Lukas Prantl, Benjamin Ummenhofer, Vladlen Koltun et al.
We present a novel method for guaranteeing linear momentum in learned physics simulations. Unlike existing methods, we enforce conservation of momentum with a hard constraint, which we realize via antisymmetrical continuous convolutional layers. We combine these strict constraints with a hierarchical network architecture, a carefully constructed resampling scheme, and a training approach for temporal coherence. In combination, the proposed method allows us to increase the physical accuracy of the learned simulator substantially. In addition, the induced physical bias leads to significantly better generalization performance and makes our method more reliable in unseen test cases. We evaluate our method on a range of different, challenging fluid scenarios. Among others, we demonstrate that our approach generalizes to new scenarios with up to one million particles. Our results show that the proposed algorithm can learn complex dynamics while outperforming existing approaches in generalization and training performance. An implementation of our approach is available at https://github.com/tum-pbs/DMCF.
Turbulence in fluids, gases, and plasmas remains an open problem of both practical and fundamental importance. Its irreducible complexity usually cannot be tackled computationally in a brute-force style. Here, we combine Large Eddy Simulation (LES) techniques with Machine Learning (ML) to retain only the largest dynamics explicitly, while small-scale dynamics are described by an ML-based sub-grid-scale model. Applying this novel approach to self-driven plasma turbulence allows us to remove large parts of the inertial range, reducing the computational effort by about three orders of magnitude, while retaining the statistical physical properties of the turbulent system.
Maximilian Mueller, Robin Greif, Frank Jenko et al.
We investigate uncertainty estimation and multimodality via the non-deterministic predictions of Bayesian neural networks (BNNs) in fluid simulations. To this end, we deploy BNNs in three challenging experimental test-cases of increasing complexity: We show that BNNs, when used as surrogate models for steady-state fluid flow predictions, provide accurate physical predictions together with sensible estimates of uncertainty. Further, we experiment with perturbed temporal sequences from Navier-Stokes simulations and evaluate the capabilities of BNNs to capture multimodal evolutions. While our findings indicate that this is problematic for large perturbations, our results show that the networks learn to correctly predict high uncertainties in such situations. Finally, we study BNNs in the context of solver interactions with turbulent plasma flows. We find that BNN-based corrector networks can stabilize coarse-grained simulations and successfully create multimodal trajectories.
Benjamin J. Holzschuh, Simona Vegetti, Nils Thuerey
We propose to solve inverse problems involving the temporal evolution of physics systems by leveraging recent advances from diffusion models. Our method moves the system's current state backward in time step by step by combining an approximate inverse physics simulator and a learned correction function. A central insight of our work is that training the learned correction with a single-step loss is equivalent to a score matching objective, while recursively predicting longer parts of the trajectory during training relates to maximum likelihood training of a corresponding probability flow. We highlight the advantages of our algorithm compared to standard denoising score matching and implicit score matching, as well as fully learned baselines for a wide range of inverse physics problems. The resulting inverse solver has excellent accuracy and temporal stability and, in contrast to other learned inverse solvers, allows for sampling the posterior of the solutions.
WeatherBench is a benchmark dataset for medium-range weather forecasting of geopotential, temperature and precipitation, consisting of preprocessed data, predefined evaluation metrics and a number of baseline models. WeatherBench Probability extends this to probabilistic forecasting by adding a set of established probabilistic verification metrics (continuous ranked probability score, spread-skill ratio and rank histograms) and a state-of-the-art operational baseline using the ECWMF IFS ensemble forecast. In addition, we test three different probabilistic machine learning methods -- Monte Carlo dropout, parametric prediction and categorical prediction, in which the probability distribution is discretized. We find that plain Monte Carlo dropout severely underestimates uncertainty. The parametric and categorical models both produce fairly reliable forecasts of similar quality. The parametric models have fewer degrees of freedom while the categorical model is more flexible when it comes to predicting non-Gaussian distributions. None of the models are able to match the skill of the operational IFS model. We hope that this benchmark will enable other researchers to evaluate their probabilistic approaches.
We propose a novel approach to generate temporally coherent UV coordinates for loose clothing. Our method is not constrained by human body outlines and can capture loose garments and hair. We implemented a differentiable pipeline to learn UV mapping between a sequence of RGB inputs and textures via UV coordinates. Instead of treating the UV coordinates of each frame separately, our data generation approach connects all UV coordinates via feature matching for temporal stability. Subsequently, a generative model is trained to balance the spatial quality and temporal stability. It is driven by supervised and unsupervised losses in both UV and image spaces. Our experiments show that the trained models output high-quality UV coordinates and generalize to new poses. Once a sequence of UV coordinates has been inferred by our model, it can be used to flexibly synthesize new looks and modified visual styles. Compared to existing methods, our approach reduces the computational workload to animate new outfits by several orders of magnitude.
