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：开发了一种新的高效神经网络和有限差异混合法,在嵌入的异常界面上以跳跃不连续的方式在常规域内解决Poisson方程式,在嵌入的不规则界面上解决Poisson方程式。由于解决方案在整个界面中常规性较低,因此在对该问题应用有限差异分解时,必须使用额外的治疗方法来计算跳跃不连续现象。在这里,我们的目标是通过机器学习方法提高这种额外的努力,以方便我们的实施。关键的想法是将解决方案分解成单元和常规部分。包含特定跳跃条件的神经网络学习机制找到了单一的解决方案,而标准的五点拉普拉克分解法用于获得相关边界条件的常规解决方案。不管界面几何几何形状如何,这两项任务只需要监督地学习功能近似性,并快速直接解决Poisson方程式,使混合方法易于实施和效率。二维和三维数字结果显示,目前的混合方法保存解决方案及其衍生物的二阶精度,并且与文献中传统的浸入式界面方法相似。作为应用程序,我们用单方方方方方程式来解,用单方方方方方程式来展示坚固的方法。</s>