Stochastic approach for elliptic problems in perforated domains

TL;DR

This paper introduces a neural network-based mesh-free method utilizing stochastic representations to efficiently solve elliptic PDEs in perforated domains, capturing multiscale behaviors with robustness and accuracy.

Contribution

It presents a novel derivative-free, neural network approach that effectively handles complex perforated geometries and multiscale features in elliptic PDE problems.

Findings

Method accurately captures macroscopic behavior in perforated domains

Robustness across various perforation scales demonstrated

Efficient mesh-free approach reduces computational complexity

Abstract

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.