Stochastic approach for elliptic problems in perforated domains
Jihun Han Yoonsang Lee
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.