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

arXiv 1711.10566 ↗First posted 28 Nov 2017cs.AI cs.LG cs.NAcited by 161

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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 second part of our two-part treatise, we focus on the problem of data-driven discovery of partial differential equations. Depending on whether the available data is scattered in space-time or arranged in fixed temporal snapshots, we introduce two main classes of algorithms, namely continuous time and discrete time models. The effectiveness of our approach is demonstrated using a wide range of benchmark problems in mathematical physics, including conservation laws, incompressible fluid flow, and the propagation of nonlinear shallow-water waves.

Key facts. Cited by 161 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: 36; 2022: 26; 2023: 35; 2024: 27; 2025: 23; 2026: 14.

161 papers.

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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. 2021GRAND: Graph Neural Diffusioncited by 258
  4. 2021Multi-Objective Loss Balancing for Physics-Informed Deep Learningcited by 87
  5. 2021Beltrami Flow and Neural Diffusion on Graphscited by 84
  6. 2021POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decompositioncited by 79
  7. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  8. 2023Can Physics-Informed Neural Networks beat the Finite Element Method?cited by 68
  9. 2022When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learningcited by 65
  10. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  11. 2021Quadratic Residual Networks: A New Class of Neural Networks for Solving Forward and Inverse Problems in Physics Involving PDEscited by 43
  12. 2022Machine Learning in Aerodynamic Shape Optimizationcited by 39
38 more
  1. 2024Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networkscited by 38
  2. 2021Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow modelscited by 35
  3. 2022Applications of physics informed neural operatorscited by 33
  4. 2021A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEscited by 32
  5. 2021Latent Space Data Assimilation by using Deep Learningcited by 30
  6. 2021Adaptive deep density approximation for Fokker-Planck equationscited by 28
  7. 2022MAgNet: Mesh Agnostic Neural PDE Solvercited by 28
  8. 2021Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Spacecited by 25
  9. 2021Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)cited by 25
  10. 2021An Extensible Benchmark Suite for Learning to Simulate Physical Systemscited by 24
  11. 2023Let Quantum Neural Networks Choose Their Own Frequenciescited by 22
  12. 2022PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equationscited by 21
  13. 2021Combining Physics and Deep Learning to learn Continuous-Time Dynamics Modelscited by 20
  14. 2023Learning Reduced-Order Models for Cardiovascular Simulations with Graph Neural Networkscited by 20
  15. 2022Physics-informed neural networks for PDE-constrained optimization and controlcited by 19
  16. 2021Value Iteration in Continuous Actions, States and Timecited by 18
  17. 2023Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarkscited by 18
  18. 2023An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networkscited by 16
  19. 2022Critical Investigation of Failure Modes in Physics-informed Neural Networkscited by 15
  20. 2021Applications of deep learning in traffic congestion detection, prediction and alleviation: A surveycited by 14
  21. 2023On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Modelscited by 14
  22. 2024Knowledge-guided Machine Learning: Current Trends and Future Prospectscited by 14
  23. 2021A Deep Collocation Method for the Bending Analysis of Kirchhoff Platecited by 13
  24. 2021Physics-Augmented Learning: A New Paradigm Beyond Physics-Informed Learningcited by 11
  25. 2024Training Stiff Neural Ordinary Differential Equations with Implicit Single-Step Methodscited by 11
  26. 2021TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problemscited by 10
  27. 2022Physical Activation Functions (PAFs): An Approach for More Efficient Induction of Physics into Physics-Informed Neural Networks (PINNs)cited by 10
  28. 2023On the Generalization of PINNs outside the training domain and the Hyperparameters influencing itcited by 10
  29. 2023MINN: Learning the dynamics of differential-algebraic equations and application to battery modelingcited by 10
  30. 2021Physics-informed linear regression is competitive with two Machine Learning methods in residential building MPCcited by 9
  31. 2021NeuralPDE: Modelling Dynamical Systems from Datacited by 9
  32. 2022Deep Learning of Chaotic Systems from Partially-Observed Datacited by 9
  33. 2022Quantum-Inspired Tensor Neural Networks for Partial Differential Equationscited by 9
  34. 2023Finite Expression Methods for Discovering Physical Laws from Datacited by 9
  35. 2024Physics-Informed Neural Networks for High-Frequency and Multi-Scale Problems using Transfer Learningcited by 9
  36. 2024DUDF: Differentiable Unsigned Distance Fields with Hyperbolic Scalingcited by 9
  37. 2024Towards General Neural Surrogate Solvers with Specialized Neural Acceleratorscited by 9
  38. 2021Performance and accuracy assessments of an incompressible fluid solver coupled with a deep Convolutional Neural Networkcited by 8

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