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.

翻译：本文提出了一种混合神经网络与MAC（Marker-And-Cell，标记与网格）格式，用于求解规则域内嵌入界面上带有奇异力的Stokes方程。众所周知，解变量（压力与速度）在界面处呈现非光滑行为，因此有限差分格式中需在界面附近进行额外的离散处理，以降低局部截断误差阶数。本文提出的混合方法避免了这一额外困难：它结合了神经网络的表达能力与有限差分格式的收敛性，既简化了代码实现，又同时实现了良好的精度。其核心思想是将解分解为奇异部分和规则部分。利用给定跳跃条件的神经网络学习机制求解奇异部分解，而标准MAC格式则用于在相应边界条件下获取规则部分解。二维和三维数值结果表明，本文提出的混合方法在速度场上实现了二阶精度收敛，在压力场上实现了一阶精度收敛，其性能与文献中传统的浸入界面法相当。