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