian Particle Tracking challenge for a flow with high particle densities. Finally, the LCTI behaviour was assessed in a real jet impingement experiment. LCTI was found to be a reliable tracking tool in complex flow motions, with a strength revealed for flows with high particle concentrations.

We investigate the coupling effects of the two-phase interface, viscosity ratio, and density ratio of the dispersed phase to the continuous phase on the flow statistics in two-phase Taylor-Couette turbulence at a system Reynolds number of 6000 and a system Weber number of 10 using interface-resolved three-dimensional direct numerical simulations with the volume-of-fluid method. Our study focuses on four different scenarios: neutral droplets, low-viscosity droplets, light droplets, and low-viscosity light droplets. We find that neutral droplets and low-viscosity droplets primarily contribute to drag enhancement through the two-phase interface, while light droplets reduce the system's drag by explicitly reducing Reynolds stress due to the density dependence of Reynolds stress. Additionally, low-viscosity light droplets contribute to greater drag reduction by further reducing momentum transport near the inner cylinder and implicitly reducing Reynolds stress. While interfacial tension enhances turbulent kinetic energy (TKE) transport, drag enhancement is not strongly correlated with TKE transport for both neutral droplets and low-viscosity droplets. Light droplets primarily reduce the production term by diminishing Reynolds stress, whereas the density contrast between the phases boosts TKE transport near the inner wall. Therefore, the reduction in the dissipation rate is predominantly attributed to decreased turbulence production, causing drag reduction. For low-viscosity light droplets, the production term diminishes further, primarily due to their greater reduction in Reynolds stress, while reduced viscosity weakens the density difference's contribution to TKE transport near the inner cylinder, resulting in a more pronounced reduction in the dissipation rate and consequently stronger drag reduction. Our findings provide new insights into the turbulence modulation in two-phase flow.

Modern techniques for physical simulations rely on numerical schemes and mesh-refinement methods to address trade-offs between precision and complexity, but these handcrafted solutions are tedious and require high computational power. Data-driven methods based on large-scale machine learning promise high adaptivity by integrating long-range dependencies more directly and efficiently. In this work, we focus on fluid dynamics and address the shortcomings of a large part of the literature, which are based on fixed support for computations and predictions in the form of regular or irregular grids. We propose a novel setup to perform predictions in a continuous spatial and temporal domain while being trained on sparse observations. We formulate the task as a double observation problem and propose a solution with two interlinked dynamical systems defined on, respectively, the sparse positions and the continuous domain, which allows to forecast and interpolate a solution from the initial condition. Our practical implementation involves recurrent GNNs and a spatio-temporal attention observer capable of interpolating the solution at arbitrary locations. Our model not only generalizes to new initial conditions (as standard auto-regressive models do) but also performs evaluation at arbitrary space and time locations. We evaluate on three standard datasets in fluid dynamics and compare to strong baselines, which are outperformed both in classical settings and in the extended new task requiring continuous predictions.

Numerical simulation of non-linear partial differential equations plays a crucial role in modeling physical science and engineering phenomena, such as weather, climate, and aerodynamics. Recent Machine Learning (ML) models trained on low-resolution spatio-temporal signals have shown new promises in capturing important dynamics in high-resolution signals, under the condition that the models can effectively recover the missing details. However, this study shows that significant information is often lost in the low-resolution down-sampled features. To address such issues, we propose a new approach, namely Temporal Stencil Modeling (TSM), which combines the strengths of advanced time-series sequence modeling (with the HiPPO features) and state-of-the-art neural PDE solvers (with learnable stencil modeling). TSM aims to recover the lost information from the PDE trajectories and can be regarded as a temporal generalization of classic finite volume methods such as WENO. Our experimental results show that TSM achieves the new state-of-the-art simulation accuracy for 2-D incompressible Navier-Stokes turbulent flows: it significantly outperforms the previously reported best results by 19.9% in terms of the highly-correlated duration time and reduces the inference latency into 80%. We also show a strong generalization ability of the proposed method to various out-of-distribution turbulent flow settings. Our code is available at "https://github.com/Edward-Sun/TSM-PDE".

Implicit Neural Spatial Representation (INSR) has emerged as an effective representation of spatially-dependent vector fields. This work explores solving time-dependent PDEs with INSR. Classical PDE solvers introduce both temporal and spatial discretizations. Common spatial discretizations include meshes and meshless point clouds, where each degree-of-freedom corresponds to a location in space. While these explicit spatial correspondences are intuitive to model and understand, these representations are not necessarily optimal for accuracy, memory usage, or adaptivity. Keeping the classical temporal discretization unchanged (e.g., explicit/implicit Euler), we explore INSR as an alternative spatial discretization, where spatial information is implicitly stored in the neural network weights. The network weights then evolve over time via time integration. Our approach does not require any training data generated by existing solvers because our approach is the solver itself. We validate our approach on various PDEs with examples involving large elastic deformations, turbulent fluids, and multi-scale phenomena. While slower to compute than traditional representations, our approach exhibits higher accuracy and lower memory consumption. Whereas classical solvers can dynamically adapt their spatial representation only by resorting to complex remeshing algorithms, our INSR approach is intrinsically adaptive. By tapping into the rich literature of classic time integrators, e.g., operator-splitting schemes, our method enables challenging simulations in contact mechanics and turbulent flows where previous neural-physics approaches struggle. Videos and codes are available on the project page: http://www.cs.columbia.edu/cg/INSR-PDE/

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the L^2 difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.

We present DIRECT-3D, a diffusion-based 3D generative model for creating high-quality 3D assets (represented by Neural Radiance Fields) from text prompts. Unlike recent 3D generative models that rely on clean and well-aligned 3D data, limiting them to single or few-class generation, our model is directly trained on extensive noisy and unaligned `in-the-wild' 3D assets, mitigating the key challenge (i.e., data scarcity) in large-scale 3D generation. In particular, DIRECT-3D is a tri-plane diffusion model that integrates two innovations: 1) A novel learning framework where noisy data are filtered and aligned automatically during the training process. Specifically, after an initial warm-up phase using a small set of clean data, an iterative optimization is introduced in the diffusion process to explicitly estimate the 3D pose of objects and select beneficial data based on conditional density. 2) An efficient 3D representation that is achieved by disentangling object geometry and color features with two separate conditional diffusion models that are optimized hierarchically. Given a prompt input, our model generates high-quality, high-resolution, realistic, and complex 3D objects with accurate geometric details in seconds. We achieve state-of-the-art performance in both single-class generation and text-to-3D generation. We also demonstrate that DIRECT-3D can serve as a useful 3D geometric prior of objects, for example to alleviate the well-known Janus problem in 2D-lifting methods such as DreamFusion. The code and models are available for research purposes at: https://github.com/qihao067/direct3d.

The spatiotemporal resolution of Partial Differential Equations (PDEs) plays important roles in the mathematical description of the world's physical phenomena. In general, scientists and engineers solve PDEs numerically by the use of computationally demanding solvers. Recently, deep learning algorithms have emerged as a viable alternative for obtaining fast solutions for PDEs. Models are usually trained on synthetic data generated by solvers, stored on disk and read back for training. This paper advocates that relying on a traditional static dataset to train these models does not allow the full benefit of the solver to be used as a data generator. It proposes an open source online training framework for deep surrogate models. The framework implements several levels of parallelism focused on simultaneously generating numerical simulations and training deep neural networks. This approach suppresses the I/O and storage bottleneck associated with disk-loaded datasets, and opens the way to training on significantly larger datasets. Experiments compare the offline and online training of four surrogate models, including state-of-the-art architectures. Results indicate that exposing deep surrogate models to more dataset diversity, up to hundreds of GB, can increase model generalization capabilities. Fully connected neural networks, Fourier Neural Operator (FNO), and Message Passing PDE Solver prediction accuracy is improved by 68%, 16% and 7%, respectively.

Simulations of turbulent flows in 3D are one of the most expensive simulations in computational fluid dynamics (CFD). Many works have been written on surrogate models to replace numerical solvers for fluid flows with faster, learned, autoregressive models. However, the intricacies of turbulence in three dimensions necessitate training these models with very small time steps, while generating realistic flow states requires either long roll-outs with many steps and significant error accumulation or starting from a known, realistic flow state - something we aimed to avoid in the first place. Instead, we propose to approach turbulent flow simulation as a generative task directly learning the manifold of all possible turbulent flow states without relying on any initial flow state. For our experiments, we introduce a challenging 3D turbulence dataset of high-resolution flows and detailed vortex structures caused by various objects and derive two novel sample evaluation metrics for turbulent flows. On this dataset, we show that our generative model captures the distribution of turbulent flows caused by unseen objects and generates high-quality, realistic samples amenable for downstream applications without access to any initial state.

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering and mathematical problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, and more. While this has led to solving many complex phenomena, there are some limitations. Conventional approaches such as Finite Element Methods (FEMs) and Finite Differential Methods (FDMs) require considerable time and are computationally expensive. In contrast, data driven machine learning-based methods such as neural networks provide a faster, fairly accurate alternative, and have certain advantages such as discretization invariance and resolution invariance. This article aims to provide a comprehensive insight into how data-driven approaches can complement conventional techniques to solve engineering and physics problems, while also noting some of the major pitfalls of machine learning-based approaches. Furthermore, we highlight, a novel and fast machine learning-based approach (~1000x) to learning the solution operator of a PDE operator learning. We will note how these new computational approaches can bring immense advantages in tackling many problems in fundamental and applied physics.

Computational Fluid Dynamics (CFD) simulation by the numerical solution of the Navier-Stokes equations is an essential tool in a wide range of applications from engineering design to climate modeling. However, the computational cost and memory demand required by CFD codes may become very high for flows of practical interest, such as in aerodynamic shape optimization. This expense is associated with the complexity of the fluid flow governing equations, which include non-linear partial derivative terms that are of difficult solution, leading to long computational times and limiting the number of hypotheses that can be tested during the process of iterative design. Therefore, we propose DeepCFD: a convolutional neural network (CNN) based model that efficiently approximates solutions for the problem of non-uniform steady laminar flows. The proposed model is able to learn complete solutions of the Navier-Stokes equations, for both velocity and pressure fields, directly from ground-truth data generated using a state-of-the-art CFD code. Using DeepCFD, we found a speedup of up to 3 orders of magnitude compared to the standard CFD approach at a cost of low error rates.

Recent advances in deep learning have inspired numerous works on data-driven solutions to partial differential equation (PDE) problems. These neural PDE solvers can often be much faster than their numerical counterparts; however, each presents its unique limitations and generally balances training cost, numerical accuracy, and ease of applicability to different problem setups. To address these limitations, we introduce several methods to apply latent diffusion models to physics simulation. Firstly, we introduce a mesh autoencoder to compress arbitrarily discretized PDE data, allowing for efficient diffusion training across various physics. Furthermore, we investigate full spatio-temporal solution generation to mitigate autoregressive error accumulation. Lastly, we investigate conditioning on initial physical quantities, as well as conditioning solely on a text prompt to introduce text2PDE generation. We show that language can be a compact, interpretable, and accurate modality for generating physics simulations, paving the way for more usable and accessible PDE solvers. Through experiments on both uniform and structured grids, we show that the proposed approach is competitive with current neural PDE solvers in both accuracy and efficiency, with promising scaling behavior up to sim3 billion parameters. By introducing a scalable, accurate, and usable physics simulator, we hope to bring neural PDE solvers closer to practical use.

Learning physical simulations has been an essential and central aspect of many recent research efforts in machine learning, particularly for Navier-Stokes-based fluid mechanics. Classic numerical solvers have traditionally been computationally expensive and challenging to use in inverse problems, whereas Neural solvers aim to address both concerns through machine learning. We propose a general formulation for continuous convolutions using separable basis functions as a superset of existing methods and evaluate a large set of basis functions in the context of (a) a compressible 1D SPH simulation, (b) a weakly compressible 2D SPH simulation, and (c) an incompressible 2D SPH Simulation. We demonstrate that even and odd symmetries included in the basis functions are key aspects of stability and accuracy. Our broad evaluation shows that Fourier-based continuous convolutions outperform all other architectures regarding accuracy and generalization. Finally, using these Fourier-based networks, we show that prior inductive biases, such as window functions, are no longer necessary. An implementation of our approach, as well as complete datasets and solver implementations, is available at https://github.com/tum-pbs/SFBC.

In recent years, applying deep learning to solve physics problems has attracted much attention. Data-driven deep learning methods produce fast numerical operators that can learn approximate solutions to the whole system of partial differential equations (i.e., surrogate modeling). Although these neural networks may have lower accuracy than traditional numerical methods, they, once trained, are orders of magnitude faster at inference. Hence, one crucial feature is that these operators can generalize to unseen PDE parameters without expensive re-training.In this paper, we construct CFDBench, a benchmark tailored for evaluating the generalization ability of neural operators after training in computational fluid dynamics (CFD) problems. It features four classic CFD problems: lid-driven cavity flow, laminar boundary layer flow in circular tubes, dam flows through the steps, and periodic Karman vortex street. The data contains a total of 302K frames of velocity and pressure fields, involving 739 cases with different operating condition parameters, generated with numerical methods. We evaluate the effectiveness of popular neural operators including feed-forward networks, DeepONet, FNO, U-Net, etc. on CFDBnech by predicting flows with non-periodic boundary conditions, fluid properties, and flow domain shapes that are not seen during training. Appropriate modifications were made to apply popular deep neural networks to CFDBench and enable the accommodation of more changing inputs. Empirical results on CFDBench show many baseline models have errors as high as 300% in some problems, and severe error accumulation when performing autoregressive inference. CFDBench facilitates a more comprehensive comparison between different neural operators for CFD compared to existing benchmarks.

Acquisition of large datasets for three-dimensional (3D) partial differential equations are usually very expensive. Physics-informed neural operator (PINO) eliminates the high costs associated with generation of training datasets, and shows great potential in a variety of partial differential equations. In this work, we employ physics-informed neural operator, encoding the large-eddy simulation (LES) equations directly into the neural operator for simulating three-dimensional incompressible turbulent flows. We develop the LESnets (Large-Eddy Simulation nets) by adding large-eddy simulation equations to two different data-driven models, including Fourier neural operator (FNO) and implicit Fourier neural operator (IFNO) without using label data. Notably, by leveraging only PDE constraints to learn the spatio-temporal dynamics problem, LESnets retains the computational efficiency of data-driven approaches while obviating the necessity for data. Meanwhile, using large-eddy simulation equations as PDE constraints makes it possible to efficiently predict complex turbulence at coarse grids. We investigate the performance of the LESnets with two standard three-dimensional turbulent flows: decaying homogeneous isotropic turbulence and temporally evolving turbulent mixing layer. In the numerical experiments, the LESnets model shows a similar or even better accuracy as compared to traditional large-eddy simulation and data-driven models of FNO and IFNO. Moreover, the well-trained LESnets is significantly faster than traditional LES, and has a similar efficiency as the data-driven FNO and IFNO models. Thus, physics-informed neural operators have a strong potential for 3D nonlinear engineering applications.

The objective of this study is to develop a fully compressible magnetohydrodynamic solver for fast simulations of the global dynamo of the Sun using unstructured grids and GPUs. Accurate modeling of the Sun's convective layers is vital to predicting the Sun's behavior, including the solar dynamo and sunspot cycles. Currently, there are many efficient codes capable of conducting these large simulations; however, many assume an anealastic density distribution. The anelastic assumption is capable of producing accurate results for low mach numbers; however, it fails in regions with a higher mach number and a fully compressible flow must be considered. To avoid these issues, Wang et al. [1] created a Compressible High-ORder Unstructured Spectral difference (CHORUS) code for simulating fluid dynamics inside stars and planets. CHORUS++ augmented the CHORUS code to adopt a higher degree of polynomials by using cubed-sphere meshing and transfinite mapping to perform simulations on unstructured grids [2]. Recently, CHORUS++ was further developed for parallel magnetohydrodynamic (MHD) solutions on GPUs at Clarkson University. In this study the solar benchmark problems presented by Chen et al. [2] are extended to unsteady solar dynamo problems, with two different density scale heights. The CHORUS-MHD code is further accelerated by multiple GPUs and used to successfully solve these solar dynamo benchmark problems. [1] Wang, J., Liang, C., and Miesch, M. S., "A Compressible High-Order Unstructured Spectral Difference Code for Stratified Convection in Rotating Spherical Shells," Journal of Computational Physics, Vol. 290, 2015, pp. 90-111. [2] Chen, K., Liang, C., and Wan, M., "Arbitrarily high-order accurate simulations of compressible rotationally constrained convection using a transfinite mapping on cubed-sphere grids," Physics of Fluids, Vol. 35, 2023, p. 086120.

Recent ODE/SDE-based generative models, such as diffusion models, rectified flows, and flow matching, define a generative process as a time reversal of a fixed forward process. Even though these models show impressive performance on large-scale datasets, numerical simulation requires multiple evaluations of a neural network, leading to a slow sampling speed. We attribute the reason to the high curvature of the learned generative trajectories, as it is directly related to the truncation error of a numerical solver. Based on the relationship between the forward process and the curvature, here we present an efficient method of training the forward process to minimize the curvature of generative trajectories without any ODE/SDE simulation. Experiments show that our method achieves a lower curvature than previous models and, therefore, decreased sampling costs while maintaining competitive performance. Code is available at https://github.com/sangyun884/fast-ode.

Recently, neural networks have been extensively employed to solve partial differential equations (PDEs) in physical system modeling. While major studies focus on learning system evolution on predefined static mesh discretizations, some methods utilize reinforcement learning or supervised learning techniques to create adaptive and dynamic meshes, due to the dynamic nature of these systems. However, these approaches face two primary challenges: (1) the need for expensive optimal mesh data, and (2) the change of the solution space's degree of freedom and topology during mesh refinement. To address these challenges, this paper proposes a neural PDE solver with a neural mesh adapter. To begin with, we introduce a novel data-free neural mesh adaptor, called Data-free Mesh Mover (DMM), with two main innovations. Firstly, it is an operator that maps the solution to adaptive meshes and is trained using the Monge-Amp\`ere equation without optimal mesh data. Secondly, it dynamically changes the mesh by moving existing nodes rather than adding or deleting nodes and edges. Theoretical analysis shows that meshes generated by DMM have the lowest interpolation error bound. Based on DMM, to efficiently and accurately model dynamic systems, we develop a moving mesh based neural PDE solver (MM-PDE) that embeds the moving mesh with a two-branch architecture and a learnable interpolation framework to preserve information within the data. Empirical experiments demonstrate that our method generates suitable meshes and considerably enhances accuracy when modeling widely considered PDE systems. The code can be found at: https://github.com/Peiyannn/MM-PDE.git.

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.

