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

作　　者：Fengjiang Fu Xiaoqun Wang 

机构地区：[1]Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China

出　　处：《Numerical Mathematics(Theory,Methods and Applications)》2023年第3期668-700,共33页高等学校计算数学学报（英文版）

基　　金：supported by the National Natural Science Foundation of China(Grant No.72071119).

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 Schr¨odinger 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.

关 键 词：Deep Ritz method quasi-Monte Carlo Poisson equation static Schrodinger equation error bound 

分 类 号：O17[理学—数学]