The loss functions of many learning problems contain multiple additive terms that can disagree and yield conflicting update directions. For Physics-Informed Neural Networks (PINNs), loss terms on initial/boundary conditions and physics equations are particularly interesting as they are well-established as highly difficult tasks. To improve learning the challenging multi-objective task posed by PINNs, we propose the ConFIG method, which provides conflict-free updates by ensuring a positive dot product between the final update and each loss-specific gradient. It also maintains consistent optimization rates for all loss terms and dynamically adjusts gradient magnitudes based on conflict levels. We additionally leverage momentum to accelerate optimizations by alternating the back-propagation of different loss terms. We provide a mathematical proof showing the convergence of the ConFIG method, and it is evaluated across a range of challenging PINN scenarios. ConFIG consistently shows superior performance and runtime compared to baseline methods. We also test the proposed method in a classic multi-task benchmark, where the ConFIG method likewise exhibits a highly promising performance. Source code is available at https://tum-pbs.github.io/ConFIG
Aleksandra Franz, Barbara Solenthaler, Nils Thuerey
We address the challenging problem of jointly inferring the 3D flow and volumetric densities moving in a fluid from a monocular input video with a deep neural network. Despite the complexity of this task, we show that it is possible to train the corresponding networks without requiring any 3D ground truth for training. In the absence of ground truth data we can train our model with observations from real-world capture setups instead of relying on synthetic reconstructions. We make this unsupervised training approach possible by first generating an initial prototype volume which is then moved and transported over time without the need for volumetric supervision. Our approach relies purely on image-based losses, an adversarial discriminator network, and regularization. Our method can estimate long-term sequences in a stable manner, while achieving closely matching targets for inputs such as rising smoke plumes.
Lukas Prantl, Jan Bender, Tassilo Kugelstadt et al.
Generating highly detailed, complex data is a long-standing and frequently considered problem in the machine learning field. However, developing detail-aware generators remains an challenging and open problem. Generative adversarial networks are the basis of many state-of-the-art methods. However, they introduce a second network to be trained as a loss function, making the interpretation of the learned functions much more difficult. As an alternative, we present a new method based on a wavelet loss formulation, which remains transparent in terms of what is optimized. The wavelet-based loss function is used to overcome the limitations of conventional distance metrics, such as L1 or L2 distances, when it comes to generate data with high-frequency details. We show that our method can successfully reconstruct high-frequency details in an illustrative synthetic test case. Additionally, we evaluate the performance when applied to more complex surfaces based on physical simulations. Taking a roughly approximated simulation as input, our method infers corresponding spatial details while taking into account how they evolve. We consider this problem in terms of spatial and temporal frequencies, and leverage generative networks trained with our wavelet loss to learn the desired spatio-temporal signal for the surface dynamics. We test the capabilities of our method with a set of synthetic wave function tests and complex 2D and 3D dynamics of elasto-plastic materials.
We consider solving complex spatiotemporal dynamical systems governed by partial differential equations (PDEs) using frequency domain-based discrete learning approaches, such as Fourier neural operators. Despite their widespread use for approximating nonlinear PDEs, the majority of these methods neglect fundamental physical laws and lack interpretability. We address these shortcomings by introducing Physics-embedded Fourier Neural Networks (PeFNN) with flexible and explainable error control. PeFNN is designed to enforce momentum conservation and yields interpretable nonlinear expressions by utilizing unique multi-scale momentum-conserving Fourier (MC-Fourier) layers and an element-wise product operation. The MC-Fourier layer is by design translation- and rotation-invariant in the frequency domain, serving as a plug-and-play module that adheres to the laws of momentum conservation. PeFNN establishes a new state-of-the-art in solving widely employed spatiotemporal PDEs and generalizes well across input resolutions. Further, we demonstrate its outstanding performance for challenging real-world applications such as large-scale flood simulations.
Simulating turbulent flows is crucial for a wide range of applications, and machine learning-based solvers are gaining increasing relevance. However, achieving temporal stability when generalizing to longer rollout horizons remains a persistent challenge for learned PDE solvers. In this work, we analyze if fully data-driven fluid solvers that utilize an autoregressive rollout based on conditional diffusion models are a viable option to address this challenge. We investigate accuracy, posterior sampling, spectral behavior, and temporal stability, while requiring that methods generalize to flow parameters beyond the training regime. To quantitatively and qualitatively benchmark the performance of various flow prediction approaches, three challenging 2D scenarios including incompressible and transonic flows, as well as isotropic turbulence are employed. We find that even simple diffusion-based approaches can outperform multiple established flow prediction methods in terms of accuracy and temporal stability, while being on par with state-of-the-art stabilization techniques like unrolling at training time. Such traditional architectures are superior in terms of inference speed, however, the probabilistic nature of diffusion approaches allows for inferring multiple predictions that align with the statistics of the underlying physics. Overall, our benchmark contains three carefully chosen data sets that are suitable for probabilistic evaluation alongside various established flow prediction architectures.
Regional climate information at kilometer scales is essential for assessing the impacts of climate change, but generating it with global climate models is too expensive due to their high computational costs. Machine learning models offer a fast alternative, yet they often violate basic physical laws and degrade when applied to climates outside of their training distribution. We present Physics-Constrained Adaptive Flow Matching (PC-AFM), a generative downscaling model that addresses both problems. Building on the Adaptive Flow Matching (AFM) model of Fotiadis et al. (2025) as our baseline, we add soft conservation constraints that keep the downscaled output consistent with the large-scale input for precipitation and humidity, and use gradient surgery via the ConFIG algorithm to prevent these constraints from interfering with the generative objective. We train the model on Central Europe climate data, evaluate it on a 10-time downscaling task (63km to 6.3km) over six variables (near-surface temperature, precipitation, specific humidity, surface pressure, and horizontal wind components) across a comprehensive set of metrics including bias, ensemble skill scores, power spectra, and conservation error, and test the generalization on two held-out climate regions. Within the training distribution, PC-AFM reduces conservation errors and improves ensemble calibration while matching the baseline on standard skill metrics. Outside the training distribution, where unconstrained models develop large systematic errors by extrapolating learned statistics, PC-AFM halves precipitation wet bias, reduces conservation error and improves extreme-quantile accuracy, all without any information about the target climate at inference time. These results indicate that physical consistency is a practical requirement for deploying generative downscaling models in real-world applications.
