A wide range of applications in science and engineering involve a PDE model in a domain with perforations, such as perforated metals or air filters. Solving such perforated domain problems suffers from computational challenges related to resolving the scale imposed by the geometries of perforations. We propose a neural network-based mesh-free approach for perforated domain problems. The method is robust and efficient in capturing various configuration scales, including the averaged macroscopic behavior of the solution that involves a multiscale nature induced by small perforations. The new approach incorporates the derivative-free loss method that uses a stochastic representation or the Feynman-Kac formulation. In particular, we implement the Neumann boundary condition for the derivative-free loss method to handle the interface between the domain and perforations. A suite of stringent numerical tests is provided to support the proposed method's efficacy in handling various perforation scales.

翻译：科学与工程中的广泛应用涉及具有穿孔区域（如穿孔金属或空气过滤器）中的偏微分方程模型。解决此类穿孔区域问题面临与穿孔几何尺度相关的计算挑战。我们提出了一种基于神经网络的穿孔区域问题无网格方法。该方法在捕捉各种尺度配置（包括由小穿孔引起的多尺度性质所决定的解的宏观平均行为）方面具有鲁棒性和高效性。新方法结合了基于随机表示或Feynman-Kac公式的无导数损失方法。特别地，我们对无导数损失方法实现了Neumann边界条件，以处理区域与穿孔之间的界面。通过一系列严格的数值测试，验证了所提方法在处理各种穿孔尺度方面的有效性。