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

arXiv: https://arxiv.org/abs/2106.14473 · HTML: https://mlsanity.com/paper/2106.14473/

## 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 (most cited first, 18 of 18)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Analysis 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 Equations](https://mlsanity.com/paper/1809.03062/) (2018)
- [Deep learning observables in computational fluid dynamics](https://mlsanity.com/paper/1903.03040/) (2019)
- [Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data](https://mlsanity.com/paper/1808.04327/) (2018)
- [Physics Informed Neural Networks for Simulating Radiative Transfer](https://mlsanity.com/paper/2009.13291/) (2020)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks](https://mlsanity.com/paper/2008.05730/) (2020)
- [Space-time deep neural network approximations for high-dimensional partial differential equations](https://mlsanity.com/paper/2006.02199/) (2020)

## Built on by (most cited first, 30 of 57)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology](https://mlsanity.com/paper/2211.08939/) (2022)
- [Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.16371/) (2025)
- [Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?](https://mlsanity.com/paper/2405.04230/) (2024)
- [wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws](https://mlsanity.com/paper/2207.08483/) (2022)
- [An operator preconditioning perspective on training in physics-informed machine learning](https://mlsanity.com/paper/2310.05801/) (2023)
- [Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems](https://mlsanity.com/paper/2112.03749/) (2021)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networks](https://mlsanity.com/paper/2306.02816/) (2023)
- [Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Loss](https://mlsanity.com/paper/2402.00152/) (2024)
- [Preconditioning for Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.00531/) (2024)
- [Solving Poisson Equations using Neural Walk-on-Spheres](https://mlsanity.com/paper/2406.03494/) (2024)
- [Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Study](https://mlsanity.com/paper/2503.17379/) (2025)
- [On the Generalization of PINNs outside the training domain and the Hyperparameters influencing it](https://mlsanity.com/paper/2302.07557/) (2023)
- [Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations](https://mlsanity.com/paper/2405.14099/) (2024)
- [Binary structured physics-informed neural networks for solving equations with rapidly changing solutions](https://mlsanity.com/paper/2401.12806/) (2024)
- [Deep Neural Networks with General Activations: Super-Convergence in Sobolev Norms](https://mlsanity.com/paper/2508.05141/) (2025)
- [Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs](https://mlsanity.com/paper/2303.11205/) (2023)
- [Deep adaptive sampling for surrogate modeling without labeled data](https://mlsanity.com/paper/2402.11283/) (2024)
- [Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms](https://mlsanity.com/paper/2311.04779/) (2023)
- [Beyond Derivative Pathology of PINNs: Variable Splitting Strategy with Convergence Analysis](https://mlsanity.com/paper/2409.20383/) (2024)
- [Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles](https://mlsanity.com/paper/2505.06459/) (2025)
- [Neural tangent kernel analysis of PINN for advection-diffusion equation](https://mlsanity.com/paper/2211.11716/) (2022)
- [Physics-informed deep learning and compressive collocation for high-dimensional diffusion-reaction equations: practical existence theory and numerics](https://mlsanity.com/paper/2406.01539/) (2024)
- [Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systems](https://mlsanity.com/paper/2407.08868/) (2024)
- [Exact and approximate error bounds for physics-informed neural networks](https://mlsanity.com/paper/2411.13848/) (2024)

## 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.