A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations

TL;DR

This paper provides a theoretical analysis of the Deep Ritz Method, showing that its generalization error bounds for high-dimensional elliptic PDEs are independent of the dimension, under certain spectral regularity assumptions.

Contribution

It establishes dimension-independent generalization error bounds for the Deep Ritz Method applied to high-dimensional elliptic equations, based on spectral Barron space assumptions.

Findings

Generalization errors are dimension-independent under spectral Barron space assumptions.

Sufficient conditions are provided for solutions to be spectral Barron functions.

A new PDE solution theory on spectral Barron space is developed.

Abstract

This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schr\"odinger equation on the $d$-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension $d$, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.