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

arXiv 2007.01138 ↗First posted 29 Jun 2020math.NA cs.LG cs.NAcited by 101

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Abstract (arXiv)

Physics informed neural networks (PINNs) have recently been very successfully applied for efficiently approximating inverse problems for PDEs. We focus on a particular class of inverse problems, the so-called data assimilation or unique continuation problems, and prove rigorous estimates on the generalization error of PINNs approximating them. An abstract framework is presented and conditional stability estimates for the underlying inverse problem are employed to derive the estimate on the PINN generalization error, providing rigorous justification for the use of PINNs in this context. The abstract framework is illustrated with examples of four prototypical linear PDEs. Numerical experiments, validating the proposed theory, are also presented.

Key facts. Cited by 101 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: 10; 2022: 21; 2023: 23; 2024: 25; 2025: 12; 2026: 10.

101 papers.

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  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  2. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 470
  3. 2022U-NO: U-shaped Neural Operatorscited by 219
  4. 2021Improved architectures and training algorithms for deep operator networkscited by 85
  5. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 75
  6. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  7. 2024PirateNets: Physics-informed Deep Learning with Residual Adaptive Networkscited by 70
  8. 2023Can Physics-Informed Neural Networks beat the Finite Element Method?cited by 68
  9. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  10. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  11. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
  12. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
38 more
  1. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 41
  2. 2022Implicit Neural Spatial Representations for Time-dependent PDEscited by 35
  3. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  4. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  5. 2021A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equationscited by 28
  6. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  7. 2023Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategiescited by 20
  8. 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
  9. 2022Variationally Mimetic Operator Networkscited by 19
  10. 2022Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problemscited by 18
  11. 2021Variational Physics Informed Neural Networks: the role of quadratures and test functionscited by 17
  12. 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
  13. 2021Error Estimates for the Deep Ritz Method with Boundary Penaltycited by 14
  14. 2022Physics-Informed Neural Networks for Shell Structurescited by 12
  15. 2024Preconditioning for Physics-Informed Neural Networkscited by 12
  16. 2021An artificial neural network approach to bifurcating phenomena in computational fluid dynamicscited by 11
  17. 2023Physics-informed Neural Network Estimation of Material Properties in Soft Tissue Nonlinear Biomechanical Modelscited by 11
  18. 2021NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEscited by 10
  19. 2022Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Methodcited by 10
  20. 2023On the Generalization of PINNs outside the training domain and the Hyperparameters influencing itcited by 10
  21. 2024Learning the Hodgkin-Huxley Model with Operator Learning Techniquescited by 10
  22. 2023Neural oscillators for generalization of physics-informed machine learningcited by 9
  23. 2024Inferring biological processes with intrinsic noise from cross-sectional datacited by 9
  24. 2022An unsupervised latent/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operatorcited by 7
  25. 2022A Dimension-Augmented Physics-Informed Neural Network (DaPINN) with High Level Accuracy and Efficiencycited by 7
  26. 2023A Block-Coordinate Approach of Multi-level Optimization with an Application to Physics-Informed Neural Networkscited by 7
  27. 2024Solving The Quantum Many-Body Hamiltonian Learning Problem with Neural Differential Equationscited by 7
  28. 2022Gradient-Enhanced Physics-Informed Neural Networks for Power Systems Operational Supportcited by 6
  29. 2023ST-PINN: A Self-Training Physics-Informed Neural Network for Partial Differential Equationscited by 6
  30. 2024Energy Dissipation Preserving Physics Informed Neural Network for Allen-Cahn Equationscited by 6
  31. 2025Operator Learning at Machine Precisioncited by 6
  32. 2022Self-scalable Tanh (Stan): Faster Convergence and Better Generalization in Physics-informed Neural Networkscited by 5
  33. 2023Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performancecited by 5
  34. 2023Physics-Informed Deep Learning to Reduce the Bias in Joint Prediction of Nitrogen Oxidescited by 5
  35. 2023Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Networkscited by 5
  36. 2024Separable Physics-Informed Neural Networks for the solution of elasticity problemscited by 5
  37. 2024Two-level overlapping additive Schwarz preconditioner for training scientific machine learning applicationscited by 5
  38. 2023LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometrycited by 4

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