An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications
Year: 2023
Author: Wei-Fan Hu, Te-Sheng Lin, Yu-Hau Tseng, Ming-Chih Lai
Communications in Computational Physics, Vol. 33 (2023), Iss. 4 : pp. 1090–1105
Abstract
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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2022-0284
Communications in Computational Physics, Vol. 33 (2023), Iss. 4 : pp. 1090–1105
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 16
Keywords: Neural networks sharp interface method fast direct solver elliptic interface problem Stokes equations.
Author Details
-
Two novel discontinuity-removing PINNs for solving variable coefficient elliptic interface problems on curved surfaces
Li, Hongji | Fan, Haolong | Tan, ZhijunComputer Methods in Applied Mechanics and Engineering, Vol. 435 (2025), Iss. P.117637
https://doi.org/10.1016/j.cma.2024.117637 [Citations: 1] -
An immersed interface neural network for elliptic interface problems
Zhang, Xinru | He, QiaolinJournal of Computational and Applied Mathematics, Vol. 459 (2025), Iss. P.116372
https://doi.org/10.1016/j.cam.2024.116372 [Citations: 0] -
Novel and general discontinuity-removing PINNs for elliptic interface problems
Fan, Haolong | Tan, ZhijunJournal of Computational Physics, Vol. 529 (2025), Iss. P.113861
https://doi.org/10.1016/j.jcp.2025.113861 [Citations: 1] -
The Hard-Constraint PINNs for Interface Optimal Control Problems
Lai, Ming-Chih | Song, Yongcun | Yuan, Xiaoming | Yue, Hangrui | Zeng, TianyouSIAM Journal on Scientific Computing, Vol. 47 (2025), Iss. 3 P.C601
https://doi.org/10.1137/23M1601249 [Citations: 0] -
A High-Order Hybrid Approach Integrating Neural Networks and Fast Poisson Solvers for Elliptic Interface Problems
Ren, Yiming | Zhao, ShanComputation, Vol. 13 (2025), Iss. 4 P.83
https://doi.org/10.3390/computation13040083 [Citations: 1]