Skip to content


Folders and files

Last commit message
Last commit date

Latest commit


Repository files navigation

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: (or as PDF:

An overview of categories of physics-based deep learning methods

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.

An example flow result from tempoGAN

I15 Physics-based Deep Learning Links

How Temporal Unrolling Supports Neural Physics Simulators , PDF:

Stabilizing Backpropagation Through Time to Learn Complex Physics , PDF:

Symmetric Basis Convolutions for Learning Lagrangian Fluid Mechanics , PDF:

Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models , Project:

Turbulent Flow Simulation using Autoregressive Conditional Diffusion Models , Project:

Physics-Preserving AI-Accelerated Simulations of Plasma Turbulence , PDF:>

Unsteady Cylinder Wakes from Arbitrary Bodies with Differentiable Physics-Assisted Neural Network , Project:

Score Matching via Differentiable Physics , Project:

Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics , Project:

Learned Turbulence Modelling with Differentiable Fluid Solvers , Project:

Half-Inverse Gradients for Physical Deep Learning , Project:

Reviving Autoencoder Pretraining (Previously: Data-driven Regularization via Racecar Training for Generalizing Neural Networks), Project:

Realistic galaxy images and improved robustness in machine learning tasks from generative modelling , PDF:

Hybrid Neural Network PDE Solvers for Reacting Flows , Project:

Scale-invariant Learning by Physics Inversion (formerly "Physical Gradients") , Project:

High-accuracy transonic RANS Flow Predictions with Deep Neural Networks , Project:

Learning Meaningful Controls for Fluids , Project:

Global Transport for Fluid Reconstruction with Learned Self-Supervision , Project:

Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers , Project:

Numerical investigation of minimum drag profiles in laminar flow using deep learning surrogates , PDF:

Purely data-driven medium-range weather forecasting achieves comparable skill to physical models at similar resolution , PDF:

Latent Space Subdivision: Stable and Controllable Time Predictions for Fluid Flow , Project:

WeatherBench: A benchmark dataset for data-driven weather forecasting , Project:

Learning Similarity Metrics for Numerical Simulations (LSiM) , Project:

Learning to Control PDEs with Differentiable Physics , Project:

Lagrangian Fluid Simulation with Continuous Convolutions , PDF:

Tranquil-Clouds: Neural Networks for Learning Temporally Coherent Features in Point Clouds , Project:

ScalarFlow: A Large-Scale Volumetric Data Set of Real-world Scalar Transport Flows for Computer Animation and Machine Learning , Project:

tempoGAN: A Temporally Coherent, Volumetric GAN for Super-resolution Fluid Flow , Project:

Deep Fluids: A Generative Network for Parameterized Fluid Simulations , Project:

Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow , Project:

A Multi-Pass GAN for Fluid Flow Super-Resolution , PDF:

A Study of Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations , Project:

Data-Driven Synthesis of Smoke Flows with CNN-based Feature Descriptors , Project:

Liquid Splash Modeling with Neural Networks , Project:

Generating Liquid Simulations with Deformation-aware Neural Networks , Project:

Additional Links for Fluids

Inferring Hybrid Neural Fluid Fields from Videos , PDF:

LagrangeBench: A Lagrangian Fluid Mechanics Benchmarking Suite , Project:

CFDBench: A Comprehensive Benchmark for Machine Learning Methods in Fluid Dynamics , PDF:

Physics-guided training of GAN to improve accuracy in airfoil design synthesis , PDF:

A probabilistic, data-driven closure model for RANS simulations with aleatoric, model uncertainty , PDF:

Differentiable Turbulence , PDF:

Super-resolving sparse observations in partial differential equations: A physics-constrained convolutional neural network approach , PDF:

Reduced-order modeling of fluid flows with transformers , PDF:

Machine learning enhanced real-time aerodynamic forces prediction based on sparse pressure sensor inputs , PDF:

Reconstructing Turbulent Flows Using Physics-Aware Spatio-Temporal Dynamics and Test-Time Refinement , PDF:

Inferring Fluid Dynamics via Inverse Rendering , PDF:

Multi-scale rotation-equivariant graph neural networks for unsteady Eulerian fluid dynamics , WWW:

AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutions , PDF:

Exploring Physical Latent Spaces for Deep Learning , PDF:

Modelling spatiotemporal turbulent dynamics with the convolutional autoencoder echo state network , PDF:

Combined space-time reduced-order model with 3D deep convolution for extrapolating fluid dynamics , PDF:

NeuroFluid: Fluid Dynamics Grounding with Particle-Driven Neural Radiance Fields , Project:

Lagrangian Large Eddy Simulations via Physics Informed Machine Learning , PDF:

Learning to Estimate and Refine Fluid Motion with Physical Dynamics , PDF:

Deep Reinforcement Learning for Turbulence Modeling in Large Eddy Simulations , PDF:

Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions , PDF:

Physics Informed Neural Fields for Smoke Reconstruction with Sparse Data , Project:

Leveraging Stochastic Predictions of Bayesian Neural Networks for Fluid Simulations , PDF:

NeuroFluid: Fluid Dynamics Grounding with Particle-Driven Neural Radiance Fields , PDF:

Deep neural networks to correct sub-precision errors in CFD , PDF:

Deep learning fluid flow reconstruction around arbitrary two-dimensional objects from sparse sensors using conformal mappings , PDF:

Predicting Physics in Mesh-reduced Space with Temporal Attention , PDF:

Inferring Turbulent Parameters via Machine Learning , PDF:

Learned Coarse Models for Efficient Turbulence Simulation , PDF:

Deep Learning for Stability Analysis of a Freely Vibrating Sphere at Moderate Reynolds Number , PDF:

Predicting High-Resolution Turbulence Details in Space and Time , PDF:

Assessments of model-form uncertainty using Gaussian stochastic weight averaging for fluid-flow regression , PDF:

Reconstructing High-resolution Turbulent Flows Using Physics-Guided Neural Networks , PDF:

Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows , PDF:

SURFNet: Super-resolution of Turbulent Flows with Transfer Learning using Small Datasets , PDF:

Deep Learning for Reduced Order Modelling and Efficient Temporal Evolution of Fluid Simulations , PDF:

Learning Incompressible Fluid Dynamics from Scratch - Towards Fast, Differentiable Fluid Models that Generalize , PDF:

Scientific multi-agent reinforcement learning for wall-models of turbulent flows , PDF:

Simulating Continuum Mechanics with Multi-Scale Graph Neural Networks , PDF:

Embedded training of neural-network sub-grid-scale turbulence models , PDF:

Optimal control of point-to-point navigation in turbulent time dependent flows using Reinforcement Learning , PDF:

Machine learning accelerated computational fluid dynamics , PDF:

Neural Particle Image Velocimetry , PDF:

A turbulent eddy-viscosity surrogate modeling framework for Reynolds-Averaged Navier-Stokes simulations , Project+Code:

Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels , PDF:

A Point-Cloud Deep Learning Framework for Prediction of Fluid Flow Fields on Irregular Geometries , PDF:

Learning Mesh-Based Simulations with Graph Networks , PDF:

Using Machine Learning to Augment Coarse-Grid Computational Fluid Dynamics Simulations , PDF:

Learning to swim in potential flow , PDF:

A neural network multigrid solver for the Navier-Stokes equations , PDF:

Enhanced data efficiency using deep neural networks and Gaussian processes for aerodynamic design optimization , PDF:

Learned discretizations for passive scalar advection in a 2-D turbulent flow , PDF:

PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain , PDF:

Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction , PDF:

CFDNet: A deep learning-based accelerator for fluid simulations , PDF:

Controlling Rayleigh-Benard convection via Reinforcement Learning , PDF:

Embedding Hard Physical Constraints in Neural Network Coarse-Graining of 3D Turbulence , PDF:

Learning to Simulate Complex Physics with Graph Networks , PDF:

DPM: A deep learning PDE augmentation method (with application to large-eddy simulation) , PDF:

Towards Physics-informed Deep Learning for Turbulent Flow Prediction , PDF:

Dynamic Upsampling of Smoke through Dictionary-based Learning , PDF:

Deep unsupervised learning of turbulence for inflow generation at various Reynolds numbers , PDF:

DeepFlow: History Matching in the Space of Deep Generative Models , PDF:

Deep learning observables in computational fluid dynamics , PDF:

Compressed convolutional LSTM: An efficient deep learning framework to model high fidelity 3D turbulence , PDF:

Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data , PDF:

Deep neural networks for data-driven LES closure models , PDF:

Computing interface curvature from volume fractions: A machine learning approach , PDF:

Machine learning the kinematics of spherical particles in fluid flows , PDF:

Deep Neural Networks for Data-Driven Turbulence Models , PDF:

Deep Dynamical Modeling and Control of Unsteady Fluid Flows , PDF:

Learning Particle Dynamics for Manipulating Rigid Bodies, Deformable Objects, and Fluids , Project+Code:

Application of Convolutional Neural Network to Predict Airfoil Lift Coefficient , PDF:

Prediction of laminar vortex shedding over a cylinder using deep learning , PDF:

Lat-Net: Compressing Lattice Boltzmann Flow Simulations using Deep Neural Networks , PDF:

Reasoning About Liquids via Closed-Loop Simulation , PDF:

Prediction model of velocity field around circular cylinder over various Reynolds numbers by fusion convolutional neural networks based on pressure on the cylinder , PDF:

Accelerating Eulerian Fluid Simulation With Convolutional Networks , Project+Code:

Reynolds averaged turbulence modelling using deep neural networks with embedded invariance , PDF:

Image divider for general PDE section

Additional Links for General PDEs

Generative Learning for Forecasting the Dynamics of Complex Systems , PDF:

Micro-Macro Consistency in Multiscale Modeling: Score-Based Model Assisted Sampling of Fast/Slow Dynamical Systems , PDF:

Machine Learning for Partial Differential Equations , PDF:

Learning to Accelerate Partial Differential Equations via Latent Global Evolution , Project:

Noise-aware physics-informed machine learning for robust PDE discovery , PDF:

Learning from Predictions: Fusing Training and Autoregressive Inference for Long-Term Spatiotemporal Forecasts , PDF:

Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systems , PDF:

Continuous PDE dynamics forecasting with implicit neural representations , PDF:

Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion , PDF:

Discovering nonlinear pde from scarce data with physics encoded learning , PDF:

CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representations , PDF:

Learning to Solve PDE-constrained Inverse Problems with Graph Networks , Project:

CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method , PDF:

Physics-Aware Downsampling with Deep Learning for Scalable Flood Modeling , PDF:

Learning Functional Priors and Posteriors from Data and Physics , PDF:

Accelerating Neural ODEs Using Model Order Reduction , PDF:

Adversarial Multi-task Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations , PDF:

gradSim: Differentiable simulation for system identification and visuomotor control , Project:

Physics-aware, probabilistic model order reduction with guaranteed stability , PDF:

Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks , PDF:

Aphynity: Augmenting physical models with deep networks for complex dynamics forecasting , PDF:

Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers , PDF:

Learning Compositional Koopman Operators for Model-Based Control , Project:

Universal Differential Equations for Scientific Machine Learning , PDF:

Understanding and mitigating gradient pathologies in physics-informed neural networks , PDF:

Variational Physics-Informed Neural Networks For Solving Partial Differential Equations , PDF:

Poisson CNN: Convolutional Neural Networks for the Solution of the Poisson Equation with Varying Meshes and Dirichlet Boundary Conditions , PDF:

IDENT: Identifying Differential Equations with Numerical Time evolution , PDF:

PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network , PDF:

Data-driven discretization: a method for systematic coarse graining of partial differential equations , PDF:

Solving high-dimensional partial differential equations using deep learning , PDF:

Neural Ordinary Differential Equations , PDF:

Deep Learning the Physics of Transport Phenomena , PDF:

DGM: A deep learning algorithm for solving partial differential equations , PDF:

Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations , PDF:

Data-assisted reduced-order modeling of extreme events in complex dynamical systems , Project+Code:

PDE-Net: Learning PDEs from Data , Project+Code:

Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems , PDF:

Additional Links for Other Physics Problems and Physics-related Problems

Hybrid Modeling Design Patterns , PDF:

ClimODE: Climate Forecasting With Physics-informed Neural ODEs , PDF:

Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs , PDF:

Investigation of the generalization capability of a generative adversarial network for large eddy simulation of turbulent premixed reacting flows , PDF:

Optimal Power Flow in Highly Renewable Power System Based on Attention Neural Networks , PDF:

Neural General Circulation Models , PDF:

A neural-preconditioned poisson solver for mixed dirichlet and neumann boundary conditions , PDF:

Neural stream functions , PDF:

Stabilized Neural Differential Equations for Learning Constrained Dynamics , PDF:

PDEBench: An Extensive Benchmark for Scientific Machine Learning , Project:

Efficient Learning of Mesh-Based Physical Simulation with Bi-Stride Multi-Scale Graph Neural Network , PDF:

Combining Slow and Fast: Complementary Filtering for Dynamics Learning , PDF:

Scalable Bayesian Uncertainty Quantification for Neural Network Potentials: Promise and Pitfalls , PDF:

Breaking Bad: A Dataset for Geometric Fracture and Reassembly , Project:

Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs , PDF:

Symplectically Integrated Symbolic Regression of Hamiltonian Dynamical Systems , PDF:

Contact Points Discovery for Soft-Body Manipulations with Differentiable Physics , PDF:

Message Passing Neural PDE Solvers , PDF:

A Survey on Machine Learning Approaches for Modelling Intuitive Physics , PDF:

Fine-grained differentiable physics: a yarn-level model for fabrics , PDF:

Accurately Solving Rod Dynamics with Graph Learning , PDF:

Constraint-based graph network simulator , PDF:

Differentiable Simulation of Soft Multi-body Systems , PDF:

Learning Material Parameters and Hydrodynamics of Soft Robotic Fish via Differentiable Simulation , PDF:

Model Reduction for the Material Point Method via Learning the Deformation map and its Spatial-temporal Gradients , Project:

PhysGNN: A Physics–Driven Graph Neural Network Based Model for Predicting Soft Tissue Deformation in Image–Guided Neurosurgery , PDF:

Deep learning for surrogate modelling of 2D mantle convection , PDF:

An Extensible Benchmark Suite for Learning to Simulate Physical Systems , PDF:

Turbulent field fluctuations in gyrokinetic and fluid plasmas , PDF:

Robust Value Iteration for Continuous Control Tasks , PDF:

Fast and Feature-Complete Differentiable Physics for Articulated Rigid Bodies with Contact , PDF:

High-order Differentiable Autoencoder for Nonlinear Model Reduction , PDF:

Modeling of the nonlinear flame response of a Bunsen-type flame via multi-layer perceptron , Paper:

Deluca – A Differentiable Control Library: Environments, Methods, and Benchmarking , PDF:

Deep Energy-based Modeling of Discrete-Time Physics , PDF:

NeuralSim: Augmenting Differentiable Simulators with Neural Networks , PDF:

Fourier Neural Operator for Parametric Partial Differential Equations , PDF:

Learning Composable Energy Surrogates for PDE Order Reduction , PDF:

Transformers for Modeling Physical Systems , PDF:

Reinforcement Learning for Molecular Design Guided by Quantum Mechanics , PDF:

Scalable Differentiable Physics for Learning and Control , PDF:

Cloth in the Wind: A Case Study of Physical Measurement through Simulation , PDF:

Learning to Slide Unknown Objects with Differentiable Physics Simulations , PDF:

Physics-aware Difference Graph Networks for Sparsely-Observed Dynamics , Project:

Differentiable Molecular Simulations for Control and Learning , PDF:

Incorporating Symmetry into Deep Dynamics Models for Improved Generalization , PDF:

Learning to Measure the Static Friction Coefficient in Cloth Contact , PDF:

Learning to Simulate Complex Physics with Graph Networks , PDF:

Hamiltonian Neural Networks , PDF:

Interactive Differentiable Simulation , PDF:

DiffTaichi: Differentiable Programming for Physical Simulation , PDF:

COPHY: Counterfactual Learning of Physical Dynamics , Project:

Modeling Expectation Violation in Intuitive Physics with Coarse Probabilistic Object Representations , Project:

End-to-End Differentiable Physics for Learning and Control , Project+Code:

Stochastic seismic waveform inversion using generative adversarial networks as a geological prior , PDF:

Learning to Optimize Multigrid PDE Solvers , PDF:

Latent-space Dynamics for Reduced Deformable Simulation , Project+Code:

Learning-Based Animation of Clothing for Virtual Try-On , PDF:

Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning , PDF:

Flexible Neural Representation for Physics Prediction , Project+Code:

Robust Reference Frame Extraction from Unsteady 2D Vector Fields with Convolutional Neural Networks , PDF:

Physics-as-Inverse-Graphics: Joint Unsupervised Learning of Objects and Physics from Video , PDF:

Unsupervised Intuitive Physics from Past Experiences , PDF:

Reasoning About Physical Interactions with Object-Oriented Prediction and Planning , PDF:

Neural Material: Learning Elastic Constitutive Material and Damping Models from Sparse Data , PDF:

Discovering physical concepts with neural networks , PDF:

Fluid directed rigid body control using deep reinforcement learning , Project:

DeepMimic, Example-Guided Deep Reinforcement Learning of Physics-Based Character Skills , PDF:

Unsupervised Intuitive Physics from Visual Observations , PDF:

Graph networks as learnable physics engines for inference and control , PDF:

DeepWarp: DNN-based Nonlinear Deformation , PDF:

A proposal on machine learning via dynamical systems , Journal:

Interaction Networks for Learning about Objects, Relations and Physics , PDF:

Surveys and Overview Articles

Physics-Guided Deep Learning for Dynamical Systems: A Survey , PDF:

Integrating Physics-Based Modeling with Machine Learning: A Survey , PDF:

Integrating Machine Learning with Physics-Based Modeling , PDF:

A review on Deep Reinforcement Learning for Fluid Mechanics , PDF:

Machine Learning for Fluid Mechanics , PDF:

Simulation and Deep Learning Frameworks









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, and feel free to check out our homepage at


Links to works on deep learning algorithms for physics problems, TUM-I15 and beyond






No releases published


No packages published