An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications  

作　　者：Wei-Fan Hu Te-Sheng Lin Yu-Hau Tseng Ming-Chih Lai 

机构地区：[1]Department of Mathematics,National Central University,Taoyuan 32001,Taiwan [2]Department of Applied Mathematics,National Yang Ming Chiao Tung University,Hsinchu 30010,Taiwan [3]Department of Applied Mathematics,National University of Kaohsiung,Kaohsiung 81148,Taiwan [4]National Center for Theoretical Sciences,National Taiwan University,Taipei 10617,Taiwan

出　　处：《Communications in Computational Physics》2023年第4期1090-1105,共16页计算物理通讯（英文）

基　　金：the supports by National Science and Technology Council,Taiwan,under the research grants 111-2115-M-008-009-MY3,111-2628-M-A49-008-MY4,111-2115-M-390-002,and 110-2115-M-A49-011-MY3,respectively;the supports by National Center for Theoretical Sciences,Taiwan.

摘　　要：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.

关 键 词：Neural networks sharp interface method fast direct solver elliptic interface problem Stokes equations 

分 类 号：O241.82[理学—计算数学]