Error analysis of deep Ritz methods for elliptic equations

作者：Jiao, Yuling; Lai, Yanming^*; Lo, Yisu; Wang, Yang; Yang, Yunfei

来源：Analysis and Applications, 2024, 22(01): 57-87.

DOI：10.1142/S021953052350015X

• Deep Ritz method
• elliptic equations
• neural networks

Using deep neural networks to solve partial differential equations (PDEs) has attracted a lot of attention recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on the deep Ritz method (DRM) for second-order elliptic equations with Dirichlet, Neumann and Robin boundary conditions, respectively. We establish the first nonasymptotic convergence rate in H-1 norm for DRM using deep neural networks with smooth activation functions including logistic and hyperbolic tangent functions. Our results show how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of the number of training samples.

• 单位
  武汉大学