Machine-learning surrogate models have shown promise in accelerating aerodynamic design, yet progress toward generalizable predictors for three-dimensional wings has been limited by the scarcity and restricted diversity of existing datasets. Here, we present SuperWing, a comprehensive open dataset of transonic swept-wing aerodynamics comprising 4,239 parameterized wing geometries and 28,856 Reynolds-averaged Navier-Stokes flow field solutions. The wing shapes in the dataset are generated using a simplified yet expressive geometry parameterization that incorporates spanwise variations in airfoil shape, twist, and dihedral, allowing for an enhanced diversity without relying on perturbations of a baseline wing. All shapes are simulated under a broad range of Mach numbers and angles of attack covering the typical flight envelope. To demonstrate the dataset's utility, we benchmark two state-of-the-art Transformers that accurately predict surface flow and achieve a 2.5 drag-count error on held-out samples. Models pretrained on SuperWing further exhibit strong zero-shot generalization to complex benchmark wings such as DLR-F6 and NASA CRM, underscoring the dataset's diversity and potential for practical usage.
Simulating complex dynamics like fluids with traditional simulators is computationally challenging. Deep learning models have been proposed as an efficient alternative, extending or replacing parts of traditional simulators. We investigate graph neural networks (GNNs) for learning fluid dynamics and find that their generalization capability is more limited than previous works would suggest. We also challenge the current practice of adding random noise to the network inputs in order to improve its generalization capability and simulation stability. We find that inserting the real data distribution, e.g. by unrolling multiple simulation steps, improves accuracy and that hiding all domain-specific features from the learning model improves generalization. Our results indicate that learning models, such as GNNs, fail to learn the exact underlying dynamics unless the training set is devoid of any other problem-specific correlations that could be used as shortcuts.
Recent works in deep learning have shown that integrating differentiable physics simulators into the training process can greatly improve the quality of results. Although this combination represents a more complex optimization task than supervised neural network training, the same gradient-based optimizers are typically employed to minimize the loss function. However, the integrated physics solvers have a profound effect on the gradient flow as manipulating scales in magnitude and direction is an inherent property of many physical processes. Consequently, the gradient flow is often highly unbalanced and creates an environment in which existing gradient-based optimizers perform poorly. In this work, we analyze the characteristics of both physical and neural network optimizations to derive a new method that does not suffer from this phenomenon. Our method is based on a half-inversion of the Jacobian and combines principles of both classical network and physics optimizers to solve the combined optimization task. Compared to state-of-the-art neural network optimizers, our method converges more quickly and yields better solutions, which we demonstrate on three complex learning problems involving nonlinear oscillators, the Schroedinger equation and the Poisson problem.
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.
We explore training deep neural network models in conjunction with physics simulations via partial differential equations (PDEs), using the simulated degrees of freedom as latent space for a neural network. In contrast to previous work, this paper treats the degrees of freedom of the simulated space purely as tools to be used by the neural network. We demonstrate this concept for learning reduced representations, as it is extremely challenging to faithfully preserve correct solutions over long time-spans with traditional reduced representations, particularly for solutions with large amounts of small scale features. This work focuses on the use of such physical, reduced latent space for the restoration of fine simulations, by training models that can modify the content of the reduced physical states as much as needed to best satisfy the learning objective. This autonomy allows the neural networks to discover alternate dynamics that significantly improve the performance in the given tasks. We demonstrate this concept for various fluid flows ranging from different turbulence scenarios to rising smoke plumes.
Finding model parameters from data is an essential task in science and engineering, from weather and climate forecasts to plasma control. Previous works have employed neural networks to greatly accelerate finding solutions to inverse problems. Of particular interest are end-to-end models which utilize differentiable simulations in order to backpropagate feedback from the simulated process to the network weights and enable roll-out of multiple time steps. So far, it has been assumed that, while model inference is faster than classical optimization, this comes at the cost of a decrease in solution accuracy. We show that this is generally not true. In fact, neural networks trained to learn solutions to inverse problems can find better solutions than classical optimizers even on their training set. To demonstrate this, we perform both a theoretical analysis as well an extensive empirical evaluation on challenging problems involving local minima, chaos, and zero-gradient regions. Our findings suggest an alternative use for neural networks: rather than generalizing to new data for fast inference, they can also be used to find better solutions on known data.
Of all the vector fields surrounding the minima of recurrent learning setups, the gradient field with its exploding and vanishing updates appears a poor choice for optimization, offering little beyond efficient computability. We seek to improve this suboptimal practice in the context of physics simulations, where backpropagating feedback through many unrolled time steps is considered crucial to acquiring temporally coherent behavior. The alternative vector field we propose follows from two principles: physics simulators, unlike neural networks, have a balanced gradient flow, and certain modifications to the backpropagation pass leave the positions of the original minima unchanged. As any modification of backpropagation decouples forward and backward pass, the rotation-free character of the gradient field is lost. Therefore, we discuss the negative implications of using such a rotational vector field for optimization and how to counteract them. Our final procedure is easily implementable via a sequence of gradient stopping and component-wise comparison operations, which do not negatively affect scalability. Our experiments on three control problems show that especially as we increase the complexity of each task, the unbalanced updates from the gradient can no longer provide the precise control signals necessary while our method still solves the tasks. Our code can be found at https://github.com/tum-pbs/StableBPTT.
