Continuity-preserved deep learning method for solving elliptic interface problems

The use of deep learning methods to solve partial differential equations has recently garnered significant attention. In this paper, the continuity-preserved deep learning method with the level set augmented technique proposed in Tseng et. al. (2023) is adopted to deal with interface problems with continuous solutions across the interface. Based on the probability filling argument, the approximations are proven to converge with the rate at least M-1d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M<^>{-\frac{1}{d}}$$\end{document}, where d represents the dimensionality of the problem and M is the number of sampling points. Numerical experiments are then used to verify the convergence behavior and demonstrate advantages of the proposed method. Compared to other deep learning methods and the classic finite element methods, the proposed method not only can maintain continuities of quantities, but also have much better accuracy. Finally, as for applications, the method is applied to solve the classic implicit continuum model for predicting electrostatics and solvation free energies.