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

High-accuracy transonic RANS Flow Predictions with Deep Neural Networks , PDF: https://arxiv.org/pdf/2109.02183

Learning Meaningful Controls for Fluids , Project: https://people.mpi-inf.mpg.de/~mchu/gvv-den2vel/den2vel.html

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

Data-driven Regularization via Racecar Training for Generalizing Neural Networks , Project: https://ge.in.tum.de/publications/2020-xie-racecar/

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

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

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

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

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

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

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

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

Additional Links for Other Physics Problems and Physics-related Problems

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

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

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

tensorFlowFoam: https://github.com/argonne-lcf/TensorFlowFoam

julia-sciml: https://julialang.org/jsoc/gsoc/sciml/

gradsim: https://gradsim.github.io

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/.