When learning simulations for modeling physical phenomena in industrial designs, geometrical variabilities are of prime interest. While classical regression techniques prove effective for parameterized geometries, practical scenarios often involve the absence of shape parametrization during the inference stage, leaving us with only mesh discretizations as available data. Learning simulations from such mesh-based representations poses significant challenges, with recent advances relying heavily on deep graph neural networks to overcome the limitations of conventional machine learning approaches. Despite their promising results, graph neural networks exhibit certain drawbacks, including their dependency on extensive datasets and limitations in providing built-in predictive uncertainties or handling large meshes. In this work, we propose a machine learning method that do not rely on graph neural networks. Complex geometrical shapes and variations with fixed topology are dealt with using well-known mesh morphing onto a common support, combined with classical dimensionality reduction techniques and Gaussian processes. The proposed methodology can easily deal with large meshes without the need for explicit shape parameterization and provides crucial predictive uncertainties, which are essential for informed decision-making. In the considered numerical experiments, the proposed method is competitive with respect to existing graph neural networks, regarding training efficiency and accuracy of the predictions.

Neural networks have shown great potential in accelerating the solution of partial differential equations (PDEs). Recently, there has been a growing interest in introducing physics constraints into training neural PDE solvers to reduce the use of costly data and improve the generalization ability. However, these physics constraints, based on certain finite dimensional approximations over the function space, must resolve the smallest scaled physics to ensure the accuracy and stability of the simulation, resulting in high computational costs from large input, output, and neural networks. This paper proposes a general acceleration methodology called NeuralStagger by spatially and temporally decomposing the original learning tasks into several coarser-resolution subtasks. We define a coarse-resolution neural solver for each subtask, which requires fewer computational resources, and jointly train them with the vanilla physics-constrained loss by simply arranging their outputs to reconstruct the original solution. Due to the perfect parallelism between them, the solution is achieved as fast as a coarse-resolution neural solver. In addition, the trained solvers bring the flexibility of simulating with multiple levels of resolution. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations, which leads to an additional 10sim100times speed-up. Moreover, the experiment also shows that the learned model could be well used for optimal control.

Discovering the underlying behavior of complex systems is an important topic in many science and engineering disciplines. In this paper, we propose a novel neural network framework, finite difference neural networks (FDNet), to learn partial differential equations from data. Specifically, our proposed finite difference inspired network is designed to learn the underlying governing partial differential equations from trajectory data, and to iteratively estimate the future dynamical behavior using only a few trainable parameters. We illustrate the performance (predictive power) of our framework on the heat equation, with and without noise and/or forcing, and compare our results to the Forward Euler method. Moreover, we show the advantages of using a Hessian-Free Trust Region method to train the network.

We present a neural operator architecture to simulate Lagrangian dynamics, such as fluid flow, granular flows, and elastoplasticity. Traditional numerical methods, such as the finite element method (FEM), suffer from long run times and large memory consumption. On the other hand, approaches based on graph neural networks are faster but still suffer from long computation times on dense graphs, which are often required for high-fidelity simulations. Our model, GIOROM or Graph Interaction Operator for Reduced-Order Modeling, learns temporal dynamics within a reduced-order setting, capturing spatial features from a highly sparse graph representation of the input and generalizing to arbitrary spatial locations during inference. The model is geometry-aware and discretization-agnostic and can generalize to different initial conditions, velocities, and geometries after training. We show that point clouds of the order of 100,000 points can be inferred from sparse graphs with sim1000 points, with negligible change in computation time. We empirically evaluate our model on elastic solids, Newtonian fluids, Non-Newtonian fluids, Drucker-Prager granular flows, and von Mises elastoplasticity. On these benchmarks, our approach results in a 25times speedup compared to other neural network-based physics simulators while delivering high-fidelity predictions of complex physical systems and showing better performance on most benchmarks. The code and the demos are provided at https://github.com/HrishikeshVish/GIOROM.

We propose PROSE-FD, a zero-shot multimodal PDE foundational model for simultaneous prediction of heterogeneous two-dimensional physical systems related to distinct fluid dynamics settings. These systems include shallow water equations and the Navier-Stokes equations with incompressible and compressible flow, regular and complex geometries, and different buoyancy settings. This work presents a new transformer-based multi-operator learning approach that fuses symbolic information to perform operator-based data prediction, i.e. non-autoregressive. By incorporating multiple modalities in the inputs, the PDE foundation model builds in a pathway for including mathematical descriptions of the physical behavior. We pre-train our foundation model on 6 parametric families of equations collected from 13 datasets, including over 60K trajectories. Our model outperforms popular operator learning, computer vision, and multi-physics models, in benchmark forward prediction tasks. We test our architecture choices with ablation studies.

We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning probability measures to the spatial domain, which allows us to treat collocation grids probabilistically as random variables to be marginalised out. Adapting this spatial statistics view, we solve forward and inverse problems for parametric PDEs in a way that leads to the construction of Gaussian process models of solution fields. The implementation of these random grids poses a unique set of challenges for inverse physics informed deep learning frameworks and we propose a new architecture called Grid Invariant Convolutional Networks (GICNets) to overcome these challenges. We further show how to incorporate noisy data in a principled manner into our physics informed model to improve predictions for problems where data may be available but whose measurement location does not coincide with any fixed mesh or grid. The proposed method is tested on a nonlinear Poisson problem, Burgers equation, and Navier-Stokes equations, and we provide extensive numerical comparisons. We demonstrate significant computational advantages over current physics informed neural learning methods for parametric PDEs while improving the predictive capabilities and flexibility of these models.

In this work, we generalize the reaction-diffusion equation in statistical physics, Schr\"odinger equation in quantum mechanics, Helmholtz equation in paraxial optics into the neural partial differential equations (NPDE), which can be considered as the fundamental equations in the field of artificial intelligence research. We take finite difference method to discretize NPDE for finding numerical solution, and the basic building blocks of deep neural network architecture, including multi-layer perceptron, convolutional neural network and recurrent neural networks, are generated. The learning strategies, such as Adaptive moment estimation, L-BFGS, pseudoinverse learning algorithms and partial differential equation constrained optimization, are also presented. We believe it is of significance that presented clear physical image of interpretable deep neural networks, which makes it be possible for applying to analog computing device design, and pave the road to physical artificial intelligence.

Numerical simulations in climate, chemistry, or astrophysics are computationally too expensive for uncertainty quantification or parameter-exploration at high-resolution. Reduced-order or surrogate models are multiple orders of magnitude faster, but traditional surrogates are inflexible or inaccurate and pure machine learning (ML)-based surrogates too data-hungry. We propose a hybrid, flexible surrogate model that exploits known physics for simulating large-scale dynamics and limits learning to the hard-to-model term, which is called parametrization or closure and captures the effect of fine- onto large-scale dynamics. Leveraging neural operators, we are the first to learn grid-independent, non-local, and flexible parametrizations. Our multiscale neural operator is motivated by a rich literature in multiscale modeling, has quasilinear runtime complexity, is more accurate or flexible than state-of-the-art parametrizations and demonstrated on the chaotic equation multiscale Lorenz96.

Generating high-quality 3D assets from text and images has long been challenging, primarily due to the absence of scalable 3D representations capable of capturing intricate geometry distributions. In this work, we introduce Direct3D, a native 3D generative model scalable to in-the-wild input images, without requiring a multiview diffusion model or SDS optimization. Our approach comprises two primary components: a Direct 3D Variational Auto-Encoder (D3D-VAE) and a Direct 3D Diffusion Transformer (D3D-DiT). D3D-VAE efficiently encodes high-resolution 3D shapes into a compact and continuous latent triplane space. Notably, our method directly supervises the decoded geometry using a semi-continuous surface sampling strategy, diverging from previous methods relying on rendered images as supervision signals. D3D-DiT models the distribution of encoded 3D latents and is specifically designed to fuse positional information from the three feature maps of the triplane latent, enabling a native 3D generative model scalable to large-scale 3D datasets. Additionally, we introduce an innovative image-to-3D generation pipeline incorporating semantic and pixel-level image conditions, allowing the model to produce 3D shapes consistent with the provided conditional image input. Extensive experiments demonstrate the superiority of our large-scale pre-trained Direct3D over previous image-to-3D approaches, achieving significantly better generation quality and generalization ability, thus establishing a new state-of-the-art for 3D content creation. Project page: https://nju-3dv.github.io/projects/Direct3D/.

Learning complex dynamics driven by partial differential equations directly from data holds great promise for fast and accurate simulations of complex physical systems. In most cases, this problem can be formulated as an operator learning task, where one aims to learn the operator representing the physics of interest, which entails discretization of the continuous system. However, preserving key continuous properties at the discrete level, such as boundary conditions, and addressing physical systems with complex geometries is challenging for most existing approaches. We introduce a family of operator learning architectures, structure-preserving operator networks (SPONs), that allows to preserve key mathematical and physical properties of the continuous system by leveraging finite element (FE) discretizations of the input-output spaces. SPONs are encode-process-decode architectures that are end-to-end differentiable, where the encoder and decoder follows from the discretizations of the input-output spaces. SPONs can operate on complex geometries, enforce certain boundary conditions exactly, and offer theoretical guarantees. Our framework provides a flexible way of devising structure-preserving architectures tailored to specific applications, and offers an explicit trade-off between performance and efficiency, all thanks to the FE discretization of the input-output spaces. Additionally, we introduce a multigrid-inspired SPON architecture that yields improved performance at higher efficiency. Finally, we release a software to automate the design and training of SPON architectures.

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new Spatiotemporal Fourier Neural Operator (SFNO) that learns maps between Bochner spaces, and a new learning framework to address these issues. This new paradigm leverages wisdom from traditional numerical PDE theory and techniques to refine the pipeline of commonly adopted end-to-end neural operator training and evaluations. Specifically, in the learning problems for the turbulent flow modeling by the Navier-Stokes Equations (NSE), the proposed architecture initiates the training with a few epochs for SFNO, concluding with the freezing of most model parameters. Then, the last linear spectral convolution layer is fine-tuned without the frequency truncation. The optimization uses a negative Sobolev norm for the first time as the loss in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is almost exact thanks to the Parseval identity. This design allows the neural operators to effectively tackle low-frequency errors while the relief of the de-aliasing filter addresses high-frequency errors. Numerical experiments on commonly used benchmarks for the 2D NSE demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers.

Diffusion-based generative models use stochastic differential equations (SDEs) and their equivalent ordinary differential equations (ODEs) to establish a smooth connection between a complex data distribution and a tractable prior distribution. In this paper, we identify several intriguing trajectory properties in the ODE-based sampling process of diffusion models. We characterize an implicit denoising trajectory and discuss its vital role in forming the coupled sampling trajectory with a strong shape regularity, regardless of the generated content. We also describe a dynamic programming-based scheme to make the time schedule in sampling better fit the underlying trajectory structure. This simple strategy requires minimal modification to any given ODE-based numerical solvers and incurs negligible computational cost, while delivering superior performance in image generation, especially in 5sim 10 function evaluations.

Denoising Diffusion Probabilistic Models (DDPMs) can generate high-quality samples such as image and audio samples. However, DDPMs require hundreds to thousands of iterations to produce final samples. Several prior works have successfully accelerated DDPMs through adjusting the variance schedule (e.g., Improved Denoising Diffusion Probabilistic Models) or the denoising equation (e.g., Denoising Diffusion Implicit Models (DDIMs)). However, these acceleration methods cannot maintain the quality of samples and even introduce new noise at a high speedup rate, which limit their practicability. To accelerate the inference process while keeping the sample quality, we provide a fresh perspective that DDPMs should be treated as solving differential equations on manifolds. Under such a perspective, we propose pseudo numerical methods for diffusion models (PNDMs). Specifically, we figure out how to solve differential equations on manifolds and show that DDIMs are simple cases of pseudo numerical methods. We change several classical numerical methods to corresponding pseudo numerical methods and find that the pseudo linear multi-step method is the best in most situations. According to our experiments, by directly using pre-trained models on Cifar10, CelebA and LSUN, PNDMs can generate higher quality synthetic images with only 50 steps compared with 1000-step DDIMs (20x speedup), significantly outperform DDIMs with 250 steps (by around 0.4 in FID) and have good generalization on different variance schedules. Our implementation is available at https://github.com/luping-liu/PNDM.

We present rectified flow, a surprisingly simple approach to learning (neural) ordinary differential equation (ODE) models to transport between two empirically observed distributions \pi_0 and \pi_1, hence providing a unified solution to generative modeling and domain transfer, among various other tasks involving distribution transport. The idea of rectified flow is to learn the ODE to follow the straight paths connecting the points drawn from \pi_0 and \pi_1 as much as possible. This is achieved by solving a straightforward nonlinear least squares optimization problem, which can be easily scaled to large models without introducing extra parameters beyond standard supervised learning. The straight paths are special and preferred because they are the shortest paths between two points, and can be simulated exactly without time discretization and hence yield computationally efficient models. We show that the procedure of learning a rectified flow from data, called rectification, turns an arbitrary coupling of \pi_0 and \pi_1 to a new deterministic coupling with provably non-increasing convex transport costs. In addition, recursively applying rectification allows us to obtain a sequence of flows with increasingly straight paths, which can be simulated accurately with coarse time discretization in the inference phase. In empirical studies, we show that rectified flow performs superbly on image generation, image-to-image translation, and domain adaptation. In particular, on image generation and translation, our method yields nearly straight flows that give high quality results even with a single Euler discretization step.

In this work, we present a novel physics-based data-driven framework for reduced-order modeling of laser ignition in a model rocket combustor based on parameterized neural ordinary differential equations (PNODE). Deep neural networks are embedded as functions of high-dimensional parameters of laser ignition to predict various terms in a 0D flow model including the heat source function, pre-exponential factors, and activation energy. Using the governing equations of a 0D flow model, our PNODE needs only a limited number of training samples and predicts trajectories of various quantities such as temperature, pressure, and mass fractions of species while satisfying physical constraints. We validate our physics-based PNODE on solution snapshots of high-fidelity Computational Fluid Dynamics (CFD) simulations of laser-induced ignition in a prototype rocket combustor. We compare the performance of our physics-based PNODE with that of kernel ridge regression and fully connected neural networks. Our results show that our physics-based PNODE provides solutions with lower mean absolute errors of average temperature over time, thus improving the prediction of successful laser ignition with high-dimensional parameters.

Inverse design refers to the problem of optimizing the input of an objective function in order to enact a target outcome. For many real-world engineering problems, the objective function takes the form of a simulator that predicts how the system state will evolve over time, and the design challenge is to optimize the initial conditions that lead to a target outcome. Recent developments in learned simulation have shown that graph neural networks (GNNs) can be used for accurate, efficient, differentiable estimation of simulator dynamics, and support high-quality design optimization with gradient- or sampling-based optimization procedures. However, optimizing designs from scratch requires many expensive model queries, and these procedures exhibit basic failures on either non-convex or high-dimensional problems.In this work, we show how denoising diffusion models (DDMs) can be used to solve inverse design problems efficiently and propose a particle sampling algorithm for further improving their efficiency. We perform experiments on a number of fluid dynamics design challenges, and find that our approach substantially reduces the number of calls to the simulator compared to standard techniques.

Time-dependent partial differential equations (PDEs) are ubiquitous in science and engineering. Recently, mostly due to the high computational cost of traditional solution techniques, deep neural network based surrogates have gained increased interest. The practical utility of such neural PDE solvers relies on their ability to provide accurate, stable predictions over long time horizons, which is a notoriously hard problem. In this work, we present a large-scale analysis of common temporal rollout strategies, identifying the neglect of non-dominant spatial frequency information, often associated with high frequencies in PDE solutions, as the primary pitfall limiting stable, accurate rollout performance. Based on these insights, we draw inspiration from recent advances in diffusion models to introduce PDE-Refiner; a novel model class that enables more accurate modeling of all frequency components via a multistep refinement process. We validate PDE-Refiner on challenging benchmarks of complex fluid dynamics, demonstrating stable and accurate rollouts that consistently outperform state-of-the-art models, including neural, numerical, and hybrid neural-numerical architectures. We further demonstrate that PDE-Refiner greatly enhances data efficiency, since the denoising objective implicitly induces a novel form of spectral data augmentation. Finally, PDE-Refiner's connection to diffusion models enables an accurate and efficient assessment of the model's predictive uncertainty, allowing us to estimate when the surrogate becomes inaccurate.

We address the challenge of sound propagation simulations in 3D virtual rooms with moving sources, which have applications in virtual/augmented reality, game audio, and spatial computing. Solutions to the wave equation can describe wave phenomena such as diffraction and interference. However, simulating them using conventional numerical discretization methods with hundreds of source and receiver positions is intractable, making stimulating a sound field with moving sources impractical. To overcome this limitation, we propose using deep operator networks to approximate linear wave-equation operators. This enables the rapid prediction of sound propagation in realistic 3D acoustic scenes with moving sources, achieving millisecond-scale computations. By learning a compact surrogate model, we avoid the offline calculation and storage of impulse responses for all relevant source/listener pairs. Our experiments, including various complex scene geometries, show good agreement with reference solutions, with root mean squared errors ranging from 0.02 Pa to 0.10 Pa. Notably, our method signifies a paradigm shift as no prior machine learning approach has achieved precise predictions of complete wave fields within realistic domains. We anticipate that our findings will drive further exploration of deep neural operator methods, advancing research in immersive user experiences within virtual environments.

The past few years have witnessed the great success of Diffusion models~(DMs) in generating high-fidelity samples in generative modeling tasks. A major limitation of the DM is its notoriously slow sampling procedure which normally requires hundreds to thousands of time discretization steps of the learned diffusion process to reach the desired accuracy. Our goal is to develop a fast sampling method for DMs with a much less number of steps while retaining high sample quality. To this end, we systematically analyze the sampling procedure in DMs and identify key factors that affect the sample quality, among which the method of discretization is most crucial. By carefully examining the learned diffusion process, we propose Diffusion Exponential Integrator Sampler~(DEIS). It is based on the Exponential Integrator designed for discretizing ordinary differential equations (ODEs) and leverages a semilinear structure of the learned diffusion process to reduce the discretization error. The proposed method can be applied to any DMs and can generate high-fidelity samples in as few as 10 steps. In our experiments, it takes about 3 minutes on one A6000 GPU to generate 50k images from CIFAR10. Moreover, by directly using pre-trained DMs, we achieve the state-of-art sampling performance when the number of score function evaluation~(NFE) is limited, e.g., 4.17 FID with 10 NFEs, 3.37 FID, and 9.74 IS with only 15 NFEs on CIFAR10. Code is available at https://github.com/qsh-zh/deis

Recent years have seen a surge in deep learning approaches to accelerate numerical solvers, which provide faithful but computationally intensive simulations of the physical world. These deep surrogates are generally trained in a supervised manner from limited amounts of data slowly generated by the same solver they intend to accelerate. We propose an open-source framework that enables the online training of these models from a large ensemble run of simulations. It leverages multiple levels of parallelism to generate rich datasets. The framework avoids I/O bottlenecks and storage issues by directly streaming the generated data. A training reservoir mitigates the inherent bias of streaming while maximizing GPU throughput. Experiment on training a fully connected network as a surrogate for the heat equation shows the proposed approach enables training on 8TB of data in 2 hours with an accuracy improved by 47% and a batch throughput multiplied by 13 compared to a traditional offline procedure.

Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and require large amounts of {\em labeled} training data. We propose the Deep Generative Neural Operator (DGNO), a physics-aware framework that addresses these challenges by leveraging a deep, generative, probabilistic model in combination with a set of lower-dimensional, latent variables that simultaneously encode PDE-inputs and PDE-outputs. This formulation can make use of unlabeled data and significantly improves inverse problem-solving, particularly for discontinuous or discrete-valued input functions. DGNO enforces physics constraints without labeled data by incorporating as virtual observables, weak-form residuals based on compactly supported radial basis functions (CSRBFs). These relax regularity constraints and eliminate higher-order derivatives from the objective function. We also introduce MultiONet, a novel neural operator architecture, which is a more expressive generalization of the popular DeepONet that significantly enhances the approximating power of the proposed model. These innovations make DGNO particularly effective for challenging forward and inverse, PDE-based problems, such as those involving multi-phase media. Numerical experiments demonstrate that DGNO achieves higher accuracy across multiple benchmarks while exhibiting robustness to noise and strong generalization to out-of-distribution cases. Its adaptability, and the ability to handle sparse, noisy data while providing probabilistic estimates, make DGNO a powerful tool for scientific and engineering applications.

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO.

Recent advances in generative AI have unveiled significant potential for the creation of 3D content. However, current methods either apply a pre-trained 2D diffusion model with the time-consuming score distillation sampling (SDS), or a direct 3D diffusion model trained on limited 3D data losing generation diversity. In this work, we approach the problem by employing a multi-view 2.5D diffusion fine-tuned from a pre-trained 2D diffusion model. The multi-view 2.5D diffusion directly models the structural distribution of 3D data, while still maintaining the strong generalization ability of the original 2D diffusion model, filling the gap between 2D diffusion-based and direct 3D diffusion-based methods for 3D content generation. During inference, multi-view normal maps are generated using the 2.5D diffusion, and a novel differentiable rasterization scheme is introduced to fuse the almost consistent multi-view normal maps into a consistent 3D model. We further design a normal-conditioned multi-view image generation module for fast appearance generation given the 3D geometry. Our method is a one-pass diffusion process and does not require any SDS optimization as post-processing. We demonstrate through extensive experiments that, our direct 2.5D generation with the specially-designed fusion scheme can achieve diverse, mode-seeking-free, and high-fidelity 3D content generation in only 10 seconds. Project page: https://nju-3dv.github.io/projects/direct25.

Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored the multiscale architectures and various operator designs, they are limited to learning the operators as a whole in the coordinate space. In real physical science problems, PDEs are complex coupled equations with numerical solvers relying on discretization into high-dimensional coordinate space, which cannot be precisely approximated by a single operator nor efficiently learned due to the curse of dimensionality. We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Going beyond the coordinate space, LSM enables an attention-based hierarchical projection network to reduce the high-dimensional data into a compact latent space in linear time. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space that approximates complex input-output mappings via learning multiple basis operators, enjoying nice theoretical guarantees for convergence and approximation. Experimentally, LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks covering both solid and fluid physics. Code is available at https://github.com/thuml/Latent-Spectral-Models.

Distillation techniques have substantially improved the sampling speed of diffusion models, allowing of the generation within only one step or a few steps. However, these distillation methods require extensive training for each dataset, sampler, and network, which limits their practical applicability. To address this limitation, we propose a straightforward distillation approach, Distilled-ODE solvers (D-ODE solvers), that optimizes the ODE solver rather than training the denoising network. D-ODE solvers are formulated by simply applying a single parameter adjustment to existing ODE solvers. Subsequently, D-ODE solvers with smaller steps are optimized by ODE solvers with larger steps through distillation over a batch of samples. Our comprehensive experiments indicate that D-ODE solvers outperform existing ODE solvers, including DDIM, PNDM, DPM-Solver, DEIS, and EDM, especially when generating samples with fewer steps. Our method incur negligible computational overhead compared to previous distillation techniques, enabling simple and rapid integration with previous samplers. Qualitative analysis further shows that D-ODE solvers enhance image quality while preserving the sampling trajectory of ODE solvers.

In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the d-dimensional advection-diffusion equation, our proposal is a two-level algorithm that merges the solutions obtained from the discretization of the equation over highly anisotropic submeshes to compute an initial approximation of the fine solution. The algorithm then iteratively refines the initial guess by leveraging the structure of the residual. Performing costly calculations on anisotropic submeshes enable us to reduce the dimensionality of the problem by one, and the merging process, which involves the computation of solutions over disjoint domains, allows for parallel implementation.

Learning radiance fields (NeRF) with powerful 2D diffusion models has garnered popularity for text-to-3D generation. Nevertheless, the implicit 3D representations of NeRF lack explicit modeling of meshes and textures over surfaces, and such surface-undefined way may suffer from the issues, e.g., noisy surfaces with ambiguous texture details or cross-view inconsistency. To alleviate this, we present DreamMesh, a novel text-to-3D architecture that pivots on well-defined surfaces (triangle meshes) to generate high-fidelity explicit 3D model. Technically, DreamMesh capitalizes on a distinctive coarse-to-fine scheme. In the coarse stage, the mesh is first deformed by text-guided Jacobians and then DreamMesh textures the mesh with an interlaced use of 2D diffusion models in a tuning free manner from multiple viewpoints. In the fine stage, DreamMesh jointly manipulates the mesh and refines the texture map, leading to high-quality triangle meshes with high-fidelity textured materials. Extensive experiments demonstrate that DreamMesh significantly outperforms state-of-the-art text-to-3D methods in faithfully generating 3D content with richer textual details and enhanced geometry. Our project page is available at https://dreammesh.github.io.

Surrogate models are necessary to optimize meaningful quantities in physical dynamics as their recursive numerical resolutions are often prohibitively expensive. It is mainly the case for fluid dynamics and the resolution of Navier-Stokes equations. However, despite the fast-growing field of data-driven models for physical systems, reference datasets representing real-world phenomena are lacking. In this work, we develop AirfRANS, a dataset for studying the two-dimensional incompressible steady-state Reynolds-Averaged Navier-Stokes equations over airfoils at a subsonic regime and for different angles of attacks. We also introduce metrics on the stress forces at the surface of geometries and visualization of boundary layers to assess the capabilities of models to accurately predict the meaningful information of the problem. Finally, we propose deep learning baselines on four machine learning tasks to study AirfRANS under different constraints for generalization considerations: big and scarce data regime, Reynolds number, and angle of attack extrapolation.

In this work, we propose a framework that constructs reduced order models for nonlinear structural mechanics in a nonintrusive fashion, and can handle large scale simulations. We identify three steps that are carried out separately in time, and possibly on different devices: (i) the production of high-fidelity solutions by a commercial software, (ii) the offline stage of the model reduction and (iii) the online stage where the reduced order model is exploited. The nonintrusivity assumes that only the displacement field solution is known, and relies on operations on simulation data during the offline phase by using an in-house code. The compatibility with a new commercial code only needs the implementation of a routine converting the mesh and result format into our in-house data format. The nonintrusive capabilities of the framework are demonstrated on numerical experiments using commercial versions of the finite element softwares Zset and Ansys Mechanical. The nonlinear constitutive equations are evaluated by using the same external plugins as for Zset or Ansys Mechanical. The large scale simulations are handled using domain decomposition and parallel computing with distributed memory. The features and performances of the framework are evaluated on two numerical applications involving elastoviscoplastic materials: the second one involves a model of high-pressure blade, where the framework is used to extrapolate cyclic loadings in 6.5 hours, whereas the reference high-fidelity computation would take 9.5 days.

Solving complex Partial Differential Equations (PDEs) accurately and efficiently is an essential and challenging problem in all scientific and engineering disciplines. Mesh movement methods provide the capability to improve the accuracy of the numerical solution without increasing the overall mesh degree of freedom count. Conventional sophisticated mesh movement methods are extremely expensive and struggle to handle scenarios with complex boundary geometries. However, existing learning-based methods require re-training from scratch given a different PDE type or boundary geometry, which limits their applicability, and also often suffer from robustness issues in the form of inverted elements. In this paper, we introduce the Universal Mesh Movement Network (UM2N), which -- once trained -- can be applied in a non-intrusive, zero-shot manner to move meshes with different size distributions and structures, for solvers applicable to different PDE types and boundary geometries. UM2N consists of a Graph Transformer (GT) encoder for extracting features and a Graph Attention Network (GAT) based decoder for moving the mesh. We evaluate our method on advection and Navier-Stokes based examples, as well as a real-world tsunami simulation case. Our method outperforms existing learning-based mesh movement methods in terms of the benchmarks described above. In comparison to the conventional sophisticated Monge-Amp\`ere PDE-solver based method, our approach not only significantly accelerates mesh movement, but also proves effective in scenarios where the conventional method fails. Our project page is at https://erizmr.github.io/UM2N/.

Solving partial differential equations (PDEs) is a central task in scientific computing. Recently, neural network approximation of PDEs has received increasing attention due to its flexible meshless discretization and its potential for high-dimensional problems. One fundamental numerical difficulty is that random samples in the training set introduce statistical errors into the discretization of loss functional which may become the dominant error in the final approximation, and therefore overshadow the modeling capability of the neural network. In this work, we propose a new minmax formulation to optimize simultaneously the approximate solution, given by a neural network model, and the random samples in the training set, provided by a deep generative model. The key idea is to use a deep generative model to adjust random samples in the training set such that the residual induced by the approximate PDE solution can maintain a smooth profile when it is being minimized. Such an idea is achieved by implicitly embedding the Wasserstein distance between the residual-induced distribution and the uniform distribution into the loss, which is then minimized together with the residual. A nearly uniform residual profile means that its variance is small for any normalized weight function such that the Monte Carlo approximation error of the loss functional is reduced significantly for a certain sample size. The adversarial adaptive sampling (AAS) approach proposed in this work is the first attempt to formulate two essential components, minimizing the residual and seeking the optimal training set, into one minmax objective functional for the neural network approximation of PDEs.

Although diffusion models can generate remarkably high-quality samples, they are intrinsically bottlenecked by their expensive iterative sampling procedure. Consistency models (CMs) have recently emerged as a promising diffusion model distillation method, reducing the cost of sampling by generating high-fidelity samples in just a few iterations. Consistency model distillation aims to solve the probability flow ordinary differential equation (ODE) defined by an existing diffusion model. CMs are not directly trained to minimize error against an ODE solver, rather they use a more computationally tractable objective. As a way to study how effectively CMs solve the probability flow ODE, and the effect that any induced error has on the quality of generated samples, we introduce Direct CMs, which directly minimize this error. Intriguingly, we find that Direct CMs reduce the ODE solving error compared to CMs but also result in significantly worse sample quality, calling into question why exactly CMs work well in the first place. Full code is available at: https://github.com/layer6ai-labs/direct-cms.

This paper introduces PDEformer, a neural solver for partial differential equations (PDEs) capable of simultaneously addressing various types of PDEs. We advocate representing the PDE in the form of a computational graph, facilitating the seamless integration of both symbolic and numerical information inherent in a PDE. A graph Transformer and an implicit neural representation (INR) are employed to generate mesh-free predicted solutions. Following pretraining on data exhibiting a certain level of diversity, our model achieves zero-shot accuracies on benchmark datasets that surpass those of adequately trained expert models. Additionally, PDEformer demonstrates promising results in the inverse problem of PDE coefficient recovery.

Machine learning for differential equations paves the way for computationally efficient alternatives to numerical solvers, with potentially broad impacts in science and engineering. Though current algorithms typically require simulated training data tailored to a given setting, one may instead wish to learn useful information from heterogeneous sources, or from real dynamical systems observations that are messy or incomplete. In this work, we learn general-purpose representations of PDEs from heterogeneous data by implementing joint embedding methods for self-supervised learning (SSL), a framework for unsupervised representation learning that has had notable success in computer vision. Our representation outperforms baseline approaches to invariant tasks, such as regressing the coefficients of a PDE, while also improving the time-stepping performance of neural solvers. We hope that our proposed methodology will prove useful in the eventual development of general-purpose foundation models for PDEs.

Directly generating scenes from satellite imagery offers exciting possibilities for integration into applications like games and map services. However, challenges arise from significant view changes and scene scale. Previous efforts mainly focused on image or video generation, lacking exploration into the adaptability of scene generation for arbitrary views. Existing 3D generation works either operate at the object level or are difficult to utilize the geometry obtained from satellite imagery. To overcome these limitations, we propose a novel architecture for direct 3D scene generation by introducing diffusion models into 3D sparse representations and combining them with neural rendering techniques. Specifically, our approach generates texture colors at the point level for a given geometry using a 3D diffusion model first, which is then transformed into a scene representation in a feed-forward manner. The representation can be utilized to render arbitrary views which would excel in both single-frame quality and inter-frame consistency. Experiments in two city-scale datasets show that our model demonstrates proficiency in generating photo-realistic street-view image sequences and cross-view urban scenes from satellite imagery.

Score Distillation Sampling (SDS) has made significant strides in distilling image-generative models for 3D generation. However, its maximum-likelihood-seeking behavior often leads to degraded visual quality and diversity, limiting its effectiveness in 3D applications. In this work, we propose Consistent Flow Distillation (CFD), which addresses these limitations. We begin by leveraging the gradient of the diffusion ODE or SDE sampling process to guide the 3D generation. From the gradient-based sampling perspective, we find that the consistency of 2D image flows across different viewpoints is important for high-quality 3D generation. To achieve this, we introduce multi-view consistent Gaussian noise on the 3D object, which can be rendered from various viewpoints to compute the flow gradient. Our experiments demonstrate that CFD, through consistent flows, significantly outperforms previous methods in text-to-3D generation.

Embedding polygonal mesh assets within photorealistic Neural Radience Fields (NeRF) volumes, such that they can be rendered and their dynamics simulated in a physically consistent manner with the NeRF, is under-explored from the system perspective of integrating NeRF into the traditional graphics pipeline. This paper designs a two-way coupling between mesh and NeRF during rendering and simulation. We first review the light transport equations for both mesh and NeRF, then distill them into an efficient algorithm for updating radiance and throughput along a cast ray with an arbitrary number of bounces. To resolve the discrepancy between the linear color space that the path tracer assumes and the sRGB color space that standard NeRF uses, we train NeRF with High Dynamic Range (HDR) images. We also present a strategy to estimate light sources and cast shadows on the NeRF. Finally, we consider how the hybrid surface-volumetric formulation can be efficiently integrated with a high-performance physics simulator that supports cloth, rigid and soft bodies. The full rendering and simulation system can be run on a GPU at interactive rates. We show that a hybrid system approach outperforms alternatives in visual realism for mesh insertion, because it allows realistic light transport from volumetric NeRF media onto surfaces, which affects the appearance of reflective/refractive surfaces and illumination of diffuse surfaces informed by the dynamic scene.

Data-driven deep learning models are transforming global weather forecasting. It is an open question if this success can extend to climate modeling, where the complexity of the data and long inference rollouts pose significant challenges. Here, we present the first conditional generative model that produces accurate and physically consistent global climate ensemble simulations by emulating a coarse version of the United States' primary operational global forecast model, FV3GFS. Our model integrates the dynamics-informed diffusion framework (DYffusion) with the Spherical Fourier Neural Operator (SFNO) architecture, enabling stable 100-year simulations at 6-hourly timesteps while maintaining low computational overhead compared to single-step deterministic baselines. The model achieves near gold-standard performance for climate model emulation, outperforming existing approaches and demonstrating promising ensemble skill. This work represents a significant advance towards efficient, data-driven climate simulations that can enhance our understanding of the climate system and inform adaptation strategies.

We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral coefficients. In contrast to current machine learning approaches which enforce PDE constraints by minimizing the numerical quadrature of the residuals in the spatiotemporal domain, we leverage Parseval's identity and introduce a new training strategy through a spectral loss. Our spectral loss enables more efficient differentiation through the neural network, and substantially reduces training complexity. At inference time, the computational cost of our method remains constant, regardless of the spatiotemporal resolution of the domain. Our experimental results demonstrate that our method significantly outperforms previous machine learning approaches in terms of speed and accuracy by one to two orders of magnitude on multiple different problems. When compared to numerical solvers of the same accuracy, our method demonstrates a 10times increase in performance speed.

We consider the dictionary-based ROM-net (Reduced Order Model) framework [T. Daniel, F. Casenave, N. Akkari, D. Ryckelynck, Model order reduction assisted by deep neural networks (ROM-net), Advanced modeling and Simulation in Engineering Sciences 7 (16), 2020] and summarize the underlying methodologies and their recent improvements. The main contribution of this work is the application of the complete workflow to a real-life industrial model of an elastoviscoplastic high-pressure turbine blade subjected to thermal, centrifugal and pressure loadings, for the quantification of the uncertainty on dual quantities (such as the accumulated plastic strain and the stress tensor), generated by the uncertainty on the temperature loading field. The dictionary-based ROM-net computes predictions of dual quantities of interest for 1008 Monte Carlo draws of the temperature loading field in 2 hours and 48 minutes, which corresponds to a speedup greater than 600 with respect to a reference parallel solver using domain decomposition, with a relative error in the order of 2%. Another contribution of this work consists in the derivation of a meta-model to reconstruct the dual quantities of interest over the complete mesh from their values on the reduced integration points.

Despite the success of physics-informed neural networks (PINNs) in approximating partial differential equations (PDEs), PINNs can sometimes fail to converge to the correct solution in problems involving complicated PDEs. This is reflected in several recent studies on characterizing the "failure modes" of PINNs, although a thorough understanding of the connection between PINN failure modes and sampling strategies is missing. In this paper, we provide a novel perspective of failure modes of PINNs by hypothesizing that training PINNs relies on successful "propagation" of solution from initial and/or boundary condition points to interior points. We show that PINNs with poor sampling strategies can get stuck at trivial solutions if there are propagation failures, characterized by highly imbalanced PDE residual fields. To mitigate propagation failures, we propose a novel Retain-Resample-Release sampling (R3) algorithm that can incrementally accumulate collocation points in regions of high PDE residuals with little to no computational overhead. We provide an extension of R3 sampling to respect the principle of causality while solving time-dependent PDEs. We theoretically analyze the behavior of R3 sampling and empirically demonstrate its efficacy and efficiency in comparison with baselines on a variety of PDE problems.

