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Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

arXiv 1711.10561 ↗First posted 28 Nov 2017cs.AI cs.LG cs.NAcited by 373

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Abstract (arXiv)

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this two part treatise, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct classes of algorithms, namely continuous time and discrete time models. The resulting neural networks form a new class of data-efficient universal function approximators that naturally encode any underlying physical laws as prior information. In this first part, we demonstrate how these networks can be used to infer solutions to partial differential equations, and obtain physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters.

Key facts. Cited by 373 papers in the map (data through 14 Oct 2026). First posted on arXiv 28 Nov 2017.

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Citing papers per year: 2021: 70; 2022: 55; 2023: 63; 2024: 64; 2025: 72; 2026: 49.

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  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  2. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  3. 2022An optimal control perspective on diffusion-based generative modelingcited by 147
  4. 2021Efficient training of physics-informed neural networks via importance samplingcited by 110
  5. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 108
  6. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 97
  7. 2021Learned Coarse Models for Efficient Turbulence Simulationcited by 91
  8. 2024Formal Mathematical Reasoning: A New Frontier in AIcited by 81
  9. 2021POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decompositioncited by 79
  10. 2022Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Samplingcited by 73
  11. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  12. 2022Neural Conservation Laws: A Divergence-Free Perspectivecited by 70
38 more
  1. 2023Can Physics-Informed Neural Networks beat the Finite Element Method?cited by 68
  2. 2021Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systemscited by 67
  3. 2022When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learningcited by 65
  4. 2023PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networkscited by 64
  5. 2023Graph Neural Networks for temporal graphs: State of the art, open challenges, and opportunitiescited by 61
  6. 2024On conditional diffusion models for PDE simulationscited by 60
  7. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  8. 2025Evaluating Newtonian Mechanics in Video Generative Models with Real Physical Systemscited by 48
  9. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 44
  10. 2021Quadratic Residual Networks: A New Class of Neural Networks for Solving Forward and Inverse Problems in Physics Involving PDEscited by 43
  11. 2021Neural graphical modelling in continuous-time: consistency guarantees and algorithmscited by 42
  12. 2022Machine Learning in Aerodynamic Shape Optimizationcited by 39
  13. 2025T2VPhysBench: A First-Principles Benchmark for Physical Consistency in Text-to-Video Generationcited by 39
  14. 2021Meta-Learning Dynamics Forecasting Using Task Inferencecited by 38
  15. 2024Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networkscited by 38
  16. 2021Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow modelscited by 35
  17. 2022Applications of physics informed neural operatorscited by 33
  18. 2021A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEscited by 32
  19. 2021Physics-Informed Neural Nets for Control of Dynamical Systemscited by 31
  20. 2021Latent Space Data Assimilation by using Deep Learningcited by 30
  21. 2022Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimizationcited by 29
  22. 2021Adaptive deep density approximation for Fokker-Planck equationscited by 28
  23. 2022MAgNet: Mesh Agnostic Neural PDE Solvercited by 28
  24. 2023Physics-Informed Computer Vision: A Review and Perspectivescited by 28
  25. 2024AirPhyNet: Harnessing Physics-Guided Neural Networks for Air Quality Predictioncited by 26
  26. 2024Air Quality Prediction with Physics-Guided Dual Neural ODEs in Open Systemscited by 26
  27. 2021Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Spacecited by 25
  28. 2021Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)cited by 25
  29. 2022Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computingcited by 24
  30. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  31. 2022PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equationscited by 21
  32. 2021Combining Physics and Deep Learning to learn Continuous-Time Dynamics Modelscited by 20
  33. 2022Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equationscited by 20
  34. 2023Learning Reduced-Order Models for Cardiovascular Simulations with Graph Neural Networkscited by 20
  35. 2024Dynamical Measure Transport and Neural PDE Solvers for Samplingcited by 20
  36. 2021Parametric Complexity Bounds for Approximating PDEs with Neural Networkscited by 19
  37. 2022Physics-informed neural networks for PDE-constrained optimization and controlcited by 19
  38. 2021Value Iteration in Continuous Actions, States and Timecited by 18

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