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 finite difference method 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,在嵌入的不规则界面上以跳动不连续的方式在常规域中解决Poisson方程式,在嵌入的不规则界面上解决跳跃不连续的Poisson方程式。由于解决方案在整个界面中具有低常态性,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解释跳跃不连续的问题。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,而标准有限差异方法则用来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,这两项任务只需要在监督下学习功能近似近似和Poisson方程式的快速直接解析器,使混合方法易于实施和效率。二维和三维的数值结果显示,目前的混合方法保持解决方案及其衍生物的二阶精确性,并且与文献中传统的混合界面方法相似。作为一个应用,我们用奇形方程式用目前的方法解算方方方方方方程式解决。