We investigate the use of deep neural networks to control complex nonlinear dynamical systems, specifically the movement of a rigid body immersed in a fluid. We solve the Navier Stokes equations with two way coupling, which gives rise to nonlinear perturbations that make the control task very challenging. Neural networks are trained in an unsupervised way to act as controllers with desired characteristics through a process of learning from a differentiable simulator. Here we introduce a set of physically interpretable loss terms to let the networks learn robust and stable interactions. We demonstrate that controllers trained in a canonical setting with quiescent initial conditions reliably generalize to varied and challenging environments such as previously unseen inflow conditions and forcing, although they do not have any fluid information as input. Further, we show that controllers trained with our approach outperform a variety of classical and learned alternatives in terms of evaluation metrics and generalization capabilities.
Despite decades of advancements, the simulation of fluids remains one of the most challenging areas of in scientific computing. Supported by the necessity of gradient information in deep learning, differentiable simulators have emerged as an effective tool for optimization and learning in physics simulations. In this work, we present our fluid simulator PICT, a differentiable pressure-implicit solver coded in PyTorch with Graphics-processing-unit (GPU) support. We first verify the accuracy of both the forward simulation and our derived gradients in various established benchmarks like lid-driven cavities and turbulent channel flows before we show that the gradients provided by our solver can be used to learn complicated turbulence models in 2D and 3D. We apply both supervised and unsupervised training regimes using physical priors to match flow statistics. In particular, we learn a stable sub-grid scale (SGS) model for a 3D turbulent channel flow purely based on reference statistics. The low-resolution corrector trained with our solver runs substantially faster than the highly resolved references, while keeping or even surpassing their accuracy. Finally, we give additional insights into the physical interpretation of different solver gradients, and motivate a physically informed regularization technique. To ensure that the full potential of PICT can be leveraged, it is published as open source: https://github.com/tum-pbs/PICT.
When simulating partial differential equations, hybrid solvers combine coarse numerical solvers with learned correctors. They promise accelerated simulations while adhering to physical constraints. However, as shown in our theoretical framework, directly applying learned corrections to solver outputs leads to significant autoregressive errors, which originate from amplified perturbations that accumulate during long-term rollouts, especially in chaotic regimes. To overcome this, we propose the Indirect Neural Corrector ($\mathrm{INC}$), which integrates learned corrections into the governing equations rather than applying direct state updates. Our key insight is that $\mathrm{INC}$ reduces the error amplification on the order of $Δt^{-1} + L$, where $Δt$ is the timestep and $L$ the Lipschitz constant. At the same time, our framework poses no architectural requirements and integrates seamlessly with arbitrary neural networks and solvers. We test $\mathrm{INC}$ in extensive benchmarks, covering numerous differentiable solvers, neural backbones, and test cases ranging from a 1D chaotic system to 3D turbulence. $\mathrm{INC}$ improves the long-term trajectory performance ($R^2$) by up to 158.7%, stabilizes blowups under aggressive coarsening, and for complex 3D turbulence cases yields speed-ups of several orders of magnitude. $\mathrm{INC}$ thus enables stable, efficient PDE emulation with formal error reduction, paving the way for faster scientific and engineering simulations with reliable physics guarantees. Our source code is available at https://github.com/tum-pbs/INC
Leveraging neural networks as surrogate models for turbulence simulation is a topic of growing interest. At the same time, embodying the inherent uncertainty of simulations in the predictions of surrogate models remains very challenging. The present study makes a first attempt to use denoising diffusion probabilistic models (DDPMs) to train an uncertainty-aware surrogate model for turbulence simulations. Due to its prevalence, the simulation of flows around airfoils with various shapes, Reynolds numbers, and angles of attack is chosen as the learning objective. Our results show that DDPMs can successfully capture the whole distribution of solutions and, as a consequence, accurately estimate the uncertainty of the simulations. The performance of DDPMs is also compared with varying baselines in the form of Bayesian neural networks and heteroscedastic models. Experiments demonstrate that DDPMs outperform the other methods regarding a variety of accuracy metrics. Besides, it offers the advantage of providing access to the complete distributions of uncertainties rather than providing a set of parameters. As such, it can yield realistic and detailed samples from the distribution of solutions. We also evaluate an emerging generative modeling variant, flow matching, in comparison to regular diffusion models. The results demonstrate that flow matching addresses the problem of slow sampling speed typically associated with diffusion models. As such, it offers a promising new paradigm for uncertainty quantification with generative models.
Physical systems with complex unsteady dynamics, such as fluid flows, are often poorly represented by a single mean solution. For many practical applications, it is crucial to access the full distribution of possible states, from which relevant statistics (e.g., RMS and two-point correlations) can be derived. Here, we propose a graph-based latent diffusion (or alternatively, flow-matching) model that enables direct sampling of states from their equilibrium distribution, given a mesh discretization of the system and its physical parameters. This allows for the efficient computation of flow statistics without running long and expensive numerical simulations. The graph-based structure enables operations on unstructured meshes, which is critical for representing complex geometries with spatially localized high gradients, while latent-space diffusion modeling with a multi-scale GNN allows for efficient learning and inference of entire distributions of solutions. A key finding is that the proposed networks can accurately learn full distributions even when trained on incomplete data from relatively short simulations. We apply this method to a range of fluid dynamics tasks, such as predicting pressure distributions on 3D wing models in turbulent flow, demonstrating both accuracy and computational efficiency in challenging scenarios. The ability to directly sample accurate solutions, and capturing their diversity from short ground-truth simulations, is highly promising for complex scientific modeling tasks.
Felix Koehler, Simon Niedermayr, Rüdiger Westermann et al.