As an alternative to classical numerical solvers for partial differential equations (PDEs) subject to boundary value constraints, there has been a surge of interest in investigating neural networks that can solve such problems efficiently. In this work, we design a general solution operator for two different time-independent PDEs using graph neural networks (GNNs) and spectral graph convolutions. We train the networks on simulated data from a finite elements solver on a variety of shapes and inhomogeneities. In contrast to previous works, we focus on the ability of the trained operator to generalize to previously unseen scenarios. Specifically, we test generalization to meshes with different shapes and superposition of solutions for a different number of inhomogeneities. We find that training on a diverse dataset with lots of variation in the finite element meshes is a key ingredient for achieving good generalization results in all cases. With this, we believe that GNNs can be used to learn solution operators that generalize over a range of properties and produce solutions much faster than a generic solver. Our dataset, which we make publicly available, can be used and extended to verify the robustness of these models under varying conditions.

The growing body of research shows how to replace classical partial differential equation (PDE) integrators with neural networks. The popular strategy is to generate the input-output pairs with a PDE solver, train the neural network in the regression setting, and use the trained model as a cheap surrogate for the solver. The bottleneck in this scheme is the number of expensive queries of a PDE solver needed to generate the dataset. To alleviate the problem, we propose a computationally cheap augmentation strategy based on general covariance and simple random coordinate transformations. Our approach relies on the fact that physical laws are independent of the coordinate choice, so the change in the coordinate system preserves the type of a parametric PDE and only changes PDE's data (e.g., initial conditions, diffusion coefficient). For tried neural networks and partial differential equations, proposed augmentation improves test error by 23% on average. The worst observed result is a 17% increase in test error for multilayer perceptron, and the best case is a 80% decrease for dilated residual network.

Machine learning approaches for solving partial differential equations require learning mappings between function spaces. While convolutional or graph neural networks are constrained to discretized functions, neural operators present a promising milestone toward mapping functions directly. Despite impressive results they still face challenges with respect to the domain geometry and typically rely on some form of discretization. In order to alleviate such limitations, we present CORAL, a new method that leverages coordinate-based networks for solving PDEs on general geometries. CORAL is designed to remove constraints on the input mesh, making it applicable to any spatial sampling and geometry. Its ability extends to diverse problem domains, including PDE solving, spatio-temporal forecasting, and inverse problems like geometric design. CORAL demonstrates robust performance across multiple resolutions and performs well in both convex and non-convex domains, surpassing or performing on par with state-of-the-art models.

The polygon mesh representation of 3D data exhibits great flexibility, fast rendering speed, and storage efficiency, which is widely preferred in various applications. However, given its unstructured graph representation, the direct generation of high-fidelity 3D meshes is challenging. Fortunately, with a pre-defined ordering strategy, 3D meshes can be represented as sequences, and the generation process can be seamlessly treated as an auto-regressive problem. In this paper, we validate the Neural Coordinate Field (NeurCF), an explicit coordinate representation with implicit neural embeddings, is a simple-yet-effective representation for large-scale sequential mesh modeling. After that, we present MeshXL, a family of generative pre-trained auto-regressive models, which addresses the process of 3D mesh generation with modern large language model approaches. Extensive experiments show that MeshXL is able to generate high-quality 3D meshes, and can also serve as foundation models for various down-stream applications.

Recently, many mesh-based graph neural network (GNN) models have been proposed for modeling complex high-dimensional physical systems. Remarkable achievements have been made in significantly reducing the solving time compared to traditional numerical solvers. These methods are typically designed to i) reduce the computational cost in solving physical dynamics and/or ii) propose techniques to enhance the solution accuracy in fluid and rigid body dynamics. However, it remains under-explored whether they are effective in addressing the challenges of flexible body dynamics, where instantaneous collisions occur within a very short timeframe. In this paper, we present Hierarchical Contact Mesh Transformer (HCMT), which uses hierarchical mesh structures and can learn long-range dependencies (occurred by collisions) among spatially distant positions of a body -- two close positions in a higher-level mesh correspond to two distant positions in a lower-level mesh. HCMT enables long-range interactions, and the hierarchical mesh structure quickly propagates collision effects to faraway positions. To this end, it consists of a contact mesh Transformer and a hierarchical mesh Transformer (CMT and HMT, respectively). Lastly, we propose a flexible body dynamics dataset, consisting of trajectories that reflect experimental settings frequently used in the display industry for product designs. We also compare the performance of several baselines using well-known benchmark datasets. Our results show that HCMT provides significant performance improvements over existing methods. Our code is available at https://github.com/yuyudeep/hcmt.

The software package Volna-OP2 is a robust and efficient code capable of simulating the complete life cycle of a tsunami whilst harnessing the latest High Performance Computing (HPC) architectures. In this paper, a comprehensive error analysis and scalability study of the GPU version of the code is presented. A novel decomposition of the numerical errors into the dispersion and dissipation components is explored. Most tsunami codes exhibit amplitude smearing and/or phase lagging/leading, so the decomposition shown here is a new approach and novel tool for explaining these occurrences. It is the first time that the errors of a tsunami code have been assessed in this manner. To date, Volna-OP2 has been widely used by the tsunami modelling community. In particular its computational efficiency has allowed various sensitivity analyses and uncertainty quantification studies. Due to the number of simulations required, there is always a trade-off between accuracy and runtime when carrying out these statistical studies. The analysis presented in this paper will guide the user towards an acceptable level of accuracy within a given runtime.

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.

We propose a new method for realistic real-time novel-view synthesis (NVS) of large scenes. Existing neural rendering methods generate realistic results, but primarily work for small scale scenes (<50 square meters) and have difficulty at large scale (>10000 square meters). Traditional graphics-based rasterization rendering is fast for large scenes but lacks realism and requires expensive manually created assets. Our approach combines the best of both worlds by taking a moderate-quality scaffold mesh as input and learning a neural texture field and shader to model view-dependant effects to enhance realism, while still using the standard graphics pipeline for real-time rendering. Our method outperforms existing neural rendering methods, providing at least 30x faster rendering with comparable or better realism for large self-driving and drone scenes. Our work is the first to enable real-time rendering of large real-world scenes.

Recent advances in 3D X-ray Computed Tomographic (CT) sensors have stimulated research efforts to unveil the extremely complex micro-scale processes that control the activity of soil microorganisms. Voxel-based description (up to hundreds millions voxels) of the pore space can be extracted, from grey level 3D CT scanner images, by means of simple image processing tools. Classical methods for numerical simulation of biological dynamics using mesh of voxels, such as Lattice Boltzmann Model (LBM), are too much time consuming. Thus, the use of more compact and reliable geometrical representations of pore space can drastically decrease the computational cost of the simulations. Several recent works propose basic analytic volume primitives (e.g. spheres, generalized cylinders, ellipsoids) to define a piece-wise approximation of pore space for numerical simulation of draining, diffusion and microbial decomposition. Such approaches work well but the drawback is that it generates approximation errors. In the present work, we study another alternative where pore space is described by means of geometrically relevant connected subsets of voxels (regions) computed from the curvilinear skeleton. Indeed, many works use the curvilinear skeleton (3D medial axis) for analyzing and partitioning 3D shapes within various domains (medicine, material sciences, petroleum engineering, etc.) but only a few ones in soil sciences. Within the context of soil sciences, most studies dealing with 3D medial axis focus on the determination of pore throats. Here, we segment pore space using curvilinear skeleton in order to achieve numerical simulation of microbial decomposition (including diffusion processes). We validate simulation outputs by comparison with other methods using different pore space geometrical representations (balls, voxels).

Score distillation sampling (SDS) has shown great promise in text-to-3D generation by distilling pretrained large-scale text-to-image diffusion models, but suffers from over-saturation, over-smoothing, and low-diversity problems. In this work, we propose to model the 3D parameter as a random variable instead of a constant as in SDS and present variational score distillation (VSD), a principled particle-based variational framework to explain and address the aforementioned issues in text-to-3D generation. We show that SDS is a special case of VSD and leads to poor samples with both small and large CFG weights. In comparison, VSD works well with various CFG weights as ancestral sampling from diffusion models and simultaneously improves the diversity and sample quality with a common CFG weight (i.e., 7.5). We further present various improvements in the design space for text-to-3D such as distillation time schedule and density initialization, which are orthogonal to the distillation algorithm yet not well explored. Our overall approach, dubbed ProlificDreamer, can generate high rendering resolution (i.e., 512times512) and high-fidelity NeRF with rich structure and complex effects (e.g., smoke and drops). Further, initialized from NeRF, meshes fine-tuned by VSD are meticulously detailed and photo-realistic. Project page: https://ml.cs.tsinghua.edu.cn/prolificdreamer/

The approximation of Partial Differential Equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and flexibility in implementing various PDEs, PINNs often suffer from limited accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which struggle to effectively learn high-frequency and non-linear components. Recently, parametric mesh representations in combination with neural networks have been investigated as a promising approach to eliminate the inductive biases of neural networks. However, they usually require very high-resolution grids and a large number of collocation points to achieve high accuracy while avoiding overfitting issues. In addition, the fixed positions of the mesh parameters restrict their flexibility, making it challenging to accurately approximate complex PDEs. To overcome these limitations, we propose Physics-Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network. Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training. This adaptability enables our model to optimally approximate PDE solutions, unlike models with fixed parameter positions. Furthermore, the proposed approach maintains the same optimization framework used in PINNs, allowing us to benefit from their excellent properties. Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs. Our project page is available at https://namgyukang.github.io/Physics-Informed-Gaussians/

Implicit neural fields, typically encoded by a multilayer perceptron (MLP) that maps from coordinates (e.g., xyz) to signals (e.g., signed distances), have shown remarkable promise as a high-fidelity and compact representation. However, the lack of a regular and explicit grid structure also makes it challenging to apply generative modeling directly on implicit neural fields in order to synthesize new data. To this end, we propose HyperDiffusion, a novel approach for unconditional generative modeling of implicit neural fields. HyperDiffusion operates directly on MLP weights and generates new neural implicit fields encoded by synthesized MLP parameters. Specifically, a collection of MLPs is first optimized to faithfully represent individual data samples. Subsequently, a diffusion process is trained in this MLP weight space to model the underlying distribution of neural implicit fields. HyperDiffusion enables diffusion modeling over a implicit, compact, and yet high-fidelity representation of complex signals across 3D shapes and 4D mesh animations within one single unified framework.

Realistic simulation of dynamic scenes requires accurately capturing diverse material properties and modeling complex object interactions grounded in physical principles. However, existing methods are constrained to basic material types with limited predictable parameters, making them insufficient to represent the complexity of real-world materials. We introduce a novel approach that leverages multi-modal foundation models and video diffusion to achieve enhanced 4D dynamic scene simulation. Our method utilizes multi-modal models to identify material types and initialize material parameters through image queries, while simultaneously inferring 3D Gaussian splats for detailed scene representation. We further refine these material parameters using video diffusion with a differentiable Material Point Method (MPM) and optical flow guidance rather than render loss or Score Distillation Sampling (SDS) loss. This integrated framework enables accurate prediction and realistic simulation of dynamic interactions in real-world scenarios, advancing both accuracy and flexibility in physics-based simulations.

With the increasing adoption of graph neural networks (GNNs) in the machine learning community, GPUs have become an essential tool to accelerate GNN training. However, training GNNs on very large graphs that do not fit in GPU memory is still a challenging task. Unlike conventional neural networks, mini-batching input samples in GNNs requires complicated tasks such as traversing neighboring nodes and gathering their feature values. While this process accounts for a significant portion of the training time, we find existing GNN implementations using popular deep neural network (DNN) libraries such as PyTorch are limited to a CPU-centric approach for the entire data preparation step. This "all-in-CPU" approach has negative impact on the overall GNN training performance as it over-utilizes CPU resources and hinders GPU acceleration of GNN training. To overcome such limitations, we introduce PyTorch-Direct, which enables a GPU-centric data accessing paradigm for GNN training. In PyTorch-Direct, GPUs are capable of efficiently accessing complicated data structures in host memory directly without CPU intervention. Our microbenchmark and end-to-end GNN training results show that PyTorch-Direct reduces data transfer time by 47.1% on average and speeds up GNN training by up to 1.6x. Furthermore, by reducing CPU utilization, PyTorch-Direct also saves system power by 12.4% to 17.5% during training. To minimize programmer effort, we introduce a new "unified tensor" type along with necessary changes to the PyTorch memory allocator, dispatch logic, and placement rules. As a result, users need to change at most two lines of their PyTorch GNN training code for each tensor object to take advantage of PyTorch-Direct.

The image-to-image translation abilities of generative learning models have recently made significant progress in the estimation of complex (steered) mappings between image distributions. While appearance based tasks like image in-painting or style transfer have been studied at length, we propose to investigate the potential of generative models in the context of physical simulations. Providing a dataset of 300k image-pairs and baseline evaluations for three different physical simulation tasks, we propose a benchmark to investigate the following research questions: i) are generative models able to learn complex physical relations from input-output image pairs? ii) what speedups can be achieved by replacing differential equation based simulations? While baseline evaluations of different current models show the potential for high speedups (ii), these results also show strong limitations toward the physical correctness (i). This underlines the need for new methods to enforce physical correctness. Data, baseline models and evaluation code http://www.physics-gen.org.

A Transformer-based deep direct sampling method is proposed for electrical impedance tomography, a well-known severely ill-posed nonlinear boundary value inverse problem. A real-time reconstruction is achieved by evaluating the learned inverse operator between carefully designed data and the reconstructed images. An effort is made to give a specific example to a fundamental question: whether and how one can benefit from the theoretical structure of a mathematical problem to develop task-oriented and structure-conforming deep neural networks? Specifically, inspired by direct sampling methods for inverse problems, the 1D boundary data in different frequencies are preprocessed by a partial differential equation-based feature map to yield 2D harmonic extensions as different input channels. Then, by introducing learnable non-local kernels, the direct sampling is recast to a modified attention mechanism. The new method achieves superior accuracy over its predecessors and contemporary operator learners and shows robustness to noises in benchmarks. This research shall strengthen the insights that, despite being invented for natural language processing tasks, the attention mechanism offers great flexibility to be modified in conformity with the a priori mathematical knowledge, which ultimately leads to the design of more physics-compatible neural architectures.

Recently, text-to-image generation has exhibited remarkable advancements, with the ability to produce visually impressive results. In contrast, text-to-3D generation has not yet reached a comparable level of quality. Existing methods primarily rely on text-guided score distillation sampling (SDS), and they encounter difficulties in transferring 2D attributes of the generated images to 3D content. In this work, we aim to develop an effective 3D generative model capable of synthesizing high-resolution textured meshes by leveraging both textual and image information. To this end, we introduce Guide3D, a zero-shot text-and-image-guided generative model for 3D avatar generation based on diffusion models. Our model involves (1) generating sparse-view images of a text-consistent character using diffusion models, and (2) jointly optimizing multi-resolution differentiable marching tetrahedral grids with pixel-aligned image features. We further propose a similarity-aware feature fusion strategy for efficiently integrating features from different views. Moreover, we introduce two novel training objectives as an alternative to calculating SDS, significantly enhancing the optimization process. We thoroughly evaluate the performance and components of our framework, which outperforms the current state-of-the-art in producing topologically and structurally correct geometry and high-resolution textures. Guide3D enables the direct transfer of 2D-generated images to the 3D space. Our code will be made publicly available.

The ability to generate highly realistic 2D images from mere text prompts has recently made huge progress in terms of speed and quality, thanks to the advent of image diffusion models. Naturally, the question arises if this can be also achieved in the generation of 3D content from such text prompts. To this end, a new line of methods recently emerged trying to harness diffusion models, trained on 2D images, for supervision of 3D model generation using view dependent prompts. While achieving impressive results, these methods, however, have two major drawbacks. First, rather than commonly used 3D meshes, they instead generate neural radiance fields (NeRFs), making them impractical for most real applications. Second, these approaches tend to produce over-saturated models, giving the output a cartoonish looking effect. Therefore, in this work we propose a novel method for generation of highly realistic-looking 3D meshes. To this end, we extend NeRF to employ an SDF backbone, leading to improved 3D mesh extraction. In addition, we propose a novel way to finetune the mesh texture, removing the effect of high saturation and improving the details of the output 3D mesh.

This paper presents a comprehensive study on the unified module for accelerating stable-diffusion processes, specifically focusing on the lcm-lora module. Stable-diffusion processes play a crucial role in various scientific and engineering domains, and their acceleration is of paramount importance for efficient computational performance. The standard iterative procedures for solving fixed-source discrete ordinates problems often exhibit slow convergence, particularly in optically thick scenarios. To address this challenge, unconditionally stable diffusion-acceleration methods have been developed, aiming to enhance the computational efficiency of transport equations and discrete ordinates problems. This study delves into the theoretical foundations and numerical results of unconditionally stable diffusion synthetic acceleration methods, providing insights into their stability and performance for model discrete ordinates problems. Furthermore, the paper explores recent advancements in diffusion model acceleration, including on device acceleration of large diffusion models via gpu aware optimizations, highlighting the potential for significantly improved inference latency. The results and analyses in this study provide important insights into stable diffusion processes and have important ramifications for the creation and application of acceleration methods specifically, the lcm-lora module in a variety of computing environments.

An experimental demonstration of directivities exceeding the fundamental Kildal limit, a phenomenon called superdirectivity, is provided for spherical high-index dielectric antennas with an electric dipole excitation. A directivity factor of about 10 with a total efficiency of more than 80\% for an antenna having a size of a third of the wavelength was measured. High directivities are shown to be associated with constructive interference of particular electric and magnetic modes of an open spherical resonator. Both analytic solution for a point dipole and a full-wave rigorous simulation for a realistic dipole antenna were employed for optimization and analysis, yielding an excellent agreement between experimentally measured and numerically predicted directivities. The use of high-index low-loss ceramics can significantly reduce the physical size of such antennas while maintaining their overall high radiation efficiency. Such antennas can be attractive for various high-frequency applications, such as antennas for the Internet of things, smart city systems, 5G network systems, and others. The demonstrated concept can be scaled in frequency.

Diffusion models (DMs) represent state-of-the-art generative models for continuous inputs. DMs work by constructing a Stochastic Differential Equation (SDE) in the input space (ie, position space), and using a neural network to reverse it. In this work, we introduce a novel generative modeling framework grounded in phase space dynamics, where a phase space is defined as {an augmented space encompassing both position and velocity.} Leveraging insights from Stochastic Optimal Control, we construct a path measure in the phase space that enables efficient sampling. {In contrast to DMs, our framework demonstrates the capability to generate realistic data points at an early stage of dynamics propagation.} This early prediction sets the stage for efficient data generation by leveraging additional velocity information along the trajectory. On standard image generation benchmarks, our model yields favorable performance over baselines in the regime of small Number of Function Evaluations (NFEs). Furthermore, our approach rivals the performance of diffusion models equipped with efficient sampling techniques, underscoring its potential as a new tool generative modeling.

