Complete Technical Paper Content — Ai In Physics
1. Executive Abstract, Problem Formulation & Background
Physics-Based Deep Learning The following collection of materials targets "Physics-Based Deep Learning" (PBDL), i.e., the field of methods with combinations of physical modeling and deep learning (DL) techniques. Here, DL will typically refer to methods based on artificial neural networks. The general direction of PBDL represents a very active and quickly growing field of research. If you're interested in a comprehensive overview, please check our digital PBDL book: https://www.physicsbaseddeeplearning.org/ (or as PDF: https://arxiv.org/pdf/2109.05237.pdf ) Within this area, we can distinguish a variety of different physics-based approaches, from targeting designs, constraints, combined methods, and optimizations to applications. More specifically, all approaches either target forward simulations (predicting state or temporal evolution) or inverse problems (e.g., obtaining a parametrization for a physical system from observations). Apart from forward or inverse, the type of integration between learning and physics gives a means for categorizing different methods: Data-driven : the data is produced by a physical system (real or simulated), but no further interaction exists. Loss-terms : the physical dynamics (or parts thereof) are encoded in the loss function, typically in the form of differentiable operations. The learning process can repeatedly evaluate the loss, and usually receives gradients from a PDE-based formulation. Interleaved : the full physical simulation is interleaved and combined with an output from a deep neural network; this requires a fully differentiable simulator and represents the tightest coupling between the physical system and the learning process. Interleaved approaches are especially important for temporal evolutions, where they can yield an estimate of future behavior of the dynamics. Thus, methods can be roughly categorized in terms of forward versus inverse solve, and how tightly the physical model is integrated into the optimization loop that trains the deep neural network. Here, especially approaches that leverage differentiable physics allow for very tight integration of deep learning and numerical simulations. This repository collects links to works on deep learning algorithms for physics problems , with a particular emphasis on fluid flow , i.e., Navier-Stokes related problems. It primarily collects links to the work of the I15 lab at TUM, as well as miscellaneous works from other groups. This is by no means a complete list, so let us know if you come across additional papers in this area. We intentionally also focus on works from the deep learning field, not machine learning in general. I15 Physics-based Deep Learning Links diffSPH: Differentiable Smoothed Particle Hydrodynamics for Adjoint Optimization and Machine Learning , Project: https://github.com/tum-pbs/diffSPH PDE-Transformer: Efficient and Versatile Transformers for Physics Simulations , Project: https://tum-pbs.github.io/pde-transformer/landing.html Flow Matching Meets PDEs – A Unified Framework for Physics-Constrained Generation , PDF: https://arxiv.org/abs/2506.08604 Component-Based Machine Learning for Indoor Flow and Temperature Fields Prediction Latent Feature Aggregation and Flow Interaction , PDF: https://arxiv.org/abs/2507.19233 PICT - A Differentiable, GPU-Accelerated Multi-Block PISO Solver for Simulation-Coupled Learning Tasks in Fluid Dynamics , Project: https://github.com/tum-pbs/PICT Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks , Project: https://github.com/tum-pbs/dgn4cfd Temporal Difference Learning: Why It Can Be Fast and How It Will Be Faster , PDF: https://openreview.net/forum?id=j3bKnEidtT Truncation Is All You Need: Improved Sampling Of Diffusion Models For Physics-Based Simulations , PDF: https://openreview.net/forum?id=0FbzC7B9xI PRDP: Progressively Refined Differentiable Physics , Project: https://github.com/tum-pbs/PRDP Temporal Difference Learning: Why It Can Be Fast and How It Will Be Faster , PDF: https://openreview.net/forum?id=j3bKnEidtT Flow Matching for Posterior Inference with Simulator Feedback , PDF: https://arxiv.org/pdf/2410.22573 APEBench: A Benchmark for Autoregressive Neural Emulators of PDEs , Project: https://github.com/tum-pbs/apebench-paper Deep learning-based predictive modelling of transonic flow over an aerofoil , PDF: https://arxiv.org/pdf/2403.17131 ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks , Project: https://tum-pbs.github.io/ConFIG/ The Unreasonable Effectiveness of Solving Inverse Problems with Neural Networks , PDF: http://arxiv.org/pdf/2408.08119 Phiflow: Differentiable Simulations for PyTorch, TensorFlow and Jax , PDF: https://openreview.net/pdf/36503358a4f388f00d587a0257c13ba2a4656098.pdf How Temporal Unrolling Supports Neural Physics Simulators , PDF: https://arxiv.org/pdf/2402.12971 Stabilizing Backpropagation Through Time to Learn Complex Physics , PDF: https://openreview.net/forum?id=bozbTTWcaw Symmetric Basis Convolutions for Learning Lagrangian Fluid Mechanics , PDF: https://openreview.net/forum?id=HKgRwNhI9R Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models , Project: https://github.com/tum-pbs/Diffusion-based-Flow-Prediction Turbulent Flow Simulation using Autoregressive Conditional Diffusion Models , Project: https://github.com/tum-pbs/autoreg-pde-diffusion Physics-Preserving AI-Accelerated Simulations of Plasma Turbulence , PDF: https://arxiv.org/pdf/2309.16400 > Unsteady Cylinder Wakes from Arbitrary Bodies with Differentiable Physics-Assisted Neural Network , Project: https://github.com/tum-pbs/DiffPhys-CylinderWakeFlow Score Matching via Differentiable Physics , Project: https://github.com/tum-pbs/SMDP Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics , Project: https://github.com/tum-pbs/DMCF Learned Turbulence Modelling with Differentiable Fluid Solvers , Project: https://github.com/tum-pbs/differentiable-piso Half-Inverse Gradients for Physical Deep Learning , Project: https://github.com/tum-pbs/half-inverse-gradients Reviving Autoencoder Pretraining (Previously: Data-driven Regularization via Racecar Training for Generalizing Neural Networks), Project: https://github.com/tum-pbs/racecar Realistic galaxy images and improved robustness in machine learning tasks from generative modelling , PDF: https://arxiv.org/pdf/2203.11956 Hybrid Neural Network PDE Solvers for Reacting Flows , Project: https://github.com/tum-pbs/Hybrid-Solver-for-Reactive-Flows Scale-invariant Learning by Physics Inversion (formerly "Physical Gradients") , Project: https://github.com/tum-pbs/SIP High-accuracy transonic RANS Flow Predictions with Deep Neural Networks , Project: https://github.com/tum-pbs/coord-trans-encoding Learning Meaningful Controls for Fluids , Project: https://rachelcmy.github.io/den2vel/ Global Transport for Fluid Reconstruction with Learned Self-Supervision , Project: https://ge.in.tum.de/publications/2021-franz-globtrans Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers , Project: https://github.com/tum-pbs/Solver-in-the-Loop Numerical investigation of minimum drag profiles in laminar flow using deep learning surrogates , PDF: https://arxiv.org/pdf/2009.14339 Purely data-driven medium-range weather forecasting achieves comparable skill to physical models at similar resolution , PDF: https://arxiv.org/pdf/2008.08626 Latent Space Subdivision: Stable and Controllable Time Predictions for Fluid Flow , Project: https://ge.in.tum.de/publications/2020-lssubdiv-wiewel WeatherBench: A benchmark dataset for data-driven weather forecasting , Project: https://github.com/pangeo-data/WeatherBench Learning Similarity Metrics for Numerical Simulations (LSiM) , Project: https://ge.in.tum.de/publications/2020-lsim-kohl Learning to Control PDEs with Differentiable Physics , Project: https://ge.in.tum.de/publications/2020-iclr-holl Lagrangian Fluid Simulation with Continuous Convolutions , PDF: https://openreview.net/forum?id=B1lDoJSYDH Tranquil-Clouds: Neural Networks for Learning Temporally Coherent Features in Point Clouds , Project: https://ge.in.tum.de/publications/2020-iclr-prantl/ ScalarFlow: A Large-Scale Volumetric Data Set of Real-world Scalar Transport Flows for Computer Animation and Machine Learning , Project: https://ge.in.tum.de/publications/2019-tog-eckert/ tempoGAN: A Temporally Coherent, Volumetric GAN for Super-resolution Fluid Flow , Project: https://ge.in.tum.de/publications/tempogan/ Deep Fluids: A Generative Network for Parameterized Fluid Simulations , Project: http://www.byungsoo.me/project/deep-fluids/ Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow , Project: https://ge.in.tum.de/publications/latent-space-physics/ A Multi-Pass GAN for Fluid Flow Super-Resolution , PDF: https://ge.in.tum.de/publications/2019-multi-pass-gan/ A Study of Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations , Project: https://github.com/thunil/Deep-Flow-Prediction Data-Driven Synthesis of Smoke Flows with CNN-based Feature Descriptors , Project: http://ge.in.tum.de/publications/2017-sig-chu/ Liquid Splash Modeling with Neural Networks , Project: https://ge.in.tum.de/publications/2018-mlflip-um/ Generating Liquid Simulations with Deformation-aware Neural Networks , Project: https://ge.in.tum.de/publications/2017-prantl-defonn/ Additional Links for Fluids Discretize first, filter next: Learning divergence-consistent closure models for large-eddy simulation , PDF: https://doi.org/10.1016/j.jcp.2024.113577 Data-Efficient Inference of Neural Fluid Fields via SciML Foundation Model , PDF: https://arxiv.org/abs/2412.13897 DeepLag: Discovering Deep Lagrangian Dynamics for Intuitive Fluid Prediction , PDF: https://arxiv.org/pdf/2402.02425 Inferring Hybrid Neural Fluid Fields from Videos , PDF: https://openreview.net/pdf?id=kRdaTkaBwC LagrangeBench: A Lagrangian Fluid Mechanics Benchmarking Suite , Project: https://github.com/tumaer/lagrangebench CFDBench: A Comprehensive Benchmark for Machine Learning Methods in Fluid Dynamics , PDF: https://arxiv.org/pdf/2310.05963.pdf Physics-guided training of GAN to improve accuracy in airfoil design synthesis , PDF: https://arxiv.org/pdf/2308.10038 A probabilistic, data-driven closure model for RANS simulations with aleatoric, model uncertainty , PDF: https://arxiv.org/pdf/2307.02432 Differentiable Turbulence , PDF: https://arxiv.org/pdf/2307.03683 Super-resolving sparse observations in partial differential equations: A physics-constrained convolutional neural network approach , PDF: https://arxiv.org/pdf/2306.10990 Reduced-order modeling of fluid flows with transformers , PDF: https://pubs.aip.org/aip/pof/article/35/5/057126/2891586 Machine learning enhanced real-time aerodynamic forces prediction based on sparse pressure sensor inputs , PDF: https://arxiv.org/pdf/2305.09199 Reconstructing Turbulent Flows Using Physics-Aware Spatio-Temporal Dynamics and Test-Time Refinement , PDF: https://arxiv.org/pdf/2304.12130 Inferring Fluid Dynamics via Inverse Rendering , PDF: https://arxiv.org/pdf/2304.04446 Multi-scale rotation-equivariant graph neural networks for unsteady Eulerian fluid dynamics , WWW: https://aip.scitation.org/doi/10.1063/5.0097679 AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutions , PDF: https://arxiv.org/pdf/2212.07564 Exploring Physical Latent Spaces for Deep Learning , PDF: https://arxiv.org/pdf/2211.11298 Modelling spatiotemporal turbulent dynamics with the convolutional autoencoder echo state network , PDF: https://arxiv.org/pdf/2211.11379 Combined space-time reduced-order model with 3D deep convolution for extrapolating fluid dynamics , PDF: https://arxiv.org/pdf/2211.00307 NeuroFluid: Fluid Dynamics Grounding with Particle-Driven Neural Radiance Fields , Project: https://github.com/syguan96/NeuroFluid Lagrangian Large Eddy Simulations via Physics Informed Machine Learning , PDF: https://arxiv.org/pdf/2207.04012 Learning to Estimate and Refine Fluid Motion with Physical Dynamics , PDF: https://arxiv.org/pdf/2206.10480.pdf Deep Reinforcement Learning for Turbulence Modeling in Large Eddy Simulations , PDF: https://arxiv.org/pdf/2206.11038 Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions , PDF: https://arxiv.org/pdf/2205.11912 Physics Informed Neural Fields for Smoke Reconstruction with Sparse Data , Project: https://rachelcmy.github.io/pinf_smoke/ Leveraging Stochastic Predictions of Bayesian Neural Networks for Fluid Simulations , PDF: https://arxiv.org/pdf/2205.01222 NeuroFluid: Fluid Dynamics Grounding with Particle-Driven Neural Radiance Fields , PDF: https://arxiv.org/pdf/2203.01762.pdf Deep neural networks to correct sub-precision errors in CFD , PDF: https://arxiv.org/pdf/2202.04233 Deep learning fluid flow reconstruction around arbitrary two-dimensional objects from sparse sensors using conformal mappings , PDF: https://arxiv.org/pdf/2202.03798.pdf Predicting Physics in Mesh-reduced Space with Temporal Attention , PDF: https://arxiv.org/pdf/2201.09113.pdf Inferring Turbulent Parameters via Machine Learning , PDF: https://arxiv.org/pdf/2201.00732 Learned Coarse Models for Efficient Turbulence Simulation , PDF: https://arxiv.org/pdf/2112.15275.pdf Deep Learning for Stability Analysis of a Freely Vibrating Sphere at Moderate Reynolds Number , PDF: https://arxiv.org/pdf/2112.09858.pdf Predicting High-Resolution Turbulence Details in Space and Time , PDF: http://geometry.caltech.edu/pubs/BWDL21.pdf Assessments of model-form uncertainty using Gaussian stochastic weight averaging for fluid-flow regression , PDF: https://arxiv.org/pdf/2109.08248.pdf Reconstructing High-resolution Turbulent Flows Using Physics-Guided Neural Networks , PDF: https://arxiv.org/pdf/2109.03327 Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows , PDF: https://arxiv.org/pdf/2109.01514.pdf SURFNet: Super-resolution of Turbulent Flows with Transfer Learning using Small Datasets , PDF: https://arxiv.org/pdf/2108.07667.pdf Deep Learning for Reduced Order Modelling and Efficient Temporal Evolution of Fluid Simulations , PDF: https://arxiv.org/pdf/2107.04556.pdf Learning Incompressible Fluid Dynamics from Scratch - Towards Fast, Differentiable Fluid Models that Generalize , PDF: https://cg.cs.uni-bonn.de/aigaion2root/attachments/Paper.pdf Scientific multi-agent reinforcement learning for wall-models of turbulent flows , PDF: https://arxiv.org/pdf/2106.11144.pdf Simulating Continuum Mechanics with Multi-Scale Graph Neural Networks , PDF: https://arxiv.org/pdf/2106.04900.pdf Embedded training of neural-network sub-grid-scale turbulence models , PDF: https://arxiv.org/pdf/2105.01030.pdf Optimal control of point-to-point navigation in turbulent time dependent flows using Reinforcement Learning , PDF: https://arxiv.org/pdf/2103.00329.pdf Machine learning accelerated computational fluid dynamics , PDF: https://arxiv.org/pdf/2102.01010.pdf Neural Particle Image Velocimetry , PDF: https://arxiv.org/pdf/2101.11950.pdf A turbulent eddy-viscosity surrogate modeling framework for Reynolds-Averaged Navier-Stokes simulations , Project+Code: https://www.sciencedirect.com/science/article/abs/pii/S0045793020303479 Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels , PDF: https://arxiv.org/pdf/2011.02364.pdf A Point-Cloud Deep Learning Framework for Prediction of Fluid Flow Fields on Irregular Geometries , PDF: https://arxiv.org/pdf/2010.09469 Learning Mesh-Based Simulations with Graph Networks , PDF: https://arxiv.org/pdf/2010.03409 Using Machine Learning to Augment Coarse-Grid Computational Fluid Dynamics Simulations , PDF: https://arxiv.org/pdf/2010.00072 Learning to swim in potential flow , PDF: https://arxiv.org/pdf/2009.14280 A neural network multigrid solver for the Navier-Stokes