Convergence of physics-informed neural networks applied to linear second-order elliptic interface problems

TL;DR

This paper provides a convergence analysis of physics-informed neural networks (PINNs) for second-order elliptic interface problems, demonstrating their ability to approximate solutions with boundary and interface jumps as sample size grows.

Contribution

The paper introduces a convergence framework for PINNs applied to elliptic interface problems, incorporating domain decomposition and gradient-enhanced strategies.

Findings

Neural network sequences converge to the true solution in $H^2$ norm.

The convergence is established as the number of samples increases.

Numerical experiments validate the theoretical results.

Abstract

With the remarkable empirical success of neural networks across diverse scientific disciplines, rigorous error and convergence analysis are also being developed and enriched. However, there has been little theoretical work focusing on neural networks in solving interface problems. In this paper, we perform a convergence analysis of physics-informed neural networks (PINNs) for solving second-order elliptic interface problems. Specifically, we consider PINNs with domain decomposition technologies and introduce gradient-enhanced strategies on the interfaces to deal with boundary and interface jump conditions. It is shown that the neural network sequence obtained by minimizing a Lipschitz regularized loss function converges to the unique solution to the interface problem in $H^2$ as the number of samples increases. Numerical experiments are provided to demonstrate our theoretical analysis.