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：本文提出一种混合神经网络与MAC（Marker-And-Cell）格式，用于求解规则域内嵌界面上含奇异力的Stokes方程。众所周知，解变量（压力与速度）在界面两侧呈现非光滑行为，因此有限差分格式需在界面附近进行额外离散处理以降低局部截断误差阶数。本文提出的混合方法规避了这一附加困难，通过结合神经网络的表达能力和有限差分格式的收敛性，在简化代码实现的同时达到良好精度。其核心思想是将解分解为奇异部分和正则部分：利用融入给定跳跃条件的神经网络学习机制求解奇异部分解，而正则部分解则通过标准MAC格式结合相应边界条件获取。二维与三维数值结果表明，所提混合方法对速度场具有二阶收敛精度，对压力场具有一阶收敛精度，与文献中传统浸入界面方法性能相当。