DreamFusion has recently demonstrated the utility of a pre-trained text-to-image diffusion model to optimize Neural Radiance Fields (NeRF), achieving remarkable text-to-3D synthesis results. However, the method has two inherent limitations: (a) extremely slow optimization of NeRF and (b) low-resolution image space supervision on NeRF, leading to low-quality 3D models with a long processing time. In this paper, we address these limitations by utilizing a two-stage optimization framework. First, we obtain a coarse model using a low-resolution diffusion prior and accelerate with a sparse 3D hash grid structure. Using the coarse representation as the initialization, we further optimize a textured 3D mesh model with an efficient differentiable renderer interacting with a high-resolution latent diffusion model. Our method, dubbed Magic3D, can create high quality 3D mesh models in 40 minutes, which is 2x faster than DreamFusion (reportedly taking 1.5 hours on average), while also achieving higher resolution. User studies show 61.7% raters to prefer our approach over DreamFusion. Together with the image-conditioned generation capabilities, we provide users with new ways to control 3D synthesis, opening up new avenues to various creative applications.

Recent methods such as Score Distillation Sampling (SDS) and Variational Score Distillation (VSD) using 2D diffusion models for text-to-3D generation have demonstrated impressive generation quality. However, the long generation time of such algorithms significantly degrades the user experience. To tackle this problem, we propose DreamPropeller, a drop-in acceleration algorithm that can be wrapped around any existing text-to-3D generation pipeline based on score distillation. Our framework generalizes Picard iterations, a classical algorithm for parallel sampling an ODE path, and can account for non-ODE paths such as momentum-based gradient updates and changes in dimensions during the optimization process as in many cases of 3D generation. We show that our algorithm trades parallel compute for wallclock time and empirically achieves up to 4.7x speedup with a negligible drop in generation quality for all tested frameworks.

Mimicking realistic dynamics in 3D garment animations is a challenging task due to the complex nature of multi-layered garments and the variety of outer forces involved. Existing approaches mostly focus on single-layered garments driven by only human bodies and struggle to handle general scenarios. In this paper, we propose a novel data-driven method, called LayersNet, to model garment-level animations as particle-wise interactions in a micro physics system. We improve simulation efficiency by representing garments as patch-level particles in a two-level structural hierarchy. Moreover, we introduce a novel Rotation Equivalent Transformation that leverages the rotation invariance and additivity of physics systems to better model outer forces. To verify the effectiveness of our approach and bridge the gap between experimental environments and real-world scenarios, we introduce a new challenging dataset, D-LAYERS, containing 700K frames of dynamics of 4,900 different combinations of multi-layered garments driven by both human bodies and randomly sampled wind. Our experiments show that LayersNet achieves superior performance both quantitatively and qualitatively. We will make the dataset and code publicly available at https://mmlab-ntu.github.io/project/layersnet/index.html .

Diffusion or flow-based models are powerful generative paradigms that are notoriously hard to sample as samples are defined as solutions to high-dimensional Ordinary or Stochastic Differential Equations (ODEs/SDEs) which require a large Number of Function Evaluations (NFE) to approximate well. Existing methods to alleviate the costly sampling process include model distillation and designing dedicated ODE solvers. However, distillation is costly to train and sometimes can deteriorate quality, while dedicated solvers still require relatively large NFE to produce high quality samples. In this paper we introduce "Bespoke solvers", a novel framework for constructing custom ODE solvers tailored to the ODE of a given pre-trained flow model. Our approach optimizes an order consistent and parameter-efficient solver (e.g., with 80 learnable parameters), is trained for roughly 1% of the GPU time required for training the pre-trained model, and significantly improves approximation and generation quality compared to dedicated solvers. For example, a Bespoke solver for a CIFAR10 model produces samples with Fr\'echet Inception Distance (FID) of 2.73 with 10 NFE, and gets to 1% of the Ground Truth (GT) FID (2.59) for this model with only 20 NFE. On the more challenging ImageNet-64times64, Bespoke samples at 2.2 FID with 10 NFE, and gets within 2% of GT FID (1.71) with 20 NFE.

Real-world data collection for robotics is costly and resource-intensive, requiring skilled operators and expensive hardware. Simulations offer a scalable alternative but often fail to achieve sim-to-real generalization due to geometric and visual gaps. To address these challenges, we propose a 3D-photorealistic real-to-sim system, namely, RE^3SIM, addressing geometric and visual sim-to-real gaps. RE^3SIM employs advanced 3D reconstruction and neural rendering techniques to faithfully recreate real-world scenarios, enabling real-time rendering of simulated cross-view cameras within a physics-based simulator. By utilizing privileged information to collect expert demonstrations efficiently in simulation, and train robot policies with imitation learning, we validate the effectiveness of the real-to-sim-to-real pipeline across various manipulation task scenarios. Notably, with only simulated data, we can achieve zero-shot sim-to-real transfer with an average success rate exceeding 58%. To push the limit of real-to-sim, we further generate a large-scale simulation dataset, demonstrating how a robust policy can be built from simulation data that generalizes across various objects. Codes and demos are available at: http://xshenhan.github.io/Re3Sim/.

This study investigates the application of deep residual networks for predicting the dynamics of interacting three-dimensional rigid bodies. We present a framework combining a 3D physics simulator implemented in C++ with a deep learning model constructed using PyTorch. The simulator generates training data encompassing linear and angular motion, elastic collisions, fluid friction, gravitational effects, and damping. Our deep residual network, consisting of an input layer, multiple residual blocks, and an output layer, is designed to handle the complexities of 3D dynamics. We evaluate the network's performance using a datasetof 10,000 simulated scenarios, each involving 3-5 interacting rigid bodies. The model achieves a mean squared error of 0.015 for position predictions and 0.022 for orientation predictions, representing a 25% improvement over baseline methods. Our results demonstrate the network's ability to capture intricate physical interactions, with particular success in predicting elastic collisions and rotational dynamics. This work significantly contributes to physics-informed machine learning by showcasing the immense potential of deep residual networks in modeling complex 3D physical systems. We discuss our approach's limitations and propose future directions for improving generalization to more diverse object shapes and materials.

Residual neural networks are state-of-the-art deep learning models. Their continuous-depth analog, neural ordinary differential equations (ODEs), are also widely used. Despite their success, the link between the discrete and continuous models still lacks a solid mathematical foundation. In this article, we take a step in this direction by establishing an implicit regularization of deep residual networks towards neural ODEs, for nonlinear networks trained with gradient flow. We prove that if the network is initialized as a discretization of a neural ODE, then such a discretization holds throughout training. Our results are valid for a finite training time, and also as the training time tends to infinity provided that the network satisfies a Polyak-Lojasiewicz condition. Importantly, this condition holds for a family of residual networks where the residuals are two-layer perceptrons with an overparameterization in width that is only linear, and implies the convergence of gradient flow to a global minimum. Numerical experiments illustrate our results.

Full-complexity Earth system models (ESMs) are computationally very expensive, limiting their use in exploring the climate outcomes of multiple emission pathways. More efficient emulators that approximate ESMs can directly map emissions onto climate outcomes, and benchmarks are being used to evaluate their accuracy on standardized tasks and datasets. We investigate a popular benchmark in data-driven climate emulation, ClimateBench, on which deep learning-based emulators are currently achieving the best performance. We implement a linear regression-based emulator, akin to pattern scaling, and find that it outperforms the incumbent 100M-parameter deep learning foundation model, ClimaX, on 3 out of 4 regionally-resolved surface-level climate variables. While emulating surface temperature is expected to be predominantly linear, this result is surprising for emulating precipitation. We identify that this outcome is a result of high levels of internal variability in the benchmark targets. To address internal variability, we update the benchmark targets with ensemble averages from the MPI-ESM1.2-LR model that contain 50 instead of 3 climate simulations per emission pathway. Using the new targets, we show that linear pattern scaling continues to be more accurate on temperature, but can be outperformed by a deep learning-based model for emulating precipitation. We publish our code, data, and an interactive tutorial at github.com/blutjens/climate-emulator.

Global climate models typically operate at a grid resolution of hundreds of kilometers and fail to resolve atmospheric mesoscale processes, e.g., clouds, precipitation, and gravity waves (GWs). Model representation of these processes and their sources is essential to the global circulation and planetary energy budget, but subgrid scale contributions from these processes are often only approximately represented in models using parameterizations. These parameterizations are subject to approximations and idealizations, which limit their capability and accuracy. The most drastic of these approximations is the "single-column approximation" which completely neglects the horizontal evolution of these processes, resulting in key biases in current climate models. With a focus on atmospheric GWs, we present the first-ever global simulation of atmospheric GW fluxes using machine learning (ML) models trained on the WINDSET dataset to emulate global GW emulation in the atmosphere, as an alternative to traditional single-column parameterizations. Using an Attention U-Net-based architecture trained on globally resolved GW momentum fluxes, we illustrate the importance and effectiveness of global nonlocality, when simulating GWs using data-driven schemes.

A popular approach to sample a diffusion-based generative model is to solve an ordinary differential equation (ODE). In existing samplers, the coefficients of the ODE solvers are pre-determined by the ODE formulation, the reverse discrete timesteps, and the employed ODE methods. In this paper, we consider accelerating several popular ODE-based sampling processes (including EDM, DDIM, and DPM-Solver) by optimizing certain coefficients via improved integration approximation (IIA). We propose to minimize, for each time step, a mean squared error (MSE) function with respect to the selected coefficients. The MSE is constructed by applying the original ODE solver for a set of fine-grained timesteps, which in principle provides a more accurate integration approximation in predicting the next diffusion state. The proposed IIA technique does not require any change of a pre-trained model, and only introduces a very small computational overhead for solving a number of quadratic optimization problems. Extensive experiments show that considerably better FID scores can be achieved by using IIA-EDM, IIA-DDIM, and IIA-DPM-Solver than the original counterparts when the neural function evaluation (NFE) is small (i.e., less than 25).

Remarkable progress has been made in 4D content generation recently. However, existing methods suffer from long optimization time, lack of motion controllability, and a low level of detail. In this paper, we introduce DreamGaussian4D, an efficient 4D generation framework that builds on 4D Gaussian Splatting representation. Our key insight is that the explicit modeling of spatial transformations in Gaussian Splatting makes it more suitable for the 4D generation setting compared with implicit representations. DreamGaussian4D reduces the optimization time from several hours to just a few minutes, allows flexible control of the generated 3D motion, and produces animated meshes that can be efficiently rendered in 3D engines.

Neural surrogates for partial differential equations (PDEs) have become popular due to their potential to quickly simulate physics. With a few exceptions, neural surrogates generally treat the forward evolution of time-dependent PDEs as a black box by directly predicting the next state. While this is a natural and easy framework for applying neural surrogates, it can be an over-simplified and rigid framework for predicting physics. In this work, we propose an alternative framework in which neural solvers predict the temporal derivative and an ODE integrator forwards the solution in time, which has little overhead and is broadly applicable across model architectures and PDEs. We find that by simply changing the training target and introducing numerical integration during inference, neural surrogates can gain accuracy and stability. Predicting temporal derivatives also allows models to not be constrained to a specific temporal discretization, allowing for flexible time-stepping during inference or training on higher-resolution PDE data. Lastly, we investigate why this new framework can be beneficial and in what situations does it work well.

Given a set of calibrated images of a scene, we present an approach that produces a simple, compact, and actionable 3D world representation by means of 3D primitives. While many approaches focus on recovering high-fidelity 3D scenes, we focus on parsing a scene into mid-level 3D representations made of a small set of textured primitives. Such representations are interpretable, easy to manipulate and suited for physics-based simulations. Moreover, unlike existing primitive decomposition methods that rely on 3D input data, our approach operates directly on images through differentiable rendering. Specifically, we model primitives as textured superquadric meshes and optimize their parameters from scratch with an image rendering loss. We highlight the importance of modeling transparency for each primitive, which is critical for optimization and also enables handling varying numbers of primitives. We show that the resulting textured primitives faithfully reconstruct the input images and accurately model the visible 3D points, while providing amodal shape completions of unseen object regions. We compare our approach to the state of the art on diverse scenes from DTU, and demonstrate its robustness on real-life captures from BlendedMVS and Nerfstudio. We also showcase how our results can be used to effortlessly edit a scene or perform physical simulations. Code and video results are available at https://www.tmonnier.com/DBW .

The accuracy of learning-based optical flow estimation models heavily relies on the realism of the training datasets. Current approaches for generating such datasets either employ synthetic data or generate images with limited realism. However, the domain gap of these data with real-world scenes constrains the generalization of the trained model to real-world applications. To address this issue, we investigate generating realistic optical flow datasets from real-world images. Firstly, to generate highly realistic new images, we construct a layered depth representation, known as multiplane images (MPI), from single-view images. This allows us to generate novel view images that are highly realistic. To generate optical flow maps that correspond accurately to the new image, we calculate the optical flows of each plane using the camera matrix and plane depths. We then project these layered optical flows into the output optical flow map with volume rendering. Secondly, to ensure the realism of motion, we present an independent object motion module that can separate the camera and dynamic object motion in MPI. This module addresses the deficiency in MPI-based single-view methods, where optical flow is generated only by camera motion and does not account for any object movement. We additionally devise a depth-aware inpainting module to merge new images with dynamic objects and address unnatural motion occlusions. We show the superior performance of our method through extensive experiments on real-world datasets. Moreover, our approach achieves state-of-the-art performance in both unsupervised and supervised training of learning-based models. The code will be made publicly available at: https://github.com/Sharpiless/MPI-Flow.

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. We demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We showcase an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.

Machine Learning (ML) has the potential to revolutionise the field of automotive aerodynamics, enabling split-second flow predictions early in the design process. However, the lack of open-source training data for realistic road cars, using high-fidelity CFD methods, represents a barrier to their development. To address this, a high-fidelity open-source (CC-BY-SA) public dataset for automotive aerodynamics has been generated, based on 500 parametrically morphed variants of the widely-used DrivAer notchback generic vehicle. Mesh generation and scale-resolving CFD was executed using consistent and validated automatic workflows representative of the industrial state-of-the-art. Geometries and rich aerodynamic data are published in open-source formats. To our knowledge, this is the first large, public-domain dataset for complex automotive configurations generated using high-fidelity CFD.

While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing PINNs formulations to respect the spatio-temporal causal structure that is inherent to the evolution of physical systems. We argue that this is a fundamental limitation and a key source of error that can ultimately steer PINN models to converge towards erroneous solutions. We address this pathology by proposing a simple re-formulation of PINNs loss functions that can explicitly account for physical causality during model training. We demonstrate that this simple modification alone is enough to introduce significant accuracy improvements, as well as a practical quantitative mechanism for assessing the convergence of a PINNs model. We provide state-of-the-art numerical results across a series of benchmarks for which existing PINNs formulations fail, including the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the Navier-Stokes equations in the turbulent regime. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such systems, introducing new opportunities for their applicability to problems of industrial complexity.

In recent years, there has been rapid development in 3D generation models, opening up new possibilities for applications such as simulating the dynamic movements of 3D objects and customizing their behaviors. However, current 3D generative models tend to focus only on surface features such as color and shape, neglecting the inherent physical properties that govern the behavior of objects in the real world. To accurately simulate physics-aligned dynamics, it is essential to predict the physical properties of materials and incorporate them into the behavior prediction process. Nonetheless, predicting the diverse materials of real-world objects is still challenging due to the complex nature of their physical attributes. In this paper, we propose Physics3D, a novel method for learning various physical properties of 3D objects through a video diffusion model. Our approach involves designing a highly generalizable physical simulation system based on a viscoelastic material model, which enables us to simulate a wide range of materials with high-fidelity capabilities. Moreover, we distill the physical priors from a video diffusion model that contains more understanding of realistic object materials. Extensive experiments demonstrate the effectiveness of our method with both elastic and plastic materials. Physics3D shows great potential for bridging the gap between the physical world and virtual neural space, providing a better integration and application of realistic physical principles in virtual environments. Project page: https://liuff19.github.io/Physics3D.

Text-guided image generation has progressed rapidly in recent years, inspiring major breakthroughs in text-guided shape generation. Recently, it has been shown that using score distillation, one can successfully text-guide a NeRF model to generate a 3D object. We adapt the score distillation to the publicly available, and computationally efficient, Latent Diffusion Models, which apply the entire diffusion process in a compact latent space of a pretrained autoencoder. As NeRFs operate in image space, a naive solution for guiding them with latent score distillation would require encoding to the latent space at each guidance step. Instead, we propose to bring the NeRF to the latent space, resulting in a Latent-NeRF. Analyzing our Latent-NeRF, we show that while Text-to-3D models can generate impressive results, they are inherently unconstrained and may lack the ability to guide or enforce a specific 3D structure. To assist and direct the 3D generation, we propose to guide our Latent-NeRF using a Sketch-Shape: an abstract geometry that defines the coarse structure of the desired object. Then, we present means to integrate such a constraint directly into a Latent-NeRF. This unique combination of text and shape guidance allows for increased control over the generation process. We also show that latent score distillation can be successfully applied directly on 3D meshes. This allows for generating high-quality textures on a given geometry. Our experiments validate the power of our different forms of guidance and the efficiency of using latent rendering. Implementation is available at https://github.com/eladrich/latent-nerf

Diffusion models suffer from slow sample generation at inference time. Therefore, developing a principled framework for fast deterministic/stochastic sampling for a broader class of diffusion models is a promising direction. We propose two complementary frameworks for accelerating sample generation in pre-trained models: Conjugate Integrators and Splitting Integrators. Conjugate integrators generalize DDIM, mapping the reverse diffusion dynamics to a more amenable space for sampling. In contrast, splitting-based integrators, commonly used in molecular dynamics, reduce the numerical simulation error by cleverly alternating between numerical updates involving the data and auxiliary variables. After extensively studying these methods empirically and theoretically, we present a hybrid method that leads to the best-reported performance for diffusion models in augmented spaces. Applied to Phase Space Langevin Diffusion [Pandey & Mandt, 2023] on CIFAR-10, our deterministic and stochastic samplers achieve FID scores of 2.11 and 2.36 in only 100 network function evaluations (NFE) as compared to 2.57 and 2.63 for the best-performing baselines, respectively. Our code and model checkpoints will be made publicly available at https://github.com/mandt-lab/PSLD.

We design a physics-aware auto-encoder to specifically reduce the dimensionality of solutions arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by a large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Moreover, the realized latent variables are often hard to interpret. Therefore, many of these methods are often dismissed in the reduced order modeling of dynamical systems governed by the partial differential equations (PDEs). Accordingly, we propose an auto-encoder type nonlinear dimensionality reduction algorithm. The unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that registers the output sequence of the PDEs on a non-uniform parameter/time-varying grid, such that the Kolmogorov n-width of the mapped data on the learned grid is minimized. We demonstrate the efficacy and interpretability of our approach to separate convection/advection from diffusion/scaling on various manufactured and physical systems.

