TY - JOUR
T1 - A Hybrid Neural-Network and MAC Scheme for Stokes Interface Problems
AU - Chang , Che-Chia
AU - Dai , Chen-Yang
AU - Hu , Wei-Fan
AU - Lin , Te-Sheng
AU - Lai , Ming-Chih
JO - East Asian Journal on Applied Mathematics
VL - 3
SP - 490
EP - 506
PY - 2024
DA - 2024/05
SN - 14
DO - http://doi.org/10.4208/eajam.2024-006.060424
UR - https://global-sci.org/intro/article_detail/eajam/23158.html
KW - Stokes interface problems, neural networks, MAC scheme, hybrid method.
AB - <p style="text-align: justify;">In this paper, we present a hybrid neural-network and MAC (Marker-And-Cell) scheme for solving Stokes equations with singular forces on an embedded interface in regular domains. As known, the solution variables (the pressure and velocity)
exhibit non-smooth behaviors across the interface so extra discretization efforts must be
paid near the interface in order to have small order of local truncation errors in finite
difference schemes. The present hybrid approach avoids such additional difficulty. It
combines the expressive power of neural networks with the convergence of finite difference schemes to ease the code implementation and to achieve good accuracy at the same
time. 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 part solution, while the standard MAC scheme is used to obtain the regular part
solution with associated boundary conditions. The two- and three-dimensional numerical results show that the present hybrid method converges with second-order accuracy
for the velocity and first-order accuracy for the pressure, and it is comparable with the
traditional immersed interface method in literature.</p>