Error estimates of residual minimization using neural networks for linear PDEs

TL;DR

This paper develops an abstract framework to analyze the convergence and error estimates of residual minimization methods, including physics-informed neural networks, for solving linear PDEs using neural networks.

Contribution

It introduces a unified theoretical framework for residual minimization methods with neural networks, covering both continuous and discrete, strong and weak formulations.

Findings

Provides error estimates for residual minimization methods with neural networks.

Includes analysis applicable to physics-informed neural networks.

Establishes convergence results for various formulations.

Abstract

We propose an abstract framework for analyzing the convergence of least-squares methods based on residual minimization when feasible solutions are neural networks. With the norm relations and compactness arguments, we derive error estimates for both continuous and discrete formulations of residual minimization in strong and weak forms. The formulations cover recently developed physics-informed neural networks based on strong and variational formulations.