An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications (Communications in Computational Physics, 2023)
A new and efficient neural-network and finite-difference hybrid method isdeveloped for solving Poisson equation in a regular domain with jump discontinuitieson embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additionaltreatment accounting for the jump discontinuities must be employed. Here, we aim toelevate such an extra effort to ease our implementation by machine learning methodology. The key idea is to decompose the solution into singular and regular parts. Theneural network learning machinery incorporating the given jump conditions finds thesingular solution, while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions. Regardless of the interface geometry, these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation, making the hybrid method easyto implement and efficient. The two- and three-dimensional numerical results showthat the present hybrid method preserves second-order accuracy for the solution andits derivatives, and it is comparable with the traditional immersed interface method inthe literature. As an application, we solve the Stokes equations with singular forces todemonstrate the robustness of the present method.
Keywords: Neural networks; sharp interface method; fast direct solver; elliptic interface problem; Stokes equations.