@Article{CiCP-33-4,
author = {Wei-Fan, Hu and Te-Sheng, Lin and Yu-Hau, Tseng and Ming-Chih, Lai},
title = {An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications},
journal = {Communications in Computational Physics},
year = {2023},
volume = {33},
number = {4},
pages = {1090--1105},
abstract = {<p style="text-align: justify;">A new and efficient neural-network and finite-difference hybrid method is
developed for solving Poisson equation in a regular domain with jump discontinuities
on embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additional
treatment accounting for the jump discontinuities must be employed. Here, we aim to
elevate 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. The
neural network learning machinery incorporating the given jump conditions finds the
singular 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 easy
to implement and efficient. The two- and three-dimensional numerical results show
that the present hybrid method preserves second-order accuracy for the solution and
its derivatives, and it is comparable with the traditional immersed interface method in
the literature. As an application, we solve the Stokes equations with singular forces to
demonstrate the robustness of the present method.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2022-0284},
url = {https://global-sci.com/article/79443/an-efficient-neural-network-and-finite-difference-hybrid-method-for-elliptic-interface-problems-with-applications}
}