Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Method

TL;DR

This paper provides a comprehensive error analysis for a three-layer neural network trained with projected gradient descent within the deep Ritz method framework, offering insights into approximation, generalization, and optimization errors for PDE solutions.

Contribution

It is the first to analyze overparameterized networks for PDEs, including error estimates and training guidance without restrictive assumptions.

Findings

Established global convergence of the training algorithm.

Derived error bounds based on sample size and network parameters.

Provided practical guidance on network design and training settings.

Abstract

Machine learning is a rapidly advancing field with diverse applications across various domains. One prominent area of research is the utilization of deep learning techniques for solving partial differential equations(PDEs). In this work, we specifically focus on employing a three-layer tanh neural network within the framework of the deep Ritz method(DRM) to solve second-order elliptic equations with three different types of boundary conditions. We perform projected gradient descent(PDG) to train the three-layer network and we establish its global convergence. To the best of our knowledge, we are the first to provide a comprehensive error analysis of using overparameterized networks to solve PDE problems, as our analysis simultaneously includes estimates for approximation error, generalization error, and optimization error. We present error bound in terms of the sample size $n$ and our work provides guidance on how to set the network depth, width, step size, and number of iterations for the projected gradient descent algorithm. Importantly, our assumptions in this work are classical and we do not require any additional assumptions on the solution of the equation. This ensures the broad applicability and generality of our results.