We introduce the Autoregressive PDE Emulator Benchmark (APEBench), a comprehensive benchmark suite to evaluate autoregressive neural emulators for solving partial differential equations. APEBench is based on JAX and provides a seamlessly integrated differentiable simulation framework employing efficient pseudo-spectral methods, enabling 46 distinct PDEs across 1D, 2D, and 3D. Facilitating systematic analysis and comparison of learned emulators, we propose a novel taxonomy for unrolled training and introduce a unique identifier for PDE dynamics that directly relates to the stability criteria of classical numerical methods. APEBench enables the evaluation of diverse neural architectures, and unlike existing benchmarks, its tight integration of the solver enables support for differentiable physics training and neural-hybrid emulators. Moreover, APEBench emphasizes rollout metrics to understand temporal generalization, providing insights into the long-term behavior of emulating PDE dynamics. In several experiments, we highlight the similarities between neural emulators and numerical simulators.
Unrolling training trajectories over time strongly influences the inference accuracy of neural network-augmented physics simulators. We analyze this in three variants of training neural time-steppers. In addition to one-step setups and fully differentiable unrolling, we include a third, less widely used variant: unrolling without temporal gradients. Comparing networks trained with these three modalities disentangles the two dominant effects of unrolling, training distribution shift and long-term gradients. We present detailed study across physical systems, network sizes and architectures, training setups, and test scenarios. It also encompasses two simulation modes: In prediction setups, we rely solely on neural networks to compute a trajectory. In contrast, correction setups include a numerical solver that is supported by a neural network. Spanning these variations, our study provides the empirical basis for our main findings: Non-differentiable but unrolled training with a numerical solver in a correction setup can yield substantial improvements over a fully differentiable prediction setup not utilizing this solver. The accuracy of models trained in a fully differentiable setup differs compared to their non-differentiable counterparts. Differentiable ones perform best in a comparison among correction networks as well as among prediction setups. For both, the accuracy of non-differentiable unrolling comes close. Furthermore, we show that these behaviors are invariant to the physical system, the network architecture and size, and the numerical scheme. These results motivate integrating non-differentiable numerical simulators into training setups even if full differentiability is unavailable. We show the convergence rate of common architectures to be low compared to numerical algorithms. This motivates correction setups combining neural and numerical parts which utilize benefits of both.
Giacomo Baldan, Qiang Liu, Alberto Guardone et al.
Generative machine learning methods, such as diffusion models and flow matching, have shown great potential in modeling complex system behaviors and building efficient surrogate models. However, these methods typically learn the underlying physics implicitly from data. We propose Physics-Based Flow Matching (PBFM), a novel generative framework that explicitly embeds physical constraints, both PDE residuals and algebraic relations, into the flow matching objective. We also introduce temporal unrolling at training time that improves the accuracy of the final, noise-free sample prediction. Our method jointly minimizes the flow matching loss and the physics-based residual loss without requiring hyperparameter tuning of their relative weights. Additionally, we analyze the role of the minimum noise level, $σ_{\min}$, in the context of physical constraints and evaluate a stochastic sampling strategy that helps to reduce physical residuals. Through extensive benchmarks on three representative PDE problems, we show that our approach yields up to an $8\times$ more accurate physical residuals compared to FM, while clearly outperforming existing algorithms in terms of distributional accuracy. PBFM thus provides a principled and efficient framework for surrogate modeling, uncertainty quantification, and accelerated simulation in physics and engineering applications.
We introduce PDE-Transformer, an improved transformer-based architecture for surrogate modeling of physics simulations on regular grids. We combine recent architectural improvements of diffusion transformers with adjustments specific for large-scale simulations to yield a more scalable and versatile general-purpose transformer architecture, which can be used as the backbone for building large-scale foundation models in physical sciences. We demonstrate that our proposed architecture outperforms state-of-the-art transformer architectures for computer vision on a large dataset of 16 different types of PDEs. We propose to embed different physical channels individually as spatio-temporal tokens, which interact via channel-wise self-attention. This helps to maintain a consistent information density of tokens when learning multiple types of PDEs simultaneously. We demonstrate that our pre-trained models achieve improved performance on several challenging downstream tasks compared to training from scratch and also beat other foundation model architectures for physics simulations.
Benjamin Holzschuh, Georg Kohl, Florian Redinger et al.
We present a scalable framework for learning deterministic and probabilistic neural surrogates for high-resolution 3D physics simulations. We introduce a hybrid CNN-Transformer backbone architecture targeted for 3D physics simulations, which significantly outperforms existing architectures in terms of speed and accuracy. Our proposed network can be pretrained on small patches of the simulation domain, which can be fused to obtain a global solution, optionally guided via a fast and scalable sequence-to-sequence model to include long-range dependencies. This setup allows for training large-scale models with reduced memory and compute requirements for high-resolution datasets. We evaluate our backbone architecture against a large set of baseline methods with the objective to simultaneously learn the dynamics of 14 different types of PDEs in 3D. We demonstrate how to scale our model to high-resolution isotropic turbulence with spatial resolutions of up to $512^3$. Finally, we demonstrate the versatility of our network by training it as a diffusion model to produce probabilistic samples of highly turbulent 3D channel flows across varying Reynolds numbers, accurately capturing the underlying flow statistics.
