Error Analysis of the Deep Mixed Residual Method for High-order Elliptic Equations

TL;DR

This paper provides an a priori error analysis of the Deep Mixed Residual method (MIM) for high-order elliptic equations, demonstrating its effectiveness and reduced regularity requirements compared to existing methods.

Contribution

It offers the first comprehensive error analysis of MIM for high-order elliptic equations, including boundedness, coercivity, and error decomposition, with results independent of dimensionality.

Findings

MIM reduces regularity requirements for activation functions.

Error bounds are derived based on training samples and network size.

MIM effectively solves high-order elliptic equations without curse of dimensionality.

Abstract

This paper presents an a priori error analysis of the Deep Mixed Residual method (MIM) for solving high-order elliptic equations with non-homogeneous boundary conditions, including Dirichlet, Neumann, and Robin conditions. We examine MIM with two types of loss functions, referred to as first-order and second-order least squares systems. By providing boundedness and coercivity analysis, we leverage C\'{e}a's Lemma to decompose the total error into the approximation, generalization, and optimization errors. Utilizing the Barron space theory and Rademacher complexity, an a priori error is derived regarding the training samples and network size that are exempt from the curse of dimensionality. Our results reveal that MIM significantly reduces the regularity requirements for activation functions compared to the deep Ritz method, implying the effectiveness of MIM in solving high-order equations.