Interface problems depict many fundamental physical phenomena and widely apply in the engineering. However, it is challenging to develop efficient fully decoupled numerical methods for solving degenerate interface problems in which the coefficient of a PDE is discontinuous and greater than or equal to zero on the interface. The main motivation in this paper is to construct fully decoupled numerical methods for solving nonlinear degenerate interface problems with ``double singularities". An efficient fully decoupled numerical method is proposed for nonlinear degenerate interface problems. The scheme combines deep neural network on the singular subdomain with finite difference method on the regular subdomain. The key of the new approach is to split nonlinear degenerate partial differential equation with interface into two independent boundary value problems based on deep learning. The outstanding advantages of the proposed schemes are that not only the convergence order of the degenerate interface problems on whole domain is determined by the finite difference scheme on the regular subdomain, but also can calculate $\mathbf{VERY}$ $\mathbf{BIG}$ jump ratio(such as $10^{12}:1$ or $1:10^{12}$) for the interface problems including degenerate and non-degenerate cases. The expansion of the solutions does not contains any undetermined parameters in the numerical method. In this way, two independent nonlinear systems are constructed in other subdomains and can be computed in parallel. The flexibility, accuracy and efficiency of the methods are validated from various experiments in both 1D and 2D. Specially, not only our method is suitable for solving degenerate interface case, but also for non-degenerate interface case. Some application examples with complicated multi-connected and sharp edge interface examples including degenerate and nondegenerate cases are also presented.

翻译：界面问题描述了众多基本物理现象，并在工程领域得到广泛应用。然而，针对偏微分方程系数在界面上不连续且大于等于零的退化界面问题，开发高效的完全解耦数值方法仍具挑战。本文的主要动机是构造求解具有"双重奇异性"的非线性退化界面问题的完全解耦数值方法。我们提出了一种求解非线性退化界面问题的高效完全解耦数值方法，该方法将奇异子域上的深度神经网络与正则子域上的有限差分法相结合。新方法的关键在于基于深度学习将带界面的非线性退化偏微分方程分裂为两个独立的边值问题。所提方案的突出优势在于：不仅全区域上退化界面问题的收敛阶由正则子域上的有限差分格式决定，而且能够计算包含退化和非退化情形的界面问题中$\mathbf{极大}$跳跃比（如$10^{12}:1$或$1:10^{12}$）。数值方法中解的展开不包含任何未定参数。通过这种方式，可在其他子域构造两个独立的非线性系统并实现并行计算。一维和二维的各类实验验证了该方法的灵活性、精度和效率。特别地，我们的方法不仅适用于退化界面情形，也适用于非退化界面情形。文中还给出了包含退化和非退化情形的复杂多连通及尖锐边界界面问题的应用实例。