The physics solvers employed for neural network training are primarily iterative, and hence, differentiating through them introduces a severe computational burden as iterations grow large. Inspired by works in bilevel optimization, we show that full accuracy of the network is achievable through physics significantly coarser than fully converged solvers. We propose Progressively Refined Differentiable Physics (PRDP), an approach that identifies the level of physics refinement sufficient for full training accuracy. By beginning with coarse physics, adaptively refining it during training, and stopping refinement at the level adequate for training, it enables significant compute savings without sacrificing network accuracy. Our focus is on differentiating iterative linear solvers for sparsely discretized differential operators, which are fundamental to scientific computing. PRDP is applicable to both unrolled and implicit differentiation. We validate its performance on a variety of learning scenarios involving differentiable physics solvers such as inverse problems, autoregressive neural emulators, and correction-based neural-hybrid solvers. In the challenging example of emulating the Navier-Stokes equations, we reduce training time by 62%.
Marc Amorós-Trepat, Luis Medrano-Navarro, Qiang Liu et al.
The reconstruction of unsteady flow fields from limited measurements is a challenging and crucial task for many engineering applications. Machine learning models are gaining popularity in solving this problem due to their ability to learn complex patterns from data and generalize across diverse conditions. Among these, diffusion models have emerged as particularly powerful in generative tasks, producing high-quality samples by iteratively refining noisy inputs. In contrast to other methods, these generative models are capable of reconstructing the smallest scales of the fluid spectrum. In this work, we introduce a novel sampling method for diffusion models that enables the reconstruction of high-fidelity samples by guiding the reverse process using the available sparse data. Moreover, we enhance the reconstructions with available physics knowledge using a conflict-free update method during training. To evaluate the effectiveness of our method, we conduct experiments on 2 and 3-dimensional turbulent flow data. Our method consistently outperforms other diffusion-based methods in predicting the fluid's structure and in pixel-wise accuracy. This study underscores the remarkable potential of diffusion models in reconstructing flow field data, paving the way for their application in Computational Fluid Dynamics research.
Accurate and efficient prediction of indoor airflow and temperature distributions is essential for building energy optimization and occupant comfort control. However, traditional CFD simulations are computationally intensive, limiting their integration into real-time or design-iterative workflows. This study proposes a component-based machine learning (CBML) surrogate modeling approach to replace conventional CFD simulation for fast prediction of indoor velocity and temperature fields. The model consists of three neural networks: a convolutional autoencoder with residual connections (CAER) to extract and compress flow features, a multilayer perceptron (MLP) to map inlet velocities to latent representations, and a convolutional neural network (CNN) as an aggregator to combine single-inlet features into dual-inlet scenarios. A two-dimensional room with varying left and right air inlet velocities is used as a benchmark case, with CFD simulations providing training and testing data. Results show that the CBML model accurately and fast predicts two-component aggregated velocity and temperature fields across both training and testing datasets.
Solving inverse problems in physics is central to understanding complex systems and advancing technologies in various fields. Iterative optimization algorithms, commonly used to solve these problems, often encounter local minima, chaos, or regions with zero gradients. This is due to their overreliance on local information and highly chaotic inverse loss landscapes governed by underlying partial differential equations (PDEs). In this work, we show that deep neural networks successfully replicate such complex loss landscapes through spatio-temporal trajectory inputs. They also offer the potential to control the underlying complexity of these chaotic loss landscapes during training through various regularization methods. We show that optimizing on network-smoothened loss landscapes leads to improved convergence in predicting optimum inverse parameters over conventional momentum-based optimizers such as BFGS on multiple challenging problems.
We introduce a single-view reconstruction technique of volumetric fields in which multiple light scattering effects are omnipresent, such as in clouds. We model the unknown distribution of volumetric fields using an unconditional diffusion model trained on a novel benchmark dataset comprising 1,000 synthetically simulated volumetric density fields. The neural diffusion model is trained on the latent codes of a novel, diffusion-friendly, monoplanar representation. The generative model is used to incorporate a tailored parametric diffusion posterior sampling technique into different reconstruction tasks. A physically-based differentiable volume renderer is employed to provide gradients with respect to light transport in the latent space. This stands in contrast to classic NeRF approaches and makes the reconstructions better aligned with observed data. Through various experiments, we demonstrate single-view reconstruction of volumetric clouds at a previously unattainable quality.
Flow-based generative modeling is a powerful tool for solving inverse problems in physical sciences that can be used for sampling and likelihood evaluation with much lower inference times than traditional methods. We propose to refine flows with additional control signals based on a simulator. Control signals can include gradients and a problem-specific cost function if the simulator is differentiable, or they can be fully learned from the simulator output. In our proposed method, we pretrain the flow network and include feedback from the simulator exclusively for finetuning, therefore requiring only a small amount of additional parameters and compute. We motivate our design choices on several benchmark problems for simulation-based inference and evaluate flow matching with simulator feedback against classical MCMC methods for modeling strong gravitational lens systems, a challenging inverse problem in astronomy. We demonstrate that including feedback from the simulator improves the accuracy by $53\%$, making it competitive with traditional techniques while being up to $67$x faster for inference.
In this paper, we train turbulence models based on convolutional neural networks. These learned turbulence models improve under-resolved low resolution solutions to the incompressible Navier-Stokes equations at simulation time. Our study involves the development of a differentiable numerical solver that supports the propagation of optimisation gradients through multiple solver steps. The significance of this property is demonstrated by the superior stability and accuracy of those models that unroll more solver steps during training. Furthermore, we introduce loss terms based on turbulence physics that further improve the model accuracy. This approach is applied to three two-dimensional turbulence flow scenarios, a homogeneous decaying turbulence case, a temporally evolving mixing layer, and a spatially evolving mixing layer. Our models achieve significant improvements of long-term a-posteriori statistics when compared to no-model simulations, without requiring these statistics to be directly included in the learning targets. At inference time, our proposed method also gains substantial performance improvements over similarly accurate, purely numerical methods.
