Stochastic approach for elliptic problems in perforated domains

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.

Figures (6)

• Figure 1: A schematic of a perforated domain
• Figure 2: Schematic diagram of numerical treatment for boundary conditions.
• Figure 3: Brownian sample distribution of colocation points based on the proximity to the boundary with Neumann and Dirichlet conditions on $\partial\mathcal{B}$ and $\partial \Omega$, respectively. Brownian samples are killed near $\partial \Omega$ and reflected near $\partial \mathcal{B}$, contrasting with the symmetric distribution of standard Brownian samples.
• Figure 4: Result of single perforation. (a): Finite element mesh, (b): FEM solution, (c): contour map of the FEM solution, (d): learning trajectory of DFLM, (e): DFLM approximation, (f): contour map of the DFLM solution, (g): pointwise difference.
• Figure 5: Result of various perforation sizes. (a): Finite element mesh, (b): FEM solution, (c): contour map of the FEM solution, (d): learning trajectory of DFLM, (e): DFLM approximation, (f): contour map of the DFLM solution, (g): pointwise difference.