The Deep Learning Galerkin Method for the General Stokes Equations

This paper considers the general stokes problems applying the Deep learning Galerkin Method (DGM) and gives the convergence of the DGM which contains two parts. First, guided by data and physical laws, depending on the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L^{2} $$\end{document} error we construct an objective function and control the performance of the approximation solution by minimizing the objective function in which the prior knowledge of PDEs and data are encoded. Then, we prove the convergence of the neural network to the exact solution. In particular, due to it is mesh free, the DGM can reduce the computational complexity and achieve the competitive results especially in face of the high dimensional problems. With this, compared with traditional numerical methods, numerical results verify the theoretical analysis and show the applicability and effectiveness of the proposed method.