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.

翻译：针对具有嵌入不规则界面上跳跃间断性的规则域中的泊松方程，提出了一种新型高效的神经网络-有限差分混合方法。由于解在界面处具有低正则性，在采用有限差分离散化处理该问题时，必须引入额外的处理来应对跳跃间断性。本文旨在通过机器学习方法消除这种额外负担，从而简化实现过程。其核心思想是将解分解为奇异部分和正则部分。利用融合给定跳跃条件的神经网络学习机制求解奇异部分，同时采用标准五点拉普拉斯离散化结合相应边界条件获得正则解。无论界面几何形状如何，这两个任务仅需通过监督学习进行函数逼近，并利用快速直接求解器求解泊松方程，使得该混合方法易于实现且高效。二维与三维数值结果表明，本混合方法对解及其导数保持了二阶精度，并与文献中传统浸入界面方法性能相当。作为应用实例，我们求解了具有奇异力的斯托克斯方程，验证了本方法的鲁棒性。