A rate of convergence of Physics Informed Neural Networks for the linear second order elliptic PDEs

TL;DR

This paper provides a theoretical analysis of the convergence rate of Physics Informed Neural Networks (PINNs) for solving second order elliptic PDEs, establishing bounds on network size and training samples needed for accuracy.

Contribution

It introduces the first rigorous convergence analysis for PINNs applied to second order elliptic PDEs, including bounds on approximation and statistical errors.

Findings

Derived upper bounds on training samples, network depth, and width for desired accuracy.

Decomposed PINN error into approximation and statistical components.

Estimated Rademacher complexity for non-Lipschitz gradient compositions.

Abstract

In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in $C^2$ norm with $\mathrm{ReLU}^{3}$ networks (deep network with activations function $\max\{0,x^3\}$) and the statistical error is estimated by Rademacher complexity. We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with $\mathrm{ReLU}^{3}$ network, which is of immense independent interest.