Simulations that produce three-dimensional data are ubiquitous in science, ranging from fluid flows to plasma physics. We propose a similarity model based on entropy, which allows for the creation of physically meaningful ground truth distances for the similarity assessment of scalar and vectorial data, produced from transport and motion-based simulations. Utilizing two data acquisition methods derived from this model, we create collections of fields from numerical PDE solvers and existing simulation data repositories. Furthermore, a multiscale CNN architecture that computes a volumetric similarity metric (VolSiM) is proposed. To the best of our knowledge this is the first learning method inherently designed to address the challenges arising for the similarity assessment of high-dimensional simulation data. Additionally, the tradeoff between a large batch size and an accurate correlation computation for correlation-based loss functions is investigated, and the metric's invariance with respect to rotation and scale operations is analyzed. Finally, the robustness and generalization of VolSiM is evaluated on a large range of test data, as well as a particularly challenging turbulence case study, that is close to potential real-world applications.
Solving inverse problems, such as parameter estimation and optimal control, is a vital part of science. Many experiments repeatedly collect data and rely on machine learning algorithms to quickly infer solutions to the associated inverse problems. We find that state-of-the-art training techniques are not well-suited to many problems that involve physical processes. The highly nonlinear behavior, common in physical processes, results in strongly varying gradients that lead first-order optimizers like SGD or Adam to compute suboptimal optimization directions. We propose a novel hybrid training approach that combines higher-order optimization methods with machine learning techniques. We take updates from a scale-invariant inverse problem solver and embed them into the gradient-descent-based learning pipeline, replacing the regular gradient of the physical process. We demonstrate the capabilities of our method on a variety of canonical physical systems, showing that it yields significant improvements on a wide range of optimization and learning problems.
This document is a hands-on, comprehensive guide to deep learning in the realm of physical simulations. Rather than just theory, we emphasize practical application: every concept is paired with interactive Jupyter notebooks to get you up and running quickly. Beyond traditional supervised learning, we dive into physical loss-constraints, differentiable simulations, diffusion-based approaches for probabilistic generative AI, as well as reinforcement learning and advanced neural network architectures. These foundations are paving the way for the next generation of scientific foundation models. We are living in an era of rapid transformation. These methods have the potential to redefine what's possible in computational science.
The present study investigates the accurate inference of Reynolds-averaged Navier-Stokes solutions for the compressible flow over aerofoils in two dimensions with a deep neural network. Our approach yields networks that learn to generate precise flow fields for varying body-fitted, structured grids by providing them with an encoding of the corresponding mapping to a canonical space for the solutions. We apply the deep neural network model to a benchmark case of incompressible flow at randomly given angles of attack and Reynolds numbers and achieve an improvement of more than an order of magnitude compared to previous work. Further, for transonic flow cases, the deep neural network model accurately predicts complex flow behaviour at high Reynolds numbers, such as shock wave/boundary layer interaction, and quantitative distributions like pressure coefficient, skin friction coefficient as well as wake total pressure profiles downstream of aerofoils. The proposed deep learning method significantly speeds up the predictions of flow fields and shows promise for enabling fast aerodynamic designs.
Igor Santesteban, Nils Thuerey, Miguel A. Otaduy et al.
We propose a new generative model for 3D garment deformations that enables us to learn, for the first time, a data-driven method for virtual try-on that effectively addresses garment-body collisions. In contrast to existing methods that require an undesirable postprocessing step to fix garment-body interpenetrations at test time, our approach directly outputs 3D garment configurations that do not collide with the underlying body. Key to our success is a new canonical space for garments that removes pose-and-shape deformations already captured by a new diffused human body model, which extrapolates body surface properties such as skinning weights and blendshapes to any 3D point. We leverage this representation to train a generative model with a novel self-supervised collision term that learns to reliably solve garment-body interpenetrations. We extensively evaluate and compare our results with recently proposed data-driven methods, and show that our method is the first to successfully address garment-body contact in unseen body shapes and motions, without compromising realism and detail.
Aleksandra Franz, Barbara Solenthaler, Nils Thuerey
We propose a novel method to reconstruct volumetric flows from sparse views via a global transport formulation. Instead of obtaining the space-time function of the observations, we reconstruct its motion based on a single initial state. In addition we introduce a learned self-supervision that constrains observations from unseen angles. These visual constraints are coupled via the transport constraints and a differentiable rendering step to arrive at a robust end-to-end reconstruction algorithm. This makes the reconstruction of highly realistic flow motions possible, even from only a single input view. We show with a variety of synthetic and real flows that the proposed global reconstruction of the transport process yields an improved reconstruction of the fluid motion.
Recent implicit neural rendering methods have demonstrated that it is possible to learn accurate view synthesis for complex scenes by predicting their volumetric density and color supervised solely by a set of RGB images. However, existing methods are restricted to learning efficient representations of static scenes that encode all scene objects into a single neural network, and lack the ability to represent dynamic scenes and decompositions into individual scene objects. In this work, we present the first neural rendering method that decomposes dynamic scenes into scene graphs. We propose a learned scene graph representation, which encodes object transformation and radiance, to efficiently render novel arrangements and views of the scene. To this end, we learn implicitly encoded scenes, combined with a jointly learned latent representation to describe objects with a single implicit function. We assess the proposed method on synthetic and real automotive data, validating that our approach learns dynamic scenes -- only by observing a video of this scene -- and allows for rendering novel photo-realistic views of novel scene compositions with unseen sets of objects at unseen poses.