Neural Radiance Fields (NeRF) methods have proved effective as compact, high-quality and versatile representations for 3D scenes, and enable downstream tasks such as editing, retrieval, navigation, etc. Various neural architectures are vying for the core structure of NeRF, including the plain Multi-Layer Perceptron (MLP), sparse tensors, low-rank tensors, hashtables and their compositions. Each of these representations has its particular set of trade-offs. For example, the hashtable-based representations admit faster training and rendering but their lack of clear geometric meaning hampers downstream tasks like spatial-relation-aware editing. In this paper, we propose Progressive Volume Distillation (PVD), a systematic distillation method that allows any-to-any conversions between different architectures, including MLP, sparse or low-rank tensors, hashtables and their compositions. PVD consequently empowers downstream applications to optimally adapt the neural representations for the task at hand in a post hoc fashion. The conversions are fast, as distillation is progressively performed on different levels of volume representations, from shallower to deeper. We also employ special treatment of density to deal with its specific numerical instability problem. Empirical evidence is presented to validate our method on the NeRF-Synthetic, LLFF and TanksAndTemples datasets. For example, with PVD, an MLP-based NeRF model can be distilled from a hashtable-based Instant-NGP model at a 10X~20X faster speed than being trained the original NeRF from scratch, while achieving a superior level of synthesis quality. Code is available at https://github.com/megvii-research/AAAI2023-PVD.

Recent advancements in 2D/3D generative techniques have facilitated the generation of dynamic 3D objects from monocular videos. Previous methods mainly rely on the implicit neural radiance fields (NeRF) or explicit Gaussian Splatting as the underlying representation, and struggle to achieve satisfactory spatial-temporal consistency and surface appearance. Drawing inspiration from modern 3D animation pipelines, we introduce DreamMesh4D, a novel framework combining mesh representation with geometric skinning technique to generate high-quality 4D object from a monocular video. Instead of utilizing classical texture map for appearance, we bind Gaussian splats to triangle face of mesh for differentiable optimization of both the texture and mesh vertices. In particular, DreamMesh4D begins with a coarse mesh obtained through an image-to-3D generation procedure. Sparse points are then uniformly sampled across the mesh surface, and are used to build a deformation graph to drive the motion of the 3D object for the sake of computational efficiency and providing additional constraint. For each step, transformations of sparse control points are predicted using a deformation network, and the mesh vertices as well as the surface Gaussians are deformed via a novel geometric skinning algorithm, which is a hybrid approach combining LBS (linear blending skinning) and DQS (dual-quaternion skinning), mitigating drawbacks associated with both approaches. The static surface Gaussians and mesh vertices as well as the deformation network are learned via reference view photometric loss, score distillation loss as well as other regularizers in a two-stage manner. Extensive experiments demonstrate superior performance of our method. Furthermore, our method is compatible with modern graphic pipelines, showcasing its potential in the 3D gaming and film industry.

Biophysical modeling, particularly involving partial differential equations (PDEs), offers significant potential for tailoring disease treatment protocols to individual patients. However, the inverse problem-solving aspect of these models presents a substantial challenge, either due to the high computational requirements of model-based approaches or the limited robustness of deep learning (DL) methods. We propose a novel framework that leverages the unique strengths of both approaches in a synergistic manner. Our method incorporates a DL ensemble for initial parameter estimation, facilitating efficient downstream evolutionary sampling initialized with this DL-based prior. We showcase the effectiveness of integrating a rapid deep-learning algorithm with a high-precision evolution strategy in estimating brain tumor cell concentrations from magnetic resonance images. The DL-Prior plays a pivotal role, significantly constraining the effective sampling-parameter space. This reduction results in a fivefold convergence acceleration and a Dice-score of 95%

The analysis of parametric and non-parametric uncertainties of very large dynamical systems requires the construction of a stochastic model of said system. Linear approaches relying on random matrix theory and principal componant analysis can be used when systems undergo low-frequency vibrations. In the case of fast dynamics and wave propagation, we investigate a random generator of boundary conditions for fast submodels by using machine learning. We show that the use of non-linear techniques in machine learning and data-driven methods is highly relevant. Physics-informed neural networks is a possible choice for a data-driven method to replace linear modal analysis. An architecture that support a random component is necessary for the construction of the stochastic model of the physical system for non-parametric uncertainties, since the goal is to learn the underlying probabilistic distribution of uncertainty in the data. Generative Adversarial Networks (GANs) are suited for such applications, where the Wasserstein-GAN with gradient penalty variant offers improved convergence results for our problem. The objective of our approach is to train a GAN on data from a finite element method code (Fenics) so as to extract stochastic boundary conditions for faster finite element predictions on a submodel. The submodel and the training data have both the same geometrical support. It is a zone of interest for uncertainty quantification and relevant to engineering purposes. In the exploitation phase, the framework can be viewed as a randomized and parametrized simulation generator on the submodel, which can be used as a Monte Carlo estimator.

We introduce Gaussian-Flow, a novel point-based approach for fast dynamic scene reconstruction and real-time rendering from both multi-view and monocular videos. In contrast to the prevalent NeRF-based approaches hampered by slow training and rendering speeds, our approach harnesses recent advancements in point-based 3D Gaussian Splatting (3DGS). Specifically, a novel Dual-Domain Deformation Model (DDDM) is proposed to explicitly model attribute deformations of each Gaussian point, where the time-dependent residual of each attribute is captured by a polynomial fitting in the time domain, and a Fourier series fitting in the frequency domain. The proposed DDDM is capable of modeling complex scene deformations across long video footage, eliminating the need for training separate 3DGS for each frame or introducing an additional implicit neural field to model 3D dynamics. Moreover, the explicit deformation modeling for discretized Gaussian points ensures ultra-fast training and rendering of a 4D scene, which is comparable to the original 3DGS designed for static 3D reconstruction. Our proposed approach showcases a substantial efficiency improvement, achieving a 5times faster training speed compared to the per-frame 3DGS modeling. In addition, quantitative results demonstrate that the proposed Gaussian-Flow significantly outperforms previous leading methods in novel view rendering quality. Project page: https://nju-3dv.github.io/projects/Gaussian-Flow

To balance quality and cost, various domain areas of science and engineering run simulations at multiple levels of sophistication. Multi-fidelity active learning aims to learn a direct mapping from input parameters to simulation outputs at the highest fidelity by actively acquiring data from multiple fidelity levels. However, existing approaches based on Gaussian processes are hardly scalable to high-dimensional data. Deep learning-based methods often impose a hierarchical structure in hidden representations, which only supports passing information from low-fidelity to high-fidelity. These approaches can lead to the undesirable propagation of errors from low-fidelity representations to high-fidelity ones. We propose a novel framework called Disentangled Multi-fidelity Deep Bayesian Active Learning (D-MFDAL), which learns the surrogate models conditioned on the distribution of functions at multiple fidelities. On benchmark tasks of learning deep surrogates of partial differential equations including heat equation, Poisson's equation and fluid simulations, our approach significantly outperforms state-of-the-art in prediction accuracy and sample efficiency.

Machine learning based surrogate models offer researchers powerful tools for accelerating simulation-based workflows. However, as standard datasets in this space often cover small classes of physical behavior, it can be difficult to evaluate the efficacy of new approaches. To address this gap, we introduce the Well: a large-scale collection of datasets containing numerical simulations of a wide variety of spatiotemporal physical systems. The Well draws from domain experts and numerical software developers to provide 15TB of data across 16 datasets covering diverse domains such as biological systems, fluid dynamics, acoustic scattering, as well as magneto-hydrodynamic simulations of extra-galactic fluids or supernova explosions. These datasets can be used individually or as part of a broader benchmark suite. To facilitate usage of the Well, we provide a unified PyTorch interface for training and evaluating models. We demonstrate the function of this library by introducing example baselines that highlight the new challenges posed by the complex dynamics of the Well. The code and data is available at https://github.com/PolymathicAI/the_well.

Neural Radiance Fields (NeRFs) can be dramatically accelerated by spatial grid representations. However, they do not explicitly reason about scale and so introduce aliasing artifacts when reconstructing scenes captured at different camera distances. Mip-NeRF and its extensions propose scale-aware renderers that project volumetric frustums rather than point samples but such approaches rely on positional encodings that are not readily compatible with grid methods. We propose a simple modification to grid-based models by training model heads at different spatial grid resolutions. At render time, we simply use coarser grids to render samples that cover larger volumes. Our method can be easily applied to existing accelerated NeRF methods and significantly improves rendering quality (reducing error rates by 20-90% across synthetic and unbounded real-world scenes) while incurring minimal performance overhead (as each model head is quick to evaluate). Compared to Mip-NeRF, we reduce error rates by 20% while training over 60x faster.

We consider using deep neural networks to solve time-dependent partial differential equations (PDEs), where multi-scale processing is crucial for modeling complex, time-evolving dynamics. While the U-Net architecture with skip connections is commonly used by prior studies to enable multi-scale processing, our analysis shows that the need for features to evolve across layers results in temporally misaligned features in skip connections, which limits the model's performance. To address this limitation, we propose SineNet, consisting of multiple sequentially connected U-shaped network blocks, referred to as waves. In SineNet, high-resolution features are evolved progressively through multiple stages, thereby reducing the amount of misalignment within each stage. We furthermore analyze the role of skip connections in enabling both parallel and sequential processing of multi-scale information. Our method is rigorously tested on multiple PDE datasets, including the Navier-Stokes equations and shallow water equations, showcasing the advantages of our proposed approach over conventional U-Nets with a comparable parameter budget. We further demonstrate that increasing the number of waves in SineNet while maintaining the same number of parameters leads to a monotonically improved performance. The results highlight the effectiveness of SineNet and the potential of our approach in advancing the state-of-the-art in neural PDE solver design. Our code is available as part of AIRS (https://github.com/divelab/AIRS).

Recent advancements in 3D content generation from text or a single image struggle with limited high-quality 3D datasets and inconsistency from 2D multi-view generation. We introduce DiffSplat, a novel 3D generative framework that natively generates 3D Gaussian splats by taming large-scale text-to-image diffusion models. It differs from previous 3D generative models by effectively utilizing web-scale 2D priors while maintaining 3D consistency in a unified model. To bootstrap the training, a lightweight reconstruction model is proposed to instantly produce multi-view Gaussian splat grids for scalable dataset curation. In conjunction with the regular diffusion loss on these grids, a 3D rendering loss is introduced to facilitate 3D coherence across arbitrary views. The compatibility with image diffusion models enables seamless adaptions of numerous techniques for image generation to the 3D realm. Extensive experiments reveal the superiority of DiffSplat in text- and image-conditioned generation tasks and downstream applications. Thorough ablation studies validate the efficacy of each critical design choice and provide insights into the underlying mechanism.

Rectified flow and reflow procedures have significantly advanced fast generation by progressively straightening ordinary differential equation (ODE) flows. They operate under the assumption that image and noise pairs, known as couplings, can be approximated by straight trajectories with constant velocity. However, we observe that modeling with constant velocity and using reflow procedures have limitations in accurately learning straight trajectories between pairs, resulting in suboptimal performance in few-step generation. To address these limitations, we introduce Constant Acceleration Flow (CAF), a novel framework based on a simple constant acceleration equation. CAF introduces acceleration as an additional learnable variable, allowing for more expressive and accurate estimation of the ODE flow. Moreover, we propose two techniques to further improve estimation accuracy: initial velocity conditioning for the acceleration model and a reflow process for the initial velocity. Our comprehensive studies on toy datasets, CIFAR-10, and ImageNet 64x64 demonstrate that CAF outperforms state-of-the-art baselines for one-step generation. We also show that CAF dramatically improves few-step coupling preservation and inversion over Rectified flow. Code is available at https://github.com/mlvlab/CAF{https://github.com/mlvlab/CAF}.

Efficiently digitizing high-fidelity animatable human avatars from videos is a challenging and active research topic. Recent volume rendering-based neural representations open a new way for human digitization with their friendly usability and photo-realistic reconstruction quality. However, they are inefficient for long optimization times and slow inference speed; their implicit nature results in entangled geometry, materials, and dynamics of humans, which are hard to edit afterward. Such drawbacks prevent their direct applicability to downstream applications, especially the prominent rasterization-based graphic ones. We present EMA, a method that Efficiently learns Meshy neural fields to reconstruct animatable human Avatars. It jointly optimizes explicit triangular canonical mesh, spatial-varying material, and motion dynamics, via inverse rendering in an end-to-end fashion. Each above component is derived from separate neural fields, relaxing the requirement of a template, or rigging. The mesh representation is highly compatible with the efficient rasterization-based renderer, thus our method only takes about an hour of training and can render in real-time. Moreover, only minutes of optimization is enough for plausible reconstruction results. The disentanglement of meshes enables direct downstream applications. Extensive experiments illustrate the very competitive performance and significant speed boost against previous methods. We also showcase applications including novel pose synthesis, material editing, and relighting. The project page: https://xk-huang.github.io/ema/.

While diffusion models can successfully generate data and make predictions, they are predominantly designed for static images. We propose an approach for efficiently training diffusion models for probabilistic spatiotemporal forecasting, where generating stable and accurate rollout forecasts remains challenging, Our method, DYffusion, leverages the temporal dynamics in the data, directly coupling it with the diffusion steps in the model. We train a stochastic, time-conditioned interpolator and a forecaster network that mimic the forward and reverse processes of standard diffusion models, respectively. DYffusion naturally facilitates multi-step and long-range forecasting, allowing for highly flexible, continuous-time sampling trajectories and the ability to trade-off performance with accelerated sampling at inference time. In addition, the dynamics-informed diffusion process in DYffusion imposes a strong inductive bias and significantly improves computational efficiency compared to traditional Gaussian noise-based diffusion models. Our approach performs competitively on probabilistic forecasting of complex dynamics in sea surface temperatures, Navier-Stokes flows, and spring mesh systems.

Neural radiance fields achieve unprecedented quality for novel view synthesis, but their volumetric formulation remains expensive, requiring a huge number of samples to render high-resolution images. Volumetric encodings are essential to represent fuzzy geometry such as foliage and hair, and they are well-suited for stochastic optimization. Yet, many scenes ultimately consist largely of solid surfaces which can be accurately rendered by a single sample per pixel. Based on this insight, we propose a neural radiance formulation that smoothly transitions between volumetric- and surface-based rendering, greatly accelerating rendering speed and even improving visual fidelity. Our method constructs an explicit mesh envelope which spatially bounds a neural volumetric representation. In solid regions, the envelope nearly converges to a surface and can often be rendered with a single sample. To this end, we generalize the NeuS formulation with a learned spatially-varying kernel size which encodes the spread of the density, fitting a wide kernel to volume-like regions and a tight kernel to surface-like regions. We then extract an explicit mesh of a narrow band around the surface, with width determined by the kernel size, and fine-tune the radiance field within this band. At inference time, we cast rays against the mesh and evaluate the radiance field only within the enclosed region, greatly reducing the number of samples required. Experiments show that our approach enables efficient rendering at very high fidelity. We also demonstrate that the extracted envelope enables downstream applications such as animation and simulation.

Earth system models suffer from various structural and parametric errors in their representation of nonlinear, multi-scale processes, leading to uncertainties in their long-term projections. The effects of many of these errors (particularly those due to fast physics) can be quantified in short-term simulations, e.g., as differences between the predicted and observed states (analysis increments). With the increase in the availability of high-quality observations and simulations, learning nudging from these increments to correct model errors has become an active research area. However, most studies focus on using neural networks, which while powerful, are hard to interpret, are data-hungry, and poorly generalize out-of-distribution. Here, we show the capabilities of Model Error Discovery with Interpretability and Data Assimilation (MEDIDA), a general, data-efficient framework that uses sparsity-promoting equation-discovery techniques to learn model errors from analysis increments. Using two-layer quasi-geostrophic turbulence as the test case, MEDIDA is shown to successfully discover various linear and nonlinear structural/parametric errors when full observations are available. Discovery from spatially sparse observations is found to require highly accurate interpolation schemes. While NNs have shown success as interpolators in recent studies, here, they are found inadequate due to their inability to accurately represent small scales, a phenomenon known as spectral bias. We show that a general remedy, adding a random Fourier feature layer to the NN, resolves this issue enabling MEDIDA to successfully discover model errors from sparse observations. These promising results suggest that with further development, MEDIDA could be scaled up to models of the Earth system and real observations.

Imposing known physical constraints, such as conservation laws, during neural network training introduces an inductive bias that can improve accuracy, reliability, convergence, and data efficiency for modeling physical dynamics. While such constraints can be softly imposed via loss function penalties, recent advancements in differentiable physics and optimization improve performance by incorporating PDE-constrained optimization as individual layers in neural networks. This enables a stricter adherence to physical constraints. However, imposing hard constraints significantly increases computational and memory costs, especially for complex dynamical systems. This is because it requires solving an optimization problem over a large number of points in a mesh, representing spatial and temporal discretizations, which greatly increases the complexity of the constraint. To address this challenge, we develop a scalable approach to enforce hard physical constraints using Mixture-of-Experts (MoE), which can be used with any neural network architecture. Our approach imposes the constraint over smaller decomposed domains, each of which is solved by an "expert" through differentiable optimization. During training, each expert independently performs a localized backpropagation step by leveraging the implicit function theorem; the independence of each expert allows for parallelization across multiple GPUs. Compared to standard differentiable optimization, our scalable approach achieves greater accuracy in the neural PDE solver setting for predicting the dynamics of challenging non-linear systems. We also improve training stability and require significantly less computation time during both training and inference stages.

We present a novel deep learning based technique for volumetric ambient occlusion in the context of direct volume rendering. Our proposed Deep Volumetric Ambient Occlusion (DVAO) approach can predict per-voxel ambient occlusion in volumetric data sets, while considering global information provided through the transfer function. The proposed neural network only needs to be executed upon change of this global information, and thus supports real-time volume interaction. Accordingly, we demonstrate DVAOs ability to predict volumetric ambient occlusion, such that it can be applied interactively within direct volume rendering. To achieve the best possible results, we propose and analyze a variety of transfer function representations and injection strategies for deep neural networks. Based on the obtained results we also give recommendations applicable in similar volume learning scenarios. Lastly, we show that DVAO generalizes to a variety of modalities, despite being trained on computed tomography data only.

