Convergence Analysis of a Quasi-Monte Carlo-Based Deep Learning Algorithm for Solving Partial Differential Equations

被引:1
|
作者
Fu, Fengjiang [1 ]
Wang, Xiaoqun [1 ]
机构
[1] Tsinghua Univ, Dept Math Sci, Beijing 100084, Peoples R China
基金
中国国家自然科学基金;
关键词
Deep Ritz method; quasi-Monte Carlo; Poisson equation; static Schrodinger equation; error bound; NEURAL-NETWORKS;
D O I
10.4208/nmtma.OA-2022-0166
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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 Schrodinger 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.
引用
收藏
页码:668 / 700
页数:33
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