We propose a novel training approach for improving the generalization in neural networks. We show that in contrast to regular constraints for orthogonality, our approach represents a {\em data-dependent} orthogonality constraint, and is closely related to singular value decompositions of the weight matrices. We also show how our formulation is easy to realize in practical network architectures via a reverse pass, which aims for reconstructing the full sequence of internal states of the network. Despite being a surprisingly simple change, we demonstrate that this forward-backward training approach, which we refer to as {\em racecar} training, leads to significantly more generic features being extracted from a given data set. Networks trained with our approach show more balanced mutual information between input and output throughout all layers, yield improved explainability and, exhibit improved performance for a variety of tasks and task transfers.
Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting for effects not captured by the discretized PDE. We target the problem of reducing numerical errors of iterative PDE solvers and compare different learning approaches for finding complex correction functions. We find that previously used learning approaches are significantly outperformed by methods that integrate the solver into the training loop and thereby allow the model to interact with the PDE during training. This provides the model with realistic input distributions that take previous corrections into account, yielding improvements in accuracy with stable rollouts of several hundred recurrent evaluation steps and surpassing even tailored supervised variants. We highlight the performance of the differentiable physics networks for a wide variety of PDEs, from non-linear advection-diffusion systems to three-dimensional Navier-Stokes flows.
With several advantages and as an alternative to predict physics field, machine learning methods can be classified into two distinct types: data-driven relying on training data and physics-driven using physics law. Choosing heat conduction problem as an example, we compared the data- and physics-driven learning process with deep Convolutional Neural Networks (CNN). It shows that the convergences of the error to ground truth solution and the residual of heat conduction equation exhibit remarkable differences. Based on this observation, we propose a combined-driven method for learning acceleration and more accurate solutions. With a weighted loss function, reference data and physical equation are able to simultaneously drive the learning. Several numerical experiments are conducted to investigate the effectiveness of the combined method. For the data-driven based method, the introduction of physical equation not only is able to speed up the convergence, but also produces physically more consistent solutions. For the physics-driven based method, it is observed that the combined method is able to speed up the convergence up to 49.0\% by using a not very restrictive coarse reference.
Steffen Wiewel, Byungsoo Kim, Vinicius C. Azevedo et al.
We propose an end-to-end trained neural networkarchitecture to robustly predict the complex dynamics of fluid flows with high temporal stability. We focus on single-phase smoke simulations in 2D and 3D based on the incompressible Navier-Stokes (NS) equations, which are relevant for a wide range of practical problems. To achieve stable predictions for long-term flow sequences, a convolutional neural network (CNN) is trained for spatial compression in combination with a temporal prediction network that consists of stacked Long Short-Term Memory (LSTM) layers. Our core contribution is a novel latent space subdivision (LSS) to separate the respective input quantities into individual parts of the encoded latent space domain. This allows to distinctively alter the encoded quantities without interfering with the remaining latent space values and hence maximizes external control. By selectively overwriting parts of the predicted latent space points, our proposed method is capable to robustly predict long-term sequences of complex physics problems. In addition, we highlight the benefits of a recurrent training on the latent space creation, which is performed by the spatial compression network.
We propose a neural network-based approach that computes a stable and generalizing metric (LSiM) to compare data from a variety of numerical simulation sources. We focus on scalar time-dependent 2D data that commonly arises from motion and transport-based partial differential equations (PDEs). Our method employs a Siamese network architecture that is motivated by the mathematical properties of a metric. We leverage a controllable data generation setup with PDE solvers to create increasingly different outputs from a reference simulation in a controlled environment. A central component of our learned metric is a specialized loss function that introduces knowledge about the correlation between single data samples into the training process. To demonstrate that the proposed approach outperforms existing metrics for vector spaces and other learned, image-based metrics, we evaluate the different methods on a large range of test data. Additionally, we analyze generalization benefits of an adjustable training data difficulty and demonstrate the robustness of LSiM via an evaluation on three real-world data sets.
Predicting outcomes and planning interactions with the physical world are long-standing goals for machine learning. A variety of such tasks involves continuous physical systems, which can be described by partial differential equations (PDEs) with many degrees of freedom. Existing methods that aim to control the dynamics of such systems are typically limited to relatively short time frames or a small number of interaction parameters. We present a novel hierarchical predictor-corrector scheme which enables neural networks to learn to understand and control complex nonlinear physical systems over long time frames. We propose to split the problem into two distinct tasks: planning and control. To this end, we introduce a predictor network that plans optimal trajectories and a control network that infers the corresponding control parameters. Both stages are trained end-to-end using a differentiable PDE solver. We demonstrate that our method successfully develops an understanding of complex physical systems and learns to control them for tasks involving PDEs such as the incompressible Navier-Stokes equations.
Lukas Prantl, Nuttapong Chentanez, Stefan Jeschke et al.
Point clouds, as a form of Lagrangian representation, allow for powerful and flexible applications in a large number of computational disciplines. We propose a novel deep-learning method to learn stable and temporally coherent feature spaces for points clouds that change over time. We identify a set of inherent problems with these approaches: without knowledge of the time dimension, the inferred solutions can exhibit strong flickering, and easy solutions to suppress this flickering can result in undesirable local minima that manifest themselves as halo structures. We propose a novel temporal loss function that takes into account higher time derivatives of the point positions, and encourages mingling, i.e., to prevent the aforementioned halos. We combine these techniques in a super-resolution method with a truncation approach to flexibly adapt the size of the generated positions. We show that our method works for large, deforming point sets from different sources to demonstrate the flexibility of our approach.