mlsanityData to 14 Oct 2026

Generic bounds on the approximation error for physics-informed (and) operator learning

arXiv 2205.11393 ↗First posted 23 May 2022cs.LG cs.NA math.NAcited by 65

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Abstract (arXiv)

We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator learning. These bounds guarantee that PINNs and (physics-informed) DeepONets or FNOs will efficiently approximate the underlying solution or solution operator of generic partial differential equations (PDEs). Our framework utilizes existing neural network approximation results to obtain bounds on more involved learning architectures for PDEs. We illustrate the general framework by deriving the first rigorous bounds on the approximation error of physics-informed operator learning and by showing that PINNs (and physics-informed DeepONets and FNOs) mitigate the curse of dimensionality in approximating nonlinear parabolic PDEs.

Key facts. Cited by 65 papers in the map (data through 14 Oct 2026). Builds on 37 of its 90 references that are in the map. First posted on arXiv 23 May 2022.

Builds on

37 of 90 references.

References · 37

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  3. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 776
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  5. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  6. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  7. 2016Error bounds for approximations with deep ReLU networkscited by 574
  8. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
  9. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  10. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  11. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 349
  12. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
25 more
  1. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  2. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 191
  3. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 116
  4. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 102
  5. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  6. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 89
  7. 2021On the approximation of functions by tanh neural networkscited by 81
  8. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
  9. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  10. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 57
  11. 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
  12. 2018Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equationscited by 45
  13. 2019Deep learning observables in computational fluid dynamicscited by 38
  14. 2018Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Datacited by 36
  15. 2020Physics Informed Neural Networks for Simulating Radiative Transfercited by 35
  16. 2018A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficientscited by 31
  17. 2020On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEscited by 31
  18. 2020Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networkscited by 27
  19. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  20. 2021Parametric Complexity Bounds for Approximating PDEs with Neural Networkscited by 19
  21. 2019Full error analysis for the training of deep neural networkscited by 17
  22. 2022Certified machine learning: A posteriori error estimation for physics-informed neural networkscited by 15
  23. 2020Space-time deep neural network approximations for high-dimensional partial differential equationscited by 9
  24. 2019On Universal Approximation by Neural Networks with Uniform Guarantees on Approximation of Infinite Dimensional Mapscited by 8
  25. 2019Space-time error estimates for deep neural network approximations for differential equationscited by 6

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 7; 2023: 18; 2024: 17; 2025: 11; 2026: 12.

65 papers.

Sort:

  1. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 214
  2. 2022Physics-Informed Deep Neural Operator Networkscited by 113
  3. 2024Operator Learning: Algorithms and Analysiscited by 85
  4. 2023A Mathematical Guide to Operator Learningcited by 76
  5. 2022Reliable extrapolation of deep neural operators informed by physics or sparse observationscited by 62
  6. 2023Towards Stability of Autoregressive Neural Operatorscited by 59
  7. 2024CViT: Continuous Vision Transformer for Operator Learningcited by 51
  8. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  9. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  10. 2023Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equationcited by 32
  11. 2024PINNACLE: PINN Adaptive ColLocation and Experimental points selectioncited by 32
  12. 2022Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuitiescited by 30
38 more
  1. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  2. 2023HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetworkcited by 26
  3. 2023Rethinking materials simulations: Blending direct numerical simulations with neural operatorscited by 25
  4. 2023Recent Developments in Machine Learning Methods for Stochastic Control and Gamescited by 24
  5. 2023Error Bounds for Learning with Vector-Valued Random Featurescited by 23
  6. 2022Transformer Meets Boundary Value Inverse Problemscited by 22
  7. 2023Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problemscited by 13
  8. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 12
  9. 2023CS4ML: A general framework for active learning with arbitrary data based on Christoffel functionscited by 11
  10. 2024The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measurescited by 11
  11. 2023Efficient Error Certification for Physics-Informed Neural Networkscited by 10
  12. 2022Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNscited by 8
  13. 2025Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modelingcited by 8
  14. 2023Scattering with Neural Operatorscited by 7
  15. 2024A Pretraining-Finetuning Computational Framework for Material Homogenizationcited by 7
  16. 2024Component Fourier Neural Operator for Singularly Perturbed Differential Equationscited by 7
  17. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 7
  18. 2023ST-PINN: A Self-Training Physics-Informed Neural Network for Partial Differential Equationscited by 6
  19. 2023Size Lowerbounds for Deep Operator Networkscited by 6
  20. 2024An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learningcited by 6
  21. 2024Deep adaptive sampling for surrogate modeling without labeled datacited by 5
  22. 2025Prediction error certification for PINNs: Theory, computation, and application to Stokes flowcited by 4
  23. 2025Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundlescited by 3
  24. 2025Simultaneously Solving Infinitely Many LQ Mean Field Games In Hilbert Spaces: The Power of Neural Operatorscited by 3
  25. 2022Error analysis for deep neural network approximations of parametric hyperbolic conservation lawscited by 2
  26. 2023Auxiliary-Tasks Learning for Physics-Informed Neural Network-Based Partial Differential Equations Solvingcited by 2
  27. 2024Error estimation for physics-informed neural networks with implicit Runge-Kutta methodscited by 2
  28. 2024Leveraging Interpolation Models and Error Bounds for Verifiable Scientific Machine Learningcited by 2
  29. 2025Noisy PDE Training Requires Bigger PINNscited by 2
  30. 2023Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Modelscited by 1
  31. 2024Neural Parameter Regression for Explicit Representations of PDE Solution Operatorscited by 1
  32. 2024Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEscited by 1
  33. 2025Physics-Informed Deep B-Spline Networkscited by 1
  34. 2025Maximal Update Parametrization and Zero-Shot Hyperparameter Transfer for Fourier Neural Operatorscited by 1
  35. 2025Fredholm Neural Networks for inverse problems in elliptic PDEscited by 1
  36. 2026Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEscited by 1
  37. 2026Physics-Informed Laplace Neural Operator for Solving Partial Differential Equationscited by 1
  38. 2026A Deep Risk Estimator for Known Operator Learningcited by 1

50 most cited of 65 All 65 (JSON)