While 2D diffusion models generate realistic, high-detail images, 3D shape generation methods like Score Distillation Sampling (SDS) built on these 2D diffusion models produce cartoon-like, over-smoothed shapes. To help explain this discrepancy, we show that the image guidance used in Score Distillation can be understood as the velocity field of a 2D denoising generative process, up to the choice of a noise term. In particular, after a change of variables, SDS resembles a high-variance version of Denoising Diffusion Implicit Models (DDIM) with a differently-sampled noise term: SDS introduces noise i.i.d. randomly at each step, while DDIM infers it from the previous noise predictions. This excessive variance can lead to over-smoothing and unrealistic outputs. We show that a better noise approximation can be recovered by inverting DDIM in each SDS update step. This modification makes SDS's generative process for 2D images almost identical to DDIM. In 3D, it removes over-smoothing, preserves higher-frequency detail, and brings the generation quality closer to that of 2D samplers. Experimentally, our method achieves better or similar 3D generation quality compared to other state-of-the-art Score Distillation methods, all without training additional neural networks or multi-view supervision, and providing useful insights into relationship between 2D and 3D asset generation with diffusion models.

Recent advances in 2D image generation have achieved remarkable quality,largely driven by the capacity of diffusion models and the availability of large-scale datasets. However, direct 3D generation is still constrained by the scarcity and lower fidelity of 3D datasets. In this paper, we introduce Zero-1-to-G, a novel approach that addresses this problem by enabling direct single-view generation on Gaussian splats using pretrained 2D diffusion models. Our key insight is that Gaussian splats, a 3D representation, can be decomposed into multi-view images encoding different attributes. This reframes the challenging task of direct 3D generation within a 2D diffusion framework, allowing us to leverage the rich priors of pretrained 2D diffusion models. To incorporate 3D awareness, we introduce cross-view and cross-attribute attention layers, which capture complex correlations and enforce 3D consistency across generated splats. This makes Zero-1-to-G the first direct image-to-3D generative model to effectively utilize pretrained 2D diffusion priors, enabling efficient training and improved generalization to unseen objects. Extensive experiments on both synthetic and in-the-wild datasets demonstrate superior performance in 3D object generation, offering a new approach to high-quality 3D generation.

Diffusion probabilistic models (DPMs) have exhibited excellent performance for high-fidelity image generation while suffering from inefficient sampling. Recent works accelerate the sampling procedure by proposing fast ODE solvers that leverage the specific ODE form of DPMs. However, they highly rely on specific parameterization during inference (such as noise/data prediction), which might not be the optimal choice. In this work, we propose a novel formulation towards the optimal parameterization during sampling that minimizes the first-order discretization error of the ODE solution. Based on such formulation, we propose DPM-Solver-v3, a new fast ODE solver for DPMs by introducing several coefficients efficiently computed on the pretrained model, which we call empirical model statistics. We further incorporate multistep methods and a predictor-corrector framework, and propose some techniques for improving sample quality at small numbers of function evaluations (NFE) or large guidance scales. Experiments show that DPM-Solver-v3 achieves consistently better or comparable performance in both unconditional and conditional sampling with both pixel-space and latent-space DPMs, especially in 5sim10 NFEs. We achieve FIDs of 12.21 (5 NFE), 2.51 (10 NFE) on unconditional CIFAR10, and MSE of 0.55 (5 NFE, 7.5 guidance scale) on Stable Diffusion, bringing a speed-up of 15\%sim30\% compared to previous state-of-the-art training-free methods. Code is available at https://github.com/thu-ml/DPM-Solver-v3.

Geometric Deep Learning has recently made striking progress with the advent of continuous Deep Implicit Fields. They allow for detailed modeling of watertight surfaces of arbitrary topology while not relying on a 3D Euclidean grid, resulting in a learnable parameterization that is not limited in resolution. Unfortunately, these methods are often not suitable for applications that require an explicit mesh-based surface representation because converting an implicit field to such a representation relies on the Marching Cubes algorithm, which cannot be differentiated with respect to the underlying implicit field. In this work, we remove this limitation and introduce a differentiable way to produce explicit surface mesh representations from Deep Signed Distance Functions. Our key insight is that by reasoning on how implicit field perturbations impact local surface geometry, one can ultimately differentiate the 3D location of surface samples with respect to the underlying deep implicit field. We exploit this to define MeshSDF, an end-to-end differentiable mesh representation which can vary its topology. We use two different applications to validate our theoretical insight: Single-View Reconstruction via Differentiable Rendering and Physically-Driven Shape Optimization. In both cases our differentiable parameterization gives us an edge over state-of-the-art algorithms.

In an industrial group like Safran, numerical simulations of physical phenomena are integral to most design processes. At Safran's corporate research center, we enhance these processes by developing fast and reliable surrogate models for various physics. We focus here on two technologies developed in recent years. The first is a physical reduced-order modeling method for non-linear structural mechanics and thermal analysis, used for calculating the lifespan of high-pressure turbine blades and performing heat analysis of high-pressure compressors. The second technology involves learning physics simulations with non-parameterized geometrical variability using classical machine learning tools, such as Gaussian process regression. Finally, we present our contributions to the open-source and open-data community.

Despite the remarkable success of diffusion models in image generation, slow sampling remains a persistent issue. To accelerate the sampling process, prior studies have reformulated diffusion sampling as an ODE/SDE and introduced higher-order numerical methods. However, these methods often produce divergence artifacts, especially with a low number of sampling steps, which limits the achievable acceleration. In this paper, we investigate the potential causes of these artifacts and suggest that the small stability regions of these methods could be the principal cause. To address this issue, we propose two novel techniques. The first technique involves the incorporation of Heavy Ball (HB) momentum, a well-known technique for improving optimization, into existing diffusion numerical methods to expand their stability regions. We also prove that the resulting methods have first-order convergence. The second technique, called Generalized Heavy Ball (GHVB), constructs a new high-order method that offers a variable trade-off between accuracy and artifact suppression. Experimental results show that our techniques are highly effective in reducing artifacts and improving image quality, surpassing state-of-the-art diffusion solvers on both pixel-based and latent-based diffusion models for low-step sampling. Our research provides novel insights into the design of numerical methods for future diffusion work.

Foundations of a new projection-based model reduction approach for convection dominated nonlinear fluid flows are summarized. In this method the evolution of the flow is approximated in the Lagrangian frame of reference. Global basis functions are used to approximate both the state and the position of the Lagrangian computational domain. It is demonstrated that in this framework, certain wave-like solutions exhibit low-rank structure and thus, can be efficiently compressed using relatively few global basis. The proposed approach is successfully demonstrated for the reduction of several simple but representative problems.

Though Rectified Flows (ReFlows) with distillation offers a promising way for fast sampling, its fast inversion transforms images back to structured noise for recovery and following editing remains unsolved. This paper introduces FireFlow, a simple yet effective zero-shot approach that inherits the startling capacity of ReFlow-based models (such as FLUX) in generation while extending its capabilities to accurate inversion and editing in 8 steps. We first demonstrate that a carefully designed numerical solver is pivotal for ReFlow inversion, enabling accurate inversion and reconstruction with the precision of a second-order solver while maintaining the practical efficiency of a first-order Euler method. This solver achieves a 3times runtime speedup compared to state-of-the-art ReFlow inversion and editing techniques, while delivering smaller reconstruction errors and superior editing results in a training-free mode. The code is available at https://github.com/HolmesShuan/FireFlow{this URL}.

We study the problem of few-shot physically-aware articulated mesh generation. By observing an articulated object dataset containing only a few examples, we wish to learn a model that can generate diverse meshes with high visual fidelity and physical validity. Previous mesh generative models either have difficulties in depicting a diverse data space from only a few examples or fail to ensure physical validity of their samples. Regarding the above challenges, we propose two key innovations, including 1) a hierarchical mesh deformation-based generative model based upon the divide-and-conquer philosophy to alleviate the few-shot challenge by borrowing transferrable deformation patterns from large scale rigid meshes and 2) a physics-aware deformation correction scheme to encourage physically plausible generations. We conduct extensive experiments on 6 articulated categories to demonstrate the superiority of our method in generating articulated meshes with better diversity, higher visual fidelity, and better physical validity over previous methods in the few-shot setting. Further, we validate solid contributions of our two innovations in the ablation study. Project page with code is available at https://meowuu7.github.io/few-arti-obj-gen.

4D content generation focuses on creating dynamic 3D objects that change over time. Existing methods primarily rely on pre-trained video diffusion models, utilizing sampling processes or reference videos. However, these approaches face significant challenges. Firstly, the generated 4D content often fails to adhere to real-world physics since video diffusion models do not incorporate physical priors. Secondly, the extensive sampling process and the large number of parameters in diffusion models result in exceedingly time-consuming generation processes. To address these issues, we introduce Phy124, a novel, fast, and physics-driven method for controllable 4D content generation from a single image. Phy124 integrates physical simulation directly into the 4D generation process, ensuring that the resulting 4D content adheres to natural physical laws. Phy124 also eliminates the use of diffusion models during the 4D dynamics generation phase, significantly speeding up the process. Phy124 allows for the control of 4D dynamics, including movement speed and direction, by manipulating external forces. Extensive experiments demonstrate that Phy124 generates high-fidelity 4D content with significantly reduced inference times, achieving stateof-the-art performance. The code and generated 4D content are available at the provided link: https://anonymous.4open.science/r/BBF2/.

We present a super-fast convergence approach to reconstructing the per-scene radiance field from a set of images that capture the scene with known poses. This task, which is often applied to novel view synthesis, is recently revolutionized by Neural Radiance Field (NeRF) for its state-of-the-art quality and flexibility. However, NeRF and its variants require a lengthy training time ranging from hours to days for a single scene. In contrast, our approach achieves NeRF-comparable quality and converges rapidly from scratch in less than 15 minutes with a single GPU. We adopt a representation consisting of a density voxel grid for scene geometry and a feature voxel grid with a shallow network for complex view-dependent appearance. Modeling with explicit and discretized volume representations is not new, but we propose two simple yet non-trivial techniques that contribute to fast convergence speed and high-quality output. First, we introduce the post-activation interpolation on voxel density, which is capable of producing sharp surfaces in lower grid resolution. Second, direct voxel density optimization is prone to suboptimal geometry solutions, so we robustify the optimization process by imposing several priors. Finally, evaluation on five inward-facing benchmarks shows that our method matches, if not surpasses, NeRF's quality, yet it only takes about 15 minutes to train from scratch for a new scene.

Scientific Machine Learning (SciML) is concerned with the development of learned emulators of physical systems governed by partial differential equations (PDE). In application domains such as weather forecasting, molecular dynamics, and inverse design, ML-based surrogate models are increasingly used to augment or replace inefficient and often non-differentiable numerical simulation algorithms. While a number of ML-based methods for approximating the solutions of PDEs have been proposed in recent years, they typically do not adapt to the parameters of the PDEs, making it difficult to generalize to PDE parameters not seen during training. We propose a Channel Attention mechanism guided by PDE Parameter Embeddings (CAPE) component for neural surrogate models and a simple yet effective curriculum learning strategy. The CAPE module can be combined with neural PDE solvers allowing them to adapt to unseen PDE parameters. The curriculum learning strategy provides a seamless transition between teacher-forcing and fully auto-regressive training. We compare CAPE in conjunction with the curriculum learning strategy using a popular PDE benchmark and obtain consistent and significant improvements over the baseline models. The experiments also show several advantages of CAPE, such as its increased ability to generalize to unseen PDE parameters without large increases inference time and parameter count.

We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, are inherently unable to scale to high dimensions, or use approximations for limiting quantities that result in biased training objectives. Riemannian Flow Matching bypasses these limitations and offers several advantages over previous approaches: it is simulation-free on simple geometries, does not require divergence computation, and computes its target vector field in closed-form. The key ingredient behind RFM is the construction of a relatively simple premetric for defining target vector fields, which encompasses the existing Euclidean case. To extend to general geometries, we rely on the use of spectral decompositions to efficiently compute premetrics on the fly. Our method achieves state-of-the-art performance on many real-world non-Euclidean datasets, and we demonstrate tractable training on general geometries, including triangular meshes with highly non-trivial curvature and boundaries.

The abundance of data has given machine learning considerable momentum in natural sciences and engineering, though modeling of physical processes is often difficult. A particularly tough problem is the efficient representation of geometric boundaries. Triangularized geometric boundaries are well understood and ubiquitous in engineering applications. However, it is notoriously difficult to integrate them into machine learning approaches due to their heterogeneity with respect to size and orientation. In this work, we introduce an effective theory to model particle-boundary interactions, which leads to our new Boundary Graph Neural Networks (BGNNs) that dynamically modify graph structures to obey boundary conditions. The new BGNNs are tested on complex 3D granular flow processes of hoppers, rotating drums and mixers, which are all standard components of modern industrial machinery but still have complicated geometry. BGNNs are evaluated in terms of computational efficiency as well as prediction accuracy of particle flows and mixing entropies. BGNNs are able to accurately reproduce 3D granular flows within simulation uncertainties over hundreds of thousands of simulation timesteps. Most notably, in our experiments, particles stay within the geometric objects without using handcrafted conditions or restrictions.

A mesh motion model based on deep operator networks is presented. The model is trained on and evaluated against a biharmonic mesh motion model on a fluid-structure interaction benchmark problem and further evaluated in a setting where biharmonic mesh motion fails. The performance of the proposed mesh motion model is comparable to the biharmonic mesh motion on the test problems.

Mesh generation has become a critical topic in recent years, forming the foundation of all 3D objects used across various applications, such as virtual reality, gaming, and 3D printing. With advancements in computational resources and machine learning, neural networks have emerged as powerful tools for generating high-quality 3D object representations, enabling accurate scene and object reconstructions. Despite these advancements, many methods produce meshes that lack realism or exhibit geometric and textural flaws, necessitating additional processing to improve their quality. This research introduces a convex optimization programming called disciplined convex programming to enhance existing meshes by refining their texture and geometry with a conic solver. By focusing on a sparse set of point clouds from both the original and target meshes, this method demonstrates significant improvements in mesh quality with minimal data requirements. To evaluate the approach, the classical dolphin mesh dataset from Facebook AI was used as a case study, with optimization performed using the CVXPY library. The results reveal promising potential for streamlined and effective mesh refinement.

We study the momentum-based minimization of a diffuse perimeter functional on Euclidean spaces and on graphs with applications to semi-supervised classification tasks in machine learning. While the gradient flow in the task at hand is a parabolic partial differential equation, the momentum-method corresponds to a damped hyperbolic PDE, leading to qualitatively and quantitatively different trajectories. Using a convex-concave splitting-based FISTA-type time discretization, we demonstrate empirically that momentum can lead to faster convergence if the time step size is large but not too large. With large time steps, the PDE analysis offers only limited insight into the geometric behavior of solutions and typical hyperbolic phenomena like loss of regularity are not be observed in sample simulations.

The planning and control of a robot swarm in a complex environment have attracted increasing attention. To this end, the idea of virtual tubes has been taken up in our previous work. Specifically, a virtual tube with varying widths has been planned to avoid collisions with obstacles in a complex environment. Based on the planned virtual tube for a large number of speed-constrained robots, the average forward speed and density along the virtual tube are further planned in this paper to ensure safety and improve efficiency. Compared with the existing methods, the proposed method is based on global information and can be applied to traversing narrow spaces for speed-constrained robot swarms. Numerical simulations and experiments are conducted to show that the safety and efficiency of the passing-through process are improved. A video about simulations and experiments is available on https://youtu.be/lJHdMQMqSpc.

Inverse Stefan problem arising in modeling of laser ablation of biomedical tissues is analyzed, where information on the coefficients, heat flux on the fixed boundary, and density of heat sources are missing and must be found along with the temperature and free boundary. Optimal control framework is employed, where the missing data and the free boundary are components of the control vector, and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary. Discretization by finite differences is pursued, and convergence of the discrete optimal control problems to the original problem is proven.

Physical simulators have been widely used in robot planning and control. Among them, differentiable simulators are particularly favored, as they can be incorporated into gradient-based optimization algorithms that are efficient in solving inverse problems such as optimal control and motion planning. Simulating deformable objects is, however, more challenging compared to rigid body dynamics. The underlying physical laws of deformable objects are more complex, and the resulting systems have orders of magnitude more degrees of freedom and therefore they are significantly more computationally expensive to simulate. Computing gradients with respect to physical design or controller parameters is typically even more computationally challenging. In this paper, we propose a real-time, differentiable hybrid Lagrangian-Eulerian physical simulator for deformable objects, ChainQueen, based on the Moving Least Squares Material Point Method (MLS-MPM). MLS-MPM can simulate deformable objects including contact and can be seamlessly incorporated into inference, control and co-design systems. We demonstrate that our simulator achieves high precision in both forward simulation and backward gradient computation. We have successfully employed it in a diverse set of control tasks for soft robots, including problems with nearly 3,000 decision variables.

We present ConDiff, a novel dataset for scientific machine learning. ConDiff focuses on the parametric diffusion equation with space dependent coefficients, a fundamental problem in many applications of partial differential equations (PDEs). The main novelty of the proposed dataset is that we consider discontinuous coefficients with high contrast. These coefficient functions are sampled from a selected set of distributions. This class of problems is not only of great academic interest, but is also the basis for describing various environmental and industrial problems. In this way, ConDiff shortens the gap with real-world problems while remaining fully synthetic and easy to use. ConDiff consists of a diverse set of diffusion equations with coefficients covering a wide range of contrast levels and heterogeneity with a measurable complexity metric for clearer comparison between different coefficient functions. We baseline ConDiff on standard deep learning models in the field of scientific machine learning. By providing a large number of problem instances, each with its own coefficient function and right-hand side, we hope to encourage the development of novel physics-based deep learning approaches, such as neural operators, ultimately driving progress towards more accurate and efficient solutions of complex PDE problems.

We introduce Garment3DGen a new method to synthesize 3D garment assets from a base mesh given a single input image as guidance. Our proposed approach allows users to generate 3D textured clothes based on both real and synthetic images, such as those generated by text prompts. The generated assets can be directly draped and simulated on human bodies. First, we leverage the recent progress of image to 3D diffusion methods to generate 3D garment geometries. However, since these geometries cannot be utilized directly for downstream tasks, we propose to use them as pseudo ground-truth and set up a mesh deformation optimization procedure that deforms a base template mesh to match the generated 3D target. Second, we introduce carefully designed losses that allow the input base mesh to freely deform towards the desired target, yet preserve mesh quality and topology such that they can be simulated. Finally, a texture estimation module generates high-fidelity texture maps that are globally and locally consistent and faithfully capture the input guidance, allowing us to render the generated 3D assets. With Garment3DGen users can generate the textured 3D garment of their choice without the need of artist intervention. One can provide a textual prompt describing the garment they desire to generate a simulation-ready 3D asset. We present a plethora of quantitative and qualitative comparisons on various assets both real and generated and provide use-cases of how one can generate simulation-ready 3D garments.