equations , PDF: https://arxiv.org/pdf/2008.11520.pdf Enhanced data efficiency using deep neural networks and Gaussian processes for aerodynamic design optimization , PDF: https://arxiv.org/pdf/2008.06731 Learned discretizations for passive scalar advection in a 2-D turbulent flow , PDF: https://arxiv.org/pdf/2004.05477 PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain , PDF: https://arxiv.org/pdf/2004.13145 Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction , PDF: https://proceedings.icml.cc/static/paper_files/icml/2020/6414-Paper.pdf CFDNet: A deep learning-based accelerator for fluid simulations , PDF: https://arxiv.org/pdf/2005.04485 The neural particle method--an updated Lagrangian physics informed neural network for computational fluid dynamics , PDF: https://arxiv.org/pdf/2003.10208 Controlling Rayleigh-Benard convection via Reinforcement Learning , PDF: https://arxiv.org/pdf/2003.14358 Embedding Hard Physical Constraints in Neural Network Coarse-Graining of 3D Turbulence , PDF: https://arxiv.org/pdf/2002.00021 Learning to Simulate Complex Physics with Graph Networks , PDF: https://arxiv.org/pdf/2002.09405 DPM: A deep learning PDE augmentation method (with application to large-eddy simulation) , PDF: https://arxiv.org/pdf/1911.09145 Towards Physics-informed Deep Learning for Turbulent Flow Prediction , PDF: https://arxiv.org/pdf/1911.08655 Dynamic Upsampling of Smoke through Dictionary-based Learning , PDF: https://arxiv.org/pdf/1910.09166 Deep unsupervised learning of turbulence for inflow generation at various Reynolds numbers , PDF: https://arxiv.org/pdf/1908.10515 DeepFlow: History Matching in the Space of Deep Generative Models , PDF: https://arxiv.org/pdf/1905.05749 Deep learning observables in computational fluid dynamics , PDF: https://arxiv.org/pdf/1903.03040 Compressed convolutional LSTM: An efficient deep learning framework to model high fidelity 3D turbulence , PDF: https://arxiv.org/pdf/1903.00033 Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data , PDF: https://arxiv.org/pdf/1901.06314.pdf Deep neural networks for data-driven LES closure models , PDF: https://www.sciencedirect.com/science/article/pii/S0021999119306151 Computing interface curvature from volume fractions: A machine learning approach , PDF: https://www.sciencedirect.com/science/article/abs/pii/S0045793019302282 Machine learning the kinematics of spherical particles in fluid flows , PDF: https://sandlab.mit.edu/wp-content/uploads/18_JFM.pdf Deep Neural Networks for Data-Driven Turbulence Models , PDF: https://export.arxiv.org/pdf/1806.04482 Deep Dynamical Modeling and Control of Unsteady Fluid Flows , PDF: http://papers.nips.cc/paper/8138-deep-dynamical-modeling-and-control-of-unsteady-fluid-flows Learning Particle Dynamics for Manipulating Rigid Bodies, Deformable Objects, and Fluids , Project+Code: http://dpi.csail.mit.edu Application of Convolutional Neural Network to Predict Airfoil Lift Coefficient , PDF: https://arxiv.org/pdf/1712.10082 Prediction of laminar vortex shedding over a cylinder using deep learning , PDF: https://arxiv.org/pdf/1712.07854 Lat-Net: Compressing Lattice Boltzmann Flow Simulations using Deep Neural Networks , PDF: https://arxiv.org/pdf/1705.09036 Reasoning About Liquids via Closed-Loop Simulation , PDF: https://arxiv.org/pdf/1703.01656 Prediction model of velocity field around circular cylinder over various Reynolds numbers by fusion convolutional neural networks based on pressure on the cylinder , PDF: https://doi.org/10.1063/1.5024595 Accelerating Eulerian Fluid Simulation With Convolutional Networks , Project+Code: https://cims.nyu.edu/~schlacht/CNNFluids.htm Reynolds averaged turbulence modelling using deep neural networks with embedded invariance , PDF: https://www.labxing.com/files/lab_publications/2259-1524535041-QiPuSd6O.pdf Additional Links for General PDEs Differentiable programming across the PDE and Machine Learning barrier , PDF: https://arxiv.org/abs/2409.06085 Zero-shot forecasting of chaotic systems , PDF: https://arxiv.org/pdf/2409.15771 Generative Learning for Forecasting the Dynamics of Complex Systems , PDF: https://arxiv.org/pdf/2402.17157 Micro-Macro Consistency in Multiscale Modeling: Score-Based Model Assisted Sampling of Fast/Slow Dynamical Systems , PDF: https://arxiv.org/pdf/2312.05715 Machine Learning for Partial Differential Equations , PDF: https://arxiv.org/pdf/2303.17078.pdf Learning to Accelerate Partial Differential Equations via Latent Global Evolution , Project: http://snap.stanford.edu/le_pde/ Implicit Neural Spatial Representations for Time-dependent PDEs , PDF: https://proceedings.mlr.press/v202/chen23af/chen23af.pdf Noise-aware physics-informed machine learning for robust PDE discovery , PDF: https://iopscience.iop.org/article/10.1088/2632-2153/acb1f0/pdf Learning from Predictions: Fusing Training and Autoregressive Inference for Long-Term Spatiotemporal Forecasts , PDF: https://arxiv.org/pdf/2302.11101.pdf Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systems , PDF: https://arxiv.org/pdf/2301.10391 Continuous PDE dynamics forecasting with implicit neural representations , PDF: https://arxiv.org/pdf/2209.14855 Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion , PDF: https://arxiv.org/pdf/2208.03322 