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Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs

arXiv 2006.16144 ↗First posted 29 Jun 2020math.NA cs.LG cs.NAcited by 115

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Abstract (arXiv)

Physics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of training samples. This abstract framework is illustrated with several examples of nonlinear PDEs. Numerical experiments, validating the proposed theory, are also presented.

Key facts. Cited by 115 papers in the map (data through 7 Oct 2026). First posted on arXiv 29 Jun 2020.

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Citing papers per year: 2021: 20; 2022: 15; 2023: 20; 2024: 19; 2025: 19; 2026: 22.

115 papers.

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  1. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 470
  2. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 142
  3. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 109
  4. 2021Improved architectures and training algorithms for deep operator networkscited by 85
  5. 2021On the approximation of functions by tanh neural networkscited by 81
  6. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 78
  7. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  8. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  9. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 57
  10. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
  11. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  12. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
38 more
  1. 2024PINNACLE: PINN Adaptive ColLocation and Experimental points selectioncited by 31
  2. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  3. 2021A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equationscited by 28
  4. 2023NVFi: Neural Velocity Fields for 3D Physics Learning from Dynamic Videoscited by 28
  5. 2023An operator preconditioning perspective on training in physics-informed machine learningcited by 27
  6. 2024RoPINN: Region Optimized Physics-Informed Neural Networkscited by 26
  7. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  8. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  9. 2021Hybrid FEM-NN models: Combining artificial neural networks with the finite element methodcited by 24
  10. 2021Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Controlcited by 24
  11. 2023Physics-informed neural networks for solving forward and inverse problems in complex beam systemscited by 24
  12. 2023Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networkscited by 23
  13. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  14. 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
  15. 2022Variationally Mimetic Operator Networkscited by 19
  16. 2023Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEscited by 19
  17. 2021A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equationscited by 18
  18. 2022Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problemscited by 18
  19. 2021Variational Physics Informed Neural Networks: the role of quadratures and test functionscited by 17
  20. 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
  21. 2022IAE-Net: Integral Autoencoders for Discretization-Invariant Learningcited by 15
  22. 2022Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspectivecited by 15
  23. 2021An artificial neural network approach to bifurcating phenomena in computational fluid dynamicscited by 11
  24. 2024Physics-informed kernel learningcited by 11
  25. 2025TRACE: Learning 3D Gaussian Physical Dynamics from Multi-view Videoscited by 11
  26. 2021NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEscited by 10
  27. 2023Efficient Error Certification for Physics-Informed Neural Networkscited by 10
  28. 2021Stationary Density Estimation of Ito Diffusions Using Deep Learningcited by 9
  29. 2023Multilevel CNNs for Parametric PDEscited by 9
  30. 2023Neural oscillators for generalization of physics-informed machine learningcited by 9
  31. 2024Initialization-enhanced Physics-Informed Neural Network with Domain Decomposition (IDPINN)cited by 9
  32. 2024Physics-Informed Neural Networks and Extensionscited by 8
  33. 2022Towards Size-Independent Generalization Bounds for Deep Operator Netscited by 6
  34. 2023Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEscited by 5
  35. 2024Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networkscited by 5
  36. 2023Deep Learning and Computational Physics (Lecture Notes)cited by 4
  37. 2023Convergence Analysis of the Deep Galerkin Method for Weak Solutionscited by 4
  38. 2023h-analysis and data-parallel physics-informed neural networkscited by 4

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