Advancements in text-to-image diffusion models have led to significant progress in fast 3D content creation. One common approach is to generate a set of multi-view images of an object, and then reconstruct it into a 3D model. However, this approach bypasses the use of a native 3D representation of the object and is hence prone to geometric artifacts and limited in controllability and manipulation capabilities. An alternative approach involves native 3D generative models that directly produce 3D representations. These models, however, are typically limited in their resolution, resulting in lower quality 3D objects. In this work, we bridge the quality gap between methods that directly generate 3D representations and ones that reconstruct 3D objects from multi-view images. We introduce a multi-view to multi-view diffusion model called Sharp-It, which takes a 3D consistent set of multi-view images rendered from a low-quality object and enriches its geometric details and texture. The diffusion model operates on the multi-view set in parallel, in the sense that it shares features across the generated views. A high-quality 3D model can then be reconstructed from the enriched multi-view set. By leveraging the advantages of both 2D and 3D approaches, our method offers an efficient and controllable method for high-quality 3D content creation. We demonstrate that Sharp-It enables various 3D applications, such as fast synthesis, editing, and controlled generation, while attaining high-quality assets.

Text-to-3D generation has recently seen significant progress. To enhance its practicality in real-world applications, it is crucial to generate multiple independent objects with interactions, similar to layer-compositing in 2D image editing. However, existing text-to-3D methods struggle with this task, as they are designed to generate either non-independent objects or independent objects lacking spatially plausible interactions. Addressing this, we propose DreamDissector, a text-to-3D method capable of generating multiple independent objects with interactions. DreamDissector accepts a multi-object text-to-3D NeRF as input and produces independent textured meshes. To achieve this, we introduce the Neural Category Field (NeCF) for disentangling the input NeRF. Additionally, we present the Category Score Distillation Sampling (CSDS), facilitated by a Deep Concept Mining (DCM) module, to tackle the concept gap issue in diffusion models. By leveraging NeCF and CSDS, we can effectively derive sub-NeRFs from the original scene. Further refinement enhances geometry and texture. Our experimental results validate the effectiveness of DreamDissector, providing users with novel means to control 3D synthesis at the object level and potentially opening avenues for various creative applications in the future.

We demonstrate the feasibility of integrating physics-based animations of solids and fluids with 3D Gaussian Splatting (3DGS) to create novel effects in virtual scenes reconstructed using 3DGS. Leveraging the coherence of the Gaussian splatting and position-based dynamics (PBD) in the underlying representation, we manage rendering, view synthesis, and the dynamics of solids and fluids in a cohesive manner. Similar to Gaussian shader, we enhance each Gaussian kernel with an added normal, aligning the kernel's orientation with the surface normal to refine the PBD simulation. This approach effectively eliminates spiky noises that arise from rotational deformation in solids. It also allows us to integrate physically based rendering to augment the dynamic surface reflections on fluids. Consequently, our framework is capable of realistically reproducing surface highlights on dynamic fluids and facilitating interactions between scene objects and fluids from new views. For more information, please visit our project page at https://amysteriouscat.github.io/GaussianSplashing/.

With the introduction of Neural Radiance Fields (NeRFs), novel view synthesis has recently made a big leap forward. At the core, NeRF proposes that each 3D point can emit radiance, allowing to conduct view synthesis using differentiable volumetric rendering. While neural radiance fields can accurately represent 3D scenes for computing the image rendering, 3D meshes are still the main scene representation supported by most computer graphics and simulation pipelines, enabling tasks such as real time rendering and physics-based simulations. Obtaining 3D meshes from neural radiance fields still remains an open challenge since NeRFs are optimized for view synthesis, not enforcing an accurate underlying geometry on the radiance field. We thus propose a novel compact and flexible architecture that enables easy 3D surface reconstruction from any NeRF-driven approach. Upon having trained the radiance field, we distill the volumetric 3D representation into a Signed Surface Approximation Network, allowing easy extraction of the 3D mesh and appearance. Our final 3D mesh is physically accurate and can be rendered in real time on an array of devices.

Most advances in 3D Generative Adversarial Networks (3D GANs) largely depend on ray casting-based volume rendering, which incurs demanding rendering costs. One promising alternative is rasterization-based 3D Gaussian Splatting (3D-GS), providing a much faster rendering speed and explicit 3D representation. In this paper, we exploit Gaussian as a 3D representation for 3D GANs by leveraging its efficient and explicit characteristics. However, in an adversarial framework, we observe that a na\"ive generator architecture suffers from training instability and lacks the capability to adjust the scale of Gaussians. This leads to model divergence and visual artifacts due to the absence of proper guidance for initialized positions of Gaussians and densification to manage their scales adaptively. To address these issues, we introduce a generator architecture with a hierarchical multi-scale Gaussian representation that effectively regularizes the position and scale of generated Gaussians. Specifically, we design a hierarchy of Gaussians where finer-level Gaussians are parameterized by their coarser-level counterparts; the position of finer-level Gaussians would be located near their coarser-level counterparts, and the scale would monotonically decrease as the level becomes finer, modeling both coarse and fine details of the 3D scene. Experimental results demonstrate that ours achieves a significantly faster rendering speed (x100) compared to state-of-the-art 3D consistent GANs with comparable 3D generation capability. Project page: https://hse1032.github.io/gsgan.

Discovering the underlying physical behavior of complex systems is a crucial, but less well-understood topic in many engineering disciplines. This study proposes a finite-difference inspired convolutional neural network framework to learn hidden partial differential equations from given data and iteratively estimate future dynamical behavior. The methodology designs the filter sizes such that they mimic the finite difference between the neighboring points. By learning the governing equation, the network predicts the future evolution of the solution by using only a few trainable parameters. In this paper, we provide numerical results to compare the efficiency of the second-order Trust-Region Conjugate Gradient (TRCG) method with the first-order ADAM optimizer.

A data-driven model (DDM) suitable for regional weather forecasting applications is presented. The model extends the Artificial Intelligence Forecasting System by introducing a stretched-grid architecture that dedicates higher resolution over a regional area of interest and maintains a lower resolution elsewhere on the globe. The model is based on graph neural networks, which naturally affords arbitrary multi-resolution grid configurations. The model is applied to short-range weather prediction for the Nordics, producing forecasts at 2.5 km spatial and 6 h temporal resolution. The model is pre-trained on 43 years of global ERA5 data at 31 km resolution and is further refined using 3.3 years of 2.5 km resolution operational analyses from the MetCoOp Ensemble Prediction System (MEPS). The performance of the model is evaluated using surface observations from measurement stations across Norway and is compared to short-range weather forecasts from MEPS. The DDM outperforms both the control run and the ensemble mean of MEPS for 2 m temperature. The model also produces competitive precipitation and wind speed forecasts, but is shown to underestimate extreme events.

Gradient regularization (GR) is a method that penalizes the gradient norm of the training loss during training. While some studies have reported that GR can improve generalization performance, little attention has been paid to it from the algorithmic perspective, that is, the algorithms of GR that efficiently improve the performance. In this study, we first reveal that a specific finite-difference computation, composed of both gradient ascent and descent steps, reduces the computational cost of GR. Next, we show that the finite-difference computation also works better in the sense of generalization performance. We theoretically analyze a solvable model, a diagonal linear network, and clarify that GR has a desirable implicit bias to so-called rich regime and finite-difference computation strengthens this bias. Furthermore, finite-difference GR is closely related to some other algorithms based on iterative ascent and descent steps for exploring flat minima. In particular, we reveal that the flooding method can perform finite-difference GR in an implicit way. Thus, this work broadens our understanding of GR for both practice and theory.

Spectral analysis provides one of the most effective paradigms for information-preserving dimensionality reduction, as simple descriptions of naturally occurring signals are often obtained via few terms of periodic basis functions. In this work, we study deep neural networks designed to harness the structure in frequency domain for efficient learning of long-range correlations in space or time: frequency-domain models (FDMs). Existing FDMs are based on complex-valued transforms i.e. Fourier Transforms (FT), and layers that perform computation on the spectrum and input data separately. This design introduces considerable computational overhead: for each layer, a forward and inverse FT. Instead, this work introduces a blueprint for frequency domain learning through a single transform: transform once (T1). To enable efficient, direct learning in the frequency domain we derive a variance-preserving weight initialization scheme and investigate methods for frequency selection in reduced-order FDMs. Our results noticeably streamline the design process of FDMs, pruning redundant transforms, and leading to speedups of 3x to 10x that increase with data resolution and model size. We perform extensive experiments on learning the solution operator of spatio-temporal dynamics, including incompressible Navier-Stokes, turbulent flows around airfoils and high-resolution video of smoke. T1 models improve on the test performance of FDMs while requiring significantly less computation (5 hours instead of 32 for our large-scale experiment), with over 20% reduction in average predictive error across tasks.

Neural Radiance Fields (NeRFs) have achieved great success in the past few years. However, most current methods still require intensive resources due to ray marching-based rendering. To construct urban-level radiance fields efficiently, we design Deformable Neural Mesh Primitive~(DNMP), and propose to parameterize the entire scene with such primitives. The DNMP is a flexible and compact neural variant of classic mesh representation, which enjoys both the efficiency of rasterization-based rendering and the powerful neural representation capability for photo-realistic image synthesis. Specifically, a DNMP consists of a set of connected deformable mesh vertices with paired vertex features to parameterize the geometry and radiance information of a local area. To constrain the degree of freedom for optimization and lower the storage budgets, we enforce the shape of each primitive to be decoded from a relatively low-dimensional latent space. The rendering colors are decoded from the vertex features (interpolated with rasterization) by a view-dependent MLP. The DNMP provides a new paradigm for urban-level scene representation with appealing properties: (1) High-quality rendering. Our method achieves leading performance for novel view synthesis in urban scenarios. (2) Low computational costs. Our representation enables fast rendering (2.07ms/1k pixels) and low peak memory usage (110MB/1k pixels). We also present a lightweight version that can run 33times faster than vanilla NeRFs, and comparable to the highly-optimized Instant-NGP (0.61 vs 0.71ms/1k pixels). Project page: https://dnmp.github.io/{https://dnmp.github.io/}.

Recent years have witnessed the outstanding success of deep learning in various fields such as vision and natural language processing. This success is largely indebted to the massive size of deep learning models that is expected to increase unceasingly. This growth of the deep learning models is accompanied by issues related to their considerable energy consumption, both during the training and inference phases, as well as their scalability. Although a number of work based on unconventional physical systems have been proposed which addresses the issue of energy efficiency in the inference phase, efficient training of deep learning models has remained unaddressed. So far, training of digital deep learning models mainly relies on backpropagation, which is not suitable for physical implementation as it requires perfect knowledge of the computation performed in the so-called forward pass of the neural network. Here, we tackle this issue by proposing a simple deep neural network architecture augmented by a biologically plausible learning algorithm, referred to as "model-free forward-forward training". The proposed architecture enables training deep physical neural networks consisting of layers of physical nonlinear systems, without requiring detailed knowledge of the nonlinear physical layers' properties. We show that our method outperforms state-of-the-art hardware-aware training methods by improving training speed, decreasing digital computations, and reducing power consumption in physical systems. We demonstrate the adaptability of the proposed method, even in systems exposed to dynamic or unpredictable external perturbations. To showcase the universality of our approach, we train diverse wave-based physical neural networks that vary in the underlying wave phenomenon and the type of non-linearity they use, to perform vowel and image classification tasks experimentally.

We propose a hybrid neural network (NN) and PDE approach for learning generalizable PDE dynamics from motion observations. Many NN approaches learn an end-to-end model that implicitly models both the governing PDE and constitutive models (or material models). Without explicit PDE knowledge, these approaches cannot guarantee physical correctness and have limited generalizability. We argue that the governing PDEs are often well-known and should be explicitly enforced rather than learned. Instead, constitutive models are particularly suitable for learning due to their data-fitting nature. To this end, we introduce a new framework termed "Neural Constitutive Laws" (NCLaw), which utilizes a network architecture that strictly guarantees standard constitutive priors, including rotation equivariance and undeformed state equilibrium. We embed this network inside a differentiable simulation and train the model by minimizing a loss function based on the difference between the simulation and the motion observation. We validate NCLaw on various large-deformation dynamical systems, ranging from solids to fluids. After training on a single motion trajectory, our method generalizes to new geometries, initial/boundary conditions, temporal ranges, and even multi-physics systems. On these extremely out-of-distribution generalization tasks, NCLaw is orders-of-magnitude more accurate than previous NN approaches. Real-world experiments demonstrate our method's ability to learn constitutive laws from videos.

We describe MGARD, a software providing MultiGrid Adaptive Reduction for floating-point scientific data on structured and unstructured grids. With exceptional data compression capability and precise error control, MGARD addresses a wide range of requirements, including storage reduction, high-performance I/O, and in-situ data analysis. It features a unified application programming interface (API) that seamlessly operates across diverse computing architectures. MGARD has been optimized with highly-tuned GPU kernels and efficient memory and device management mechanisms, ensuring scalable and rapid operations.

Diffusion models have shown great promise for image and video generation, but sampling from state-of-the-art models requires expensive numerical integration of a generative ODE. One approach for tackling this problem is rectified flows, which iteratively learn smooth ODE paths that are less susceptible to truncation error. However, rectified flows still require a relatively large number of function evaluations (NFEs). In this work, we propose improved techniques for training rectified flows, allowing them to compete with knowledge distillation methods even in the low NFE setting. Our main insight is that under realistic settings, a single iteration of the Reflow algorithm for training rectified flows is sufficient to learn nearly straight trajectories; hence, the current practice of using multiple Reflow iterations is unnecessary. We thus propose techniques to improve one-round training of rectified flows, including a U-shaped timestep distribution and LPIPS-Huber premetric. With these techniques, we improve the FID of the previous 2-rectified flow by up to 72% in the 1 NFE setting on CIFAR-10. On ImageNet 64times64, our improved rectified flow outperforms the state-of-the-art distillation methods such as consistency distillation and progressive distillation in both one-step and two-step settings and rivals the performance of improved consistency training (iCT) in FID. Code is available at https://github.com/sangyun884/rfpp.

Training competitive deep video models is an order of magnitude slower than training their counterpart image models. Slow training causes long research cycles, which hinders progress in video understanding research. Following standard practice for training image models, video model training assumes a fixed mini-batch shape: a specific number of clips, frames, and spatial size. However, what is the optimal shape? High resolution models perform well, but train slowly. Low resolution models train faster, but they are inaccurate. Inspired by multigrid methods in numerical optimization, we propose to use variable mini-batch shapes with different spatial-temporal resolutions that are varied according to a schedule. The different shapes arise from resampling the training data on multiple sampling grids. Training is accelerated by scaling up the mini-batch size and learning rate when shrinking the other dimensions. We empirically demonstrate a general and robust grid schedule that yields a significant out-of-the-box training speedup without a loss in accuracy for different models (I3D, non-local, SlowFast), datasets (Kinetics, Something-Something, Charades), and training settings (with and without pre-training, 128 GPUs or 1 GPU). As an illustrative example, the proposed multigrid method trains a ResNet-50 SlowFast network 4.5x faster (wall-clock time, same hardware) while also improving accuracy (+0.8% absolute) on Kinetics-400 compared to the baseline training method. Code is available online.

AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN). We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP in terms of accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.

Neural Radiance Field training can be accelerated through the use of grid-based representations in NeRF's learned mapping from spatial coordinates to colors and volumetric density. However, these grid-based approaches lack an explicit understanding of scale and therefore often introduce aliasing, usually in the form of jaggies or missing scene content. Anti-aliasing has previously been addressed by mip-NeRF 360, which reasons about sub-volumes along a cone rather than points along a ray, but this approach is not natively compatible with current grid-based techniques. We show how ideas from rendering and signal processing can be used to construct a technique that combines mip-NeRF 360 and grid-based models such as Instant NGP to yield error rates that are 8% - 77% lower than either prior technique, and that trains 24x faster than mip-NeRF 360.

We present a novel generative 3D modeling system, coined CraftsMan, which can generate high-fidelity 3D geometries with highly varied shapes, regular mesh topologies, and detailed surfaces, and, notably, allows for refining the geometry in an interactive manner. Despite the significant advancements in 3D generation, existing methods still struggle with lengthy optimization processes, irregular mesh topologies, noisy surfaces, and difficulties in accommodating user edits, consequently impeding their widespread adoption and implementation in 3D modeling software. Our work is inspired by the craftsman, who usually roughs out the holistic figure of the work first and elaborates the surface details subsequently. Specifically, we employ a 3D native diffusion model, which operates on latent space learned from latent set-based 3D representations, to generate coarse geometries with regular mesh topology in seconds. In particular, this process takes as input a text prompt or a reference image and leverages a powerful multi-view (MV) diffusion model to generate multiple views of the coarse geometry, which are fed into our MV-conditioned 3D diffusion model for generating the 3D geometry, significantly improving robustness and generalizability. Following that, a normal-based geometry refiner is used to significantly enhance the surface details. This refinement can be performed automatically, or interactively with user-supplied edits. Extensive experiments demonstrate that our method achieves high efficacy in producing superior-quality 3D assets compared to existing methods. HomePage: https://craftsman3d.github.io/, Code: https://github.com/wyysf-98/CraftsMan

Intra-fraction motion in radiotherapy is commonly modeled using deformable image registration (DIR). However, existing methods often struggle to balance speed and accuracy, limiting their applicability in clinical scenarios. This study introduces a novel approach that harnesses Neural Graphics Primitives (NGP) to optimize the displacement vector field (DVF). Our method leverages learned primitives, processed as splats, and interpolates within space using a shallow neural network. Uniquely, it enables self-supervised optimization at an ultra-fast speed, negating the need for pre-training on extensive datasets and allowing seamless adaptation to new cases. We validated this approach on the 4D-CT lung dataset DIR-lab, achieving a target registration error (TRE) of 1.15\pm1.15 mm within a remarkable time of 1.77 seconds. Notably, our method also addresses the sliding boundary problem, a common challenge in conventional DIR methods.

The creation of photorealistic virtual worlds requires the accurate modeling of 3D surface geometry for a wide range of objects. For this, meshes are appealing since they 1) enable fast physics-based rendering with realistic material and lighting, 2) support physical simulation, and 3) are memory-efficient for modern graphics pipelines. Recent work on reconstructing and statistically modeling 3D shape, however, has critiqued meshes as being topologically inflexible. To capture a wide range of object shapes, any 3D representation must be able to model solid, watertight, shapes as well as thin, open, surfaces. Recent work has focused on the former, and methods for reconstructing open surfaces do not support fast reconstruction with material and lighting or unconditional generative modelling. Inspired by the observation that open surfaces can be seen as islands floating on watertight surfaces, we parameterize open surfaces by defining a manifold signed distance field on watertight templates. With this parameterization, we further develop a grid-based and differentiable representation that parameterizes both watertight and non-watertight meshes of arbitrary topology. Our new representation, called Ghost-on-the-Shell (G-Shell), enables two important applications: differentiable rasterization-based reconstruction from multiview images and generative modelling of non-watertight meshes. We empirically demonstrate that G-Shell achieves state-of-the-art performance on non-watertight mesh reconstruction and generation tasks, while also performing effectively for watertight meshes.