DRM Revisited: A Complete Error Analysis

TL;DR

This paper provides a comprehensive error analysis of the Deep Ritz Method, focusing on how to choose training samples, network architecture, and optimization parameters to achieve a desired precision in solving PDEs.

Contribution

It offers a complete theoretical framework for error analysis and parameter selection in the Deep Ritz Method under over-parameterization.

Findings

Guidelines for selecting training samples and network parameters

Error bounds for the approximation of PDE solutions

Analysis of convergence behavior of the gradient descent process

Abstract

In this work, we address a foundational question in the theoretical analysis of the Deep Ritz Method (DRM) under the over-parameteriztion regime: Given a target precision level, how can one determine the appropriate number of training samples, the key architectural parameters of the neural networks, the step size for the projected gradient descent optimization procedure, and the requisite number of iterations, such that the output of the gradient descent process closely approximates the true solution of the underlying partial differential equation to the specified precision?