Convergence Analysis of a Quasi-Monte Carlo-Based Deep Learning Algorithm for Solving Partial Differential Equations
Year: 2023
Author: Xiaoqun Wang, Fengjiang Fu, Xiaoqun Wang
Numerical Mathematics: Theory, Methods and Applications, Vol. 16 (2023), Iss. 3 : pp. 668–700
Abstract
Deep learning has 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 Schrödinger 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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2022-0166
Numerical Mathematics: Theory, Methods and Applications, Vol. 16 (2023), Iss. 3 : pp. 668–700
Published online: 2023-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 33
Keywords: Deep Ritz method quasi-Monte Carlo Poisson equation static Schrödinger equation error bound.