In this paper, we present a novel hybrid method for solving a Stokes interface problem in a regular domain with jump discontinuities on an interface. Our approach combines the expressive power of neural networks with the convergence of finite difference schemes to achieve efficient implementations and accurate results. The key concept of our method is to decompose the solution into two parts: the singular part and the regular part. We employ neural networks to approximate the singular part, which captures the jump discontinuities across the interface. We then utilize a finite difference scheme to approximate the regular part, which handles the smooth variations of the solution in that regular domain. To validate the effectiveness of our approach, we present two- and three-dimensional examples to demonstrate the accuracy and convergence of the proposed method, and show that our proposed hybrid method provides an innovative and reliable approach to tackle Stokes interface problems.

翻译：本文提出了一种新颖的混合方法，用于求解具有界面跳跃间断性的规则域上的斯托克斯界面问题。我们的方法结合了神经网络的表达能力与有限差分格式的收敛性，以实现高效计算与精确结果。该方法的核心思想是将解分解为两部分：奇异部分与正则部分。我们采用神经网络逼近奇异部分，以捕捉跨越界面的跳跃间断性；随后利用有限差分格式逼近正则部分，以处理规则域中解的平滑变化。为验证方法的有效性，我们给出了二维和三维算例，证明了所提方法的精确性与收敛性，并表明该混合方法为求解斯托克斯界面问题提供了一种创新且可靠的途径。