Discovering nonlinear pde from scarce data with physics encoded learning , PDF: https://arxiv.org/pdf/2201.12354.pdf CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representations , PDF: https://arxiv.org/pdf/2206.02607.pdf Learning to Solve PDE-constrained Inverse Problems with Graph Networks , Project: https://cyanzhao42.github.io/LearnInverseProblem CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method , PDF: https://arxiv.org/pdf/2110.15832 Physics-Aware Downsampling with Deep Learning for Scalable Flood Modeling , PDF: https://arxiv.org/pdf/2106.07218v1.pdf Learning Functional Priors and Posteriors from Data and Physics , PDF: https://arxiv.org/pdf/2106.05863.pdf Accelerating Neural ODEs Using Model Order Reduction , PDF: https://arxiv.org/pdf/2105.14070 Adversarial Multi-task Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations , PDF: https://arxiv.org/pdf/2104.14320 gradSim: Differentiable simulation for system identification and visuomotor control , Project: https://gradsim.github.io Physics-aware, probabilistic model order reduction with guaranteed stability , PDF: https://arxiv.org/pdf/2101.05834 Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks , PDF: https://arxiv.org/pdf/2012.03133.pdf Aphynity: Augmenting physical models with deep networks for complex dynamics forecasting , PDF: https://arxiv.org/pdf/2010.04456.pdf Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers , PDF: https://arxiv.org/pdf/2008.09768 Learning Compositional Koopman Operators for Model-Based Control , Project: http://koopman.csail.mit.edu Universal Differential Equations for Scientific Machine Learning , PDF: https://arxiv.org/pdf/2001.04385.pdf Understanding and mitigating gradient pathologies in physics-informed neural networks , PDF: https://arxiv.org/pdf/2001.04536 Variational Physics-Informed Neural Networks For Solving Partial Differential Equations , PDF: https://arxiv.org/pdf/1912.00873 Poisson CNN: Convolutional Neural Networks for the Solution of the Poisson Equation with Varying Meshes and Dirichlet Boundary Conditions , PDF: https://arxiv.org/pdf/1910.08613 IDENT: Identifying Differential Equations with Numerical Time evolution , PDF: https://arxiv.org/pdf/1904.03538 PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network , PDF: https://arxiv.org/pdf/1812.04426 Data-driven discretization: a method for systematic coarse graining of partial differential equations , PDF: https://arxiv.org/pdf/1808.04930 Solving high-dimensional partial differential equations using deep learning , PDF: https://www.pnas.org/content/115/34/8505.full.pdf Neural Ordinary Differential Equations , PDF: https://arxiv.org/pdf/1806.07366 Deep Learning the Physics of Transport Phenomena , PDF: https://arxiv.org/pdf/1709.02432 DGM: A deep learning algorithm for solving partial differential equations , PDF: https://arxiv.org/pdf/1708.07469 Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations , PDF: https://arxiv.org/pdf/1708.00588 Data-assisted reduced-order modeling of extreme events in complex dynamical systems , Project+Code: https://github.com/zhong1wan/data-assisted PDE-Net: Learning PDEs from Data , Project+Code: https://github.com/ZichaoLong/PDE-Net Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems , PDF: https://arxiv.org/pdf/1708.06850 Neural-network-based approximations for solving partial differential equations , DOI: https://doi.org/10.1002/cnm.1640100303 Additional Links for Other Physics Problems and Physics-related Problems Towards a Physics Foundation Model , PDF: https://arxiv.org/abs/2509.13805 FunDiff: diffusion models over function spaces for physics-informed generative modeling , PDF: https://arxiv.org/pdf/2506.07902 CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs , PDF: https://arxiv.org/abs/2505.12944 Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers , PDF: https://arxiv.org/abs/2505.14595 Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods , Project: https://github.com/2ailesB/neural-parametric-solver Implicit Neural Differential Model for Spatiotemporal Dynamics , PDF: https://arxiv.org/abs/2504.02260 A Multimodal PDE Foundation Model for Prediction and Scientific Text Descriptions , PDF: https://www.arxiv.org/abs/2502.06026 PINN-FEM: A Hybrid Approach for Enforcing Dirichlet Boundary Conditions in Physics-Informed Neural Networks , PDF: https://arxiv.org/abs/2501.07765 HypeRL: Parameter-Informed Reinforcement Learning for Parametric PDEs , PDF: https://arxiv.org/abs/2501.04538 Advancing Generalization in PINNs through Latent-Space Representations , PDF: https://arxiv.org/pdf/2411.19125 Text2PDE: Latent Diffusion Models for Accessible Physics Simulation , PDF: https://arxiv.org/abs/2410.01137 Active Learning for Neural PDE Solvers , PDF: https://arxiv.org/pdf/2408.01536 Physics-embedded Fourier Neural Network for Partial Differential Equations , PDF: https://arxiv.org/pdf/2407.11158 Accelerating Legacy Numerical Solvers by Non-intrusive Gradient-based Meta-solving , PDF: https://arxiv.org/pdf/2405.02952 Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations , Project: https://jhagnberger.github.io/vectorized-conditional-neural-field/ UM2N: Towards Universal Mesh Movement Networks , PDF: https://arxiv.org/pdf/2407.00382 Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous media , PDF: https://arxiv.org/pdf/2405.19019 Universal Physics Transformers: A Framework For Efficiently Scaling Neural Operators , PDF: https://arxiv.org/html/2402.12365 Hybrid Modeling Design Patterns , PDF: https://arxiv.org/pdf/2401.00033 ClimODE: Climate Forecasting With Physics-informed Neural ODEs , PDF: https://openreview.net/pdf?id=xuY33XhEGR Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs , PDF: https://arxiv.org/pdf/2403.12553 Investigation of the generalization capability of a generative adversarial network for large eddy simulation of turbulent premixed reacting flows , PDF: https://www.sciencedirect.com/science/article/pii/S1540748922002851 Optimal Power Flow in Highly Renewable Power System Based on Attention Neural Networks , PDF: https://arxiv.org/pdf/2311.13949 Neural General Circulation Models , PDF: https://arxiv.org/pdf/2311.07222 A neural-preconditioned poisson solver for mixed dirichlet and neumann boundary conditions , PDF: https://arxiv.org/pdf/2310.00177.pdf Neural stream functions , PDF: https://arxiv.org/pdf/2307.08142.pdf Stabilized Neural Differential Equations for Learning Constrained Dynamics , PDF: https://arxiv.org/pdf/2306.09739 Combining Slow and Fast: Complementary Filtering for Dynamics Learning , PDF: https://arxiv.org/pdf/2302.13754.pdf PDEBench: An Extensive Benchmark for Scientific Machine Learning , Project: https://github.com/pdebench/PDEBench Efficient Learning of Mesh-Based Physical Simulation with Bi-Stride Multi-Scale Graph Neural Network , PDF: https://arxiv.org/pdf/2210.02573.pdf Scalable Bayesian Uncertainty Quantification for Neural Network Potentials: Promise and Pitfalls , PDF: https://arxiv.org/pdf/2212.07959 Breaking Bad: A Dataset for Geometric Fracture and Reassembly , Project: https://arxiv.org/pdf/2210.11463 Probabilistic forecasts of extreme heatwaves using convolutional neural networks in a regime of lack of data , PDF: https://arxiv.org/pdf/2208.00971 Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs , PDF: https://arxiv.org/pdf/2206.11990 Symplectically Integrated Symbolic Regression of Hamiltonian Dynamical Systems , PDF: https://arxiv.org/pdf/2209.01521.pdf Contact Points Discovery for Soft-Body Manipulations with Differentiable Physics , PDF: https://arxiv.org/pdf/2205.02835.pdf Message Passing Neural PDE Solvers , PDF: https://arxiv.org/pdf/2202.03376 A Survey on Machine Learning Approaches for Modelling Intuitive Physics , PDF: https://arxiv.org/pdf/2202.06481 Fine-grained differentiable physics: a yarn-level model for fabrics , PDF: https://arxiv.org/pdf/2202.00504 Accurately Solving Rod Dynamics with Graph Learning , PDF: http://computationalsciences.org/publications/shao-2021-physical-systems-graph-learning/shao-2021-physical-systems-graph-learning.pdf Constraint-based graph network simulator , PDF: https://arxiv.org/pdf/2112.09161 Differentiable Simulation of Soft Multi-body Systems , PDF: https://arxiv.org/pdf/2205.01758 Learning Material Parameters and Hydrodynamics of Soft Robotic Fish via Differentiable Simulation , PDF: https://arxiv.org/pdf/2109.14855 Model Reduction for the Material Point Method via Learning the Deformation map and its Spatial-temporal Gradients , Project: https://peterchencyc.com/projects/rom4mpm/ PhysGNN: A Physics–Driven Graph Neural Network Based Model for Predicting Soft Tissue Deformation in Image–Guided Neurosurgery , PDF: https://arxiv.org/pdf/2109.04352.pdf Deep learning for surrogate modelling of 2D mantle convection , PDF: https://arxiv.org/pdf/2108.10105 An Extensible Benchmark Suite for Learning to Simulate Physical Systems , PDF: https://arxiv.org/pdf/2108.07799 Turbulent field fluctuations in gyrokinetic and fluid plasmas , PDF: https://arxiv.org/pdf/2107.09744.pdf Robust Value Iteration for Continuous Control Tasks , PDF: https://arxiv.org/pdf/2105.12189 Fast and Feature-Complete Differentiable Physics for Articulated Rigid Bodies with Contact , PDF: https://arxiv.org/pdf/2103.16021 High-order Differentiable Autoencoder for Nonlinear Model Reduction , PDF: https://arxiv.org/pdf/2102.11026.pdf Modeling of the nonlinear flame response of a Bunsen-type flame via multi-layer perceptron , Paper: https://www.sciencedirect.com/science/article/pii/S1540748920305666 Deluca – A Differentiable Control Library: Environments, Methods, and Benchmarking , PDF: https://montrealrobotics.ca/diffcvgp/assets/papers/1.pdf Deep Energy-based Modeling of Discrete-Time Physics , PDF: https://proceedings.neurips.cc/paper/2020/file/98b418276d571e623651fc1d471c7811-Paper.pdf NeuralSim: Augmenting Differentiable Simulators with Neural Networks , PDF: https://arxiv.org/pdf/2011.04217.pdf Fourier Neural Operator for Parametric Partial Differential Equations , PDF: https://arxiv.org/pdf/2010.08895.pdf Learning Composable Energy Surrogates for PDE Order Reduction , PDF: https://arxiv.org/pdf/2005.06549.pdf Transformers for Modeling Physical Systems , PDF: https://arxiv.org/pdf/2010.03957 Reinforcement Learning for Molecular Design Guided by Quantum Mechanics , PDF: https://proceedings.icml.cc/static/paper_files/icml/2020/1323-Paper.pdf Scalable Differentiable Physics for Learning and Control , PDF: https://proceedings.icml.cc/static/paper_files/icml/2020/15-Paper.pdf Cloth in the Wind: A Case Study of Physical Measurement through Simulation , PDF: https://arxiv.org/pdf/2003.05065 Learning to Slide Unknown Objects with Differentiable Physics Simulations , PDF: https://arxiv.org/pdf/2005.05456 Physics-aware Difference Graph Networks for Sparsely-Observed Dynamics , Project: https://github.com/USC-Melady/ICLR2020-PADGN Differentiable Molecular Simulations for Control and Learning , PDF: https://arxiv.org/pdf/2003.00868 Incorporating Symmetry into Deep Dynamics Models for Improved Generalization , PDF: https://arxiv.org/pdf/2002.03061 Learning to Measure the Static Friction Coefficient in Cloth Contact , PDF: https://hal.inria.fr/hal-02511646 Learning to Simulate Complex Physics with Graph Networks , PDF: https://arxiv.org/pdf/2002.09405 Hamiltonian Neural Networks , PDF: http://papers.nips.cc/paper/9672-hamiltonian-neural-networks.pdf Interactive Differentiable Simulation , PDF: https://arxiv.org/pdf/1905.10706 DiffTaichi: Differentiable Programming for Physical Simulation , PDF: https://arxiv.org/pdf/1910.00935 Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data , PDF: https://arxiv.org/pdf/1901.06314 COPHY: Counterfactual Learning of Physical Dynamics , Project: https://github.com/fabienbaradel/cophy Modeling Expectation Violation in Intuitive Physics with Coarse Probabilistic Object Representations , Project: http://physadept.csail.mit.edu End-to-End Differentiable Physics for Learning and Control , Project+Code: https://github.com/locuslab/lcp-physics Stochastic seismic waveform inversion using generative adversarial networks as a geological prior , PDF: https://arxiv.org/pdf/1806.03720 Learning to Optimize Multigrid PDE Solvers , PDF: http://proceedings.mlr.press/v97/greenfeld19a/greenfeld19a.pdf Latent-space Dynamics for Reduced Deformable Simulation , Project+Code: http://www.dgp.toronto.edu/projects/latent-space-dynamics/ Learning-Based Animation of Clothing for Virtual Try-On , PDF: http://www.gmrv.es/Publications/2019/SOC19/ Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning , PDF: https://openreview.net/pdf?id=BklHpjCqKm Flexible Neural Representation for Physics Prediction , Project+Code: https://neuroailab.github.io/physics/ Robust Reference Frame Extraction from Unsteady 2D Vector Fields with Convolutional Neural Networks , PDF: https://arxiv.org/pdf/1903.10255 Physics-as-Inverse-Graphics: Joint Unsupervised Learning of Objects and Physics from Video , PDF: https://arxiv.org/pdf/1905.11169 Unsupervised Intuitive Physics from Past Experiences , PDF: https://arxiv.org/pdf/1905.10793 Reasoning About Physical Interactions with Object-Oriented Prediction and Planning , PDF: https://arxiv.org/pdf/1812.10972 Neural Material: Learning Elastic Constitutive Material and Damping Models from Sparse Data , PDF: https://arxiv.org/pdf/1808.04931 Discovering physical concepts with neural networks , PDF: https://arxiv.org/pdf/1807.10300 Fluid directed rigid body control using deep reinforcement learning , Project: http://gamma.cs.unc.edu/DRL_FluidRigid/ DeepMimic, Example-Guided Deep Reinforcement Learning of Physics-Based Character Skills , PDF: https://arxiv.org/pdf/1804.02717 Unsupervised Intuitive Physics from Visual Observations , PDF: https://arxiv.org/pdf/1805.05086 Graph networks as learnable physics engines for inference and control , PDF: https://arxiv.org/pdf/1806.01242 DeepWarp: DNN-based Nonlinear Deformation , PDF: https://arxiv.org/pdf/1803.09109 A proposal on machine learning via dynamical systems , Journal: https://link.springer.com/article/10.1007/s40304-017-0103-z Interaction Networks for Learning about Objects, Relations and Physics , PDF: https://arxiv.org/pdf/1612.00222 Surveys and Overview Articles Physics-Guided Deep Learning for Dynamical Systems: A Survey , PDF: https://arxiv.org/pdf/2107.01272 Integrating Physics-Based Modeling with Machine Learning: A Survey , PDF: https://arxiv.org/pdf/2003.04919 Integrating Machine Learning with Physics-Based Modeling , PDF: https://arxiv.org/pdf/2006.02619 A review on Deep Reinforcement Learning for Fluid Mechanics , PDF: https://arxiv.org/pdf/1908.04127 Machine Learning for Fluid Mechanics , PDF: https://arxiv.org/pdf/1905.11075 Simulation and Deep Learning Frameworks PhiFlow: https://github.com/tum-pbs/phiflow Diff-Taichi: https://github.com/yuanming-hu/difftaichi Jax-Md: https://github.com/google/jax-md TensorFlow-Foam: https://github.com/argonne-lcf/TensorFlowFoam Julia-Sciml: https://sciml.ai GradSim: https://gradsim.github.io Jax-Cfd: https://github.com/google/jax-cfd Jax-Fluids: https://github.com/tumaer/JAXFLUIDS Concluding Remarks Physics-based deep learning is a very dynamic field. Please let us know if we've overlooked papers that you think should be included by sending a mail to i15ge at cs.tum.de , and feel free to check out our homepage at https://ge.in.tum.de/ .
The following collection of materials targets "Physics-Based Deep Learning" (PBDL), i.e., the field of methods with combinations of physical modeling and deep learning (DL) techniques. Here, DL will typically refer to methods based on artificial neural networks. The general direction of PBDL represents a very active and quickly growing field of research.
