A derivative-free loss method (DFLM) for solving PDEs in the perforated domains
Reference: Jihun Han, Yoonsang Lee, Stochastic approach for elliptic problems in perforated domain, Journal of Computational Physics, 519, 113426, 2024
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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.
Example of perforated domain
e.g., modeling in fluid flow in porous media, perforated metal material for sustanability, thermal dynamics of battery pack.
Brownian walker sampling for DFLM
(e.g., with Dirichlet and Neumann B.C. )
The DFLM solutