A Priori Error Estimate of Deep Mixed Residual Method for Elliptic PDEs

• Published in: Journal of scientific computing Vol. 98; no. 2; p. 44
• Main Authors: Li, Lingfeng, Tai, Xue-Cheng, Yang, Jiang, Zhu, Quanhui
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2024; Springer Nature B.V
• Subjects: Algorithms; Approximation; Computational Mathematics and Numerical Analysis; Elliptic differential equations; Elliptic functions; Error analysis; Finite element analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Neural networks; Partial differential equations; Ritz method; Theoretical; Training; Deep mixed residual method; 65N15; 68Q25; Generalization error; Rademacher complexity; Neural networks approximation
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-023-02432-x

In this work, we derive a priori error estimate of the deep mixed residual method (DMRM) when solving some elliptic partial differential equations (PDEs). DMRM is a new deep-learning based method for solving PDEs and it has been shown to be efficient and accurate in previous studies. Our work is the first theoretical study of DMRM. We prove that the neural network solutions will converge if we increase the training samples and network size without any constraint on the ratio of training samples to the network size. Besides, our results suggest that the DMRM can approximate the Laplacian of the solution by the intermediate auxiliary variable, which is dismissing in the deep Ritz method. It is verified by the numerical experiments.