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

TL;DR

This paper provides a rigorous error analysis for physics-informed neural networks (PINNs) when approximating solutions to Kolmogorov PDEs, demonstrating bounds on errors and showing polynomial growth of network size with dimension.

Contribution

The paper derives error bounds for PINNs approximating Kolmogorov PDEs and proves polynomial growth of network size and training samples with dimension, overcoming the curse of dimensionality.

Findings

PINNs can achieve arbitrarily small residuals for Kolmogorov PDEs.

Total $L^2$-error is bounded by training error with sufficient collocation points.

Network size and training samples grow polynomially with dimension.

Abstract

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 $L^2$-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.