Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equations

TL;DR

This paper demonstrates that integrating quasi-Monte Carlo methods with deep learning algorithms for PDEs improves convergence rates and reduces variance, leading to more accurate and efficient solutions.

Contribution

It introduces a QMC-based Deep Ritz Method for PDEs, providing theoretical error bounds and empirical evidence of superior convergence and variance reduction.

Findings

QMC-based DRM achieves smaller error bounds than traditional DRM.

Numerical experiments confirm faster convergence with QMC methods.

Variance of gradient estimators is significantly reduced using QMC.

Abstract

Deep learning methods have achieved great success in solving partial differential equations (PDEs), where the loss is often defined as an integral. The accuracy and efficiency of these algorithms depend greatly on the quadrature method. We propose to apply quasi-Monte Carlo (QMC) methods to the Deep Ritz Method (DRM) for solving the Neumann problems for the Poisson equation and the static Schr\"{o}dinger equation. For error estimation, we decompose the error of using the deep learning algorithm to solve PDEs into the generalization error, the approximation error and the training error. We establish the upper bounds and prove that QMC-based DRM achieves an asymptotically smaller error bound than DRM. Numerical experiments show that the proposed method converges faster in all cases and the variances of the gradient estimators of randomized QMC-based DRM are much smaller than those of DRM, which illustrates the superiority of QMC in deep learning over MC.