A Stable Generalized Finite Element Method Coupled with Deep Neural Network for Interface Problems with Discontinuities

The stable generalized finite element method (SGFEM) is an improved version of generalized or extended FEM (GFEM/XFEM), which (i) uses simple and unfitted meshes, (ii) reaches optimal convergence orders, and (iii) is stable and robust in the sense that conditioning is of the same order as that of FEM and does not get bad as interfaces approach boundaries of elements. This paper designs the SGFEM for the discontinuous interface problem (DIP) by coupling a deep neural network (DNN). The main idea is to construct a function using the DNN, which captures the discontinuous interface condition, and transform the DIP to an (approximately) equivalent continuous interface problem (CIP) based on the DNN function such that the SGFEM for CIPs can be applied. The SGFEM for the DIP is a conforming method that maintains the features (i)–(iii) of SGFEM and is free from penalty terms. The approximation error of the proposed SGFEM is analyzed mathematically, which is split into an error of SGFEM of the CIP and a learning error of the DNN. The learning dimension of DNN is one dimension less than that of the domain and can be implemented efficiently. It is known that the DNN enjoys advantages in nonlinear approximations and high-dimensional problems. Therefore, the proposed SGFEM coupled with the DNN has great potential in the high-dimensional interface problem with interfaces of complex geometries. Numerical experiments verify the efficiency and optimal convergence of the proposed method.