If you're interested in a comprehensive overview, please check our digital PBDL book: https://www.physicsbaseddeeplearning.org/ (or as PDF: https://arxiv.org/pdf/2109.05237.pdf )
Within this area, we can distinguish a variety of different physics-based approaches, from targeting designs, constraints, combined methods, and optimizations to applications. More specifically, all approaches either target forward simulations (predicting state or temporal evolution) or inverse problems (e.g., obtaining a parametrization for a physical system from observations). Apart from forward or inverse, the type of integration between learning and physics gives a means for categorizing different methods:
Data-driven : the data is produced by a physical system (real or simulated), but no further interaction exists.
Loss-terms : the physical dynamics (or parts thereof) are encoded in the loss function, typically in the form of differentiable operations. The learning process can repeatedly evaluate the loss, and usually receives gradients from a PDE-based formulation.
Interleaved : the full physical simulation is interleaved and combined with an output from a deep neural network; this requires a fully differentiable simulator and represents the tightest coupling between the physical system and the learning process. Interleaved approaches are especially important for temporal evolutions, where they can yield an estimate of future behavior of the dynamics.
2. Related Work, Literature Survey & Algorithmic Foundations
Thus, methods can be roughly categorized in terms of forward versus inverse solve, and how tightly the physical model is integrated into the optimization loop that trains the deep neural network. Here, especially approaches that leverage differentiable physics allow for very tight integration of deep learning and numerical simulations.
This repository collects links to works on deep learning algorithms for physics problems , with a particular emphasis on fluid flow , i.e., Navier-Stokes related problems. It primarily collects links to the work of the I15 lab at TUM, as well as miscellaneous works from other groups. This is by no means a complete list, so let us know if you come across additional papers in this area. We intentionally also focus on works from the deep learning field, not machine learning in general.
diffSPH: Differentiable Smoothed Particle Hydrodynamics for Adjoint Optimization and Machine Learning , Project: https://github.com/tum-pbs/diffSPH
PDE-Transformer: Efficient and Versatile Transformers for Physics Simulations , Project: https://tum-pbs.github.io/pde-transformer/landing.html
Flow Matching Meets PDEs – A Unified Framework for Physics-Constrained Generation , PDF: https://arxiv.org/abs/2506.08604
Component-Based Machine Learning for Indoor Flow and Temperature Fields Prediction Latent Feature Aggregation and Flow Interaction , PDF: https://arxiv.org/abs/2507.19233
PICT - A Differentiable, GPU-Accelerated Multi-Block PISO Solver for Simulation-Coupled Learning Tasks in Fluid Dynamics , Project: https://github.com/tum-pbs/PICT
3. Dataset Ingestion, Preprocessing & Stochastic Augmentations
Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks , Project: https://github.com/tum-pbs/dgn4cfd
Temporal Difference Learning: Why It Can Be Fast and How It Will Be Faster , PDF: https://openreview.net/forum?id=j3bKnEidtT
Truncation Is All You Need: Improved Sampling Of Diffusion Models For Physics-Based Simulations , PDF: https://openreview.net/forum?id=0FbzC7B9xI
PRDP: Progressively Refined Differentiable Physics , Project: https://github.com/tum-pbs/PRDP
Temporal Difference Learning: Why It Can Be Fast and How It Will Be Faster , PDF: https://openreview.net/forum?id=j3bKnEidtT
Flow Matching for Posterior Inference with Simulator Feedback , PDF: https://arxiv.org/pdf/2410.22573
APEBench: A Benchmark for Autoregressive Neural Emulators of PDEs , Project: https://github.com/tum-pbs/apebench-paper
4. Neural Network Backbone, Layer Specifications & Loss Formulations
Deep learning-based predictive modelling of transonic flow over an aerofoil , PDF: https://arxiv.org/pdf/2403.17131
ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks , Project: https://tum-pbs.github.io/ConFIG/
The Unreasonable Effectiveness of Solving Inverse Problems with Neural Networks , PDF: http://arxiv.org/pdf/2408.08119
Phiflow: Differentiable Simulations for PyTorch, TensorFlow and Jax , PDF: https://openreview.net/pdf/36503358a4f388f00d587a0257c13ba2a4656098.pdf
How Temporal Unrolling Supports Neural Physics Simulators , PDF: https://arxiv.org/pdf/2402.12971
Stabilizing Backpropagation Through Time to Learn Complex Physics , PDF: https://openreview.net/forum?id=bozbTTWcaw
Symmetric Basis Convolutions for Learning Lagrangian Fluid Mechanics , PDF: https://openreview.net/forum?id=HKgRwNhI9R
5. Training Protocol, Convergence Behavior & Experimental Evaluation
Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models , Project: https://github.com/tum-pbs/Diffusion-based-Flow-Prediction
Turbulent Flow Simulation using Autoregressive Conditional Diffusion Models , Project: https://github.com/tum-pbs/autoreg-pde-diffusion
Physics-Preserving AI-Accelerated Simulations of Plasma Turbulence , PDF: https://arxiv.org/pdf/2309.16400 >
Unsteady Cylinder Wakes from Arbitrary Bodies with Differentiable Physics-Assisted Neural Network , Project: https://github.com/tum-pbs/DiffPhys-CylinderWakeFlow
Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics , Project: https://github.com/tum-pbs/DMCF
Learned Turbulence Modelling with Differentiable Fluid Solvers , Project: https://github.com/tum-pbs/differentiable-piso
Half-Inverse Gradients for Physical Deep Learning , Project: https://github.com/tum-pbs/half-inverse-gradients
Model Figures & Evaluation Curves (4 Images)
Deep Learning Execution Pipeline
(Preprocessing)
(Feature Extraction)
(Self-Attention)
(Loss Optimization)