mlsanityData to 14 Oct 2026

Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs

arXiv 2106.14473 ↗First posted 28 Jun 2021math.NA cs.LG cs.NAcited by 57

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Abstract (arXiv)

Physics informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely Kolmogorov equations that include the heat equation and Black-Scholes equation of option pricing, as examples. We construct neural networks, whose PINN residual (generalization error) can be made as small as desired. We also prove that the total L2-error can be bounded by the generalization error, which in turn is bounded in terms of the training error, provided that a sufficient number of randomly chosen training (collocation) points is used. Moreover, we prove that the size of the PINNs and the number of training samples only grow polynomially with the underlying dimension, enabling PINNs to overcome the curse of dimensionality in this context. These results enable us to provide a comprehensive error analysis for PINNs in approximating Kolmogorov PDEs.

Key facts. Cited by 57 papers in the map (data through 14 Oct 2026). Builds on 18 of its 39 references that are in the map. First posted on arXiv 28 Jun 2021.

Builds on

18 of 39 references.

References · 18

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  3. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
  4. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  5. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 191
  6. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 116
  7. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 102
  8. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  9. 2021On the approximation of functions by tanh neural networkscited by 81
  10. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
  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
6 more
  1. 2019Deep learning observables in computational fluid dynamicscited by 38
  2. 2018Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Datacited by 36
  3. 2020Physics Informed Neural Networks for Simulating Radiative Transfercited by 35
  4. 2020On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEscited by 31
  5. 2020Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networkscited by 27
  6. 2020Space-time deep neural network approximations for high-dimensional partial differential equationscited by 9

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 2; 2022: 7; 2023: 7; 2024: 18; 2025: 13; 2026: 10.

57 papers.

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  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  2. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  3. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  4. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  5. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 58
  6. 2022Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodologycited by 44
  7. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 42
  8. 2024Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?cited by 31
  9. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  10. 2023An operator preconditioning perspective on training in physics-informed machine learningcited by 27
  11. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  12. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
38 more
  1. 2023MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networkscited by 17
  2. 2024Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Losscited by 14
  3. 2024Preconditioning for Physics-Informed Neural Networkscited by 13
  4. 2024Solving Poisson Equations using Neural Walk-on-Spherescited by 11
  5. 2025Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Studycited by 11
  6. 2023On the Generalization of PINNs outside the training domain and the Hyperparameters influencing itcited by 10
  7. 2024Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equationscited by 9
  8. 2024Binary structured physics-informed neural networks for solving equations with rapidly changing solutionscited by 8
  9. 2025Deep Neural Networks with General Activations: Super-Convergence in Sobolev Normscited by 6
  10. 2023Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEscited by 5
  11. 2024Deep adaptive sampling for surrogate modeling without labeled datacited by 5
  12. 2023Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Normscited by 4
  13. 2024Beyond Derivative Pathology of PINNs: Variable Splitting Strategy with Convergence Analysiscited by 4
  14. 2025Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundlescited by 3
  15. 2022Neural tangent kernel analysis of PINN for advection-diffusion equationcited by 2
  16. 2024Physics-informed deep learning and compressive collocation for high-dimensional diffusion-reaction equations: practical existence theory and numericscited by 2
  17. 2024Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systemscited by 2
  18. 2024Exact and approximate error bounds for physics-informed neural networkscited by 2
  19. 2025DPG loss functions for learning parameter-to-solution maps by neural networkscited by 2
  20. 2024Error Estimation for Physics-informed Neural Networks Approximating Semilinear Wave Equationscited by 1
  21. 2024Correctness Verification of Neural Networks Approximating Differential Equationscited by 1
  22. 2024Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Methodcited by 1
  23. 2024Astral: training physics-informed neural networks with error majorantscited by 1
  24. 2024Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEscited by 1
  25. 2025Understanding Generalization in Physics Informed Models through Affine Variety Dimensionscited by 1
  26. 2025Physics-Informed Deep B-Spline Networkscited by 1
  27. 2025Fredholm Neural Networks for inverse problems in elliptic PDEscited by 1
  28. 2025PALQO: Physics-informed Model for Accelerating Large-scale Quantum Optimizationcited by 1
  29. 2025Complexity Dependent Error Rates for Physics-informed Statistical Learning via the Small-ball Methodcited by 1
  30. 2025HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEscited by 1
  31. 2026Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEscited by 1
  32. 2026Deep Neural networks for solving high-dimensional parabolic partial differential equationscited by 1
  33. 2026Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolutioncited by 1
  34. 2026EML Trees Are Universal Approximatorscited by 1
  35. 2023A numerical approach for the fractional Laplacian via deep neural networkscited by 0
  36. 2024Feynman-Kac Operator Expectation Estimatorcited by 0
  37. 2024Parallel-in-Time Solutions with Random Projection Neural Networkscited by 0
  38. 2025Convergence of physics-informed neural networks modeling time-harmonic wave fieldscited by 0

50 most cited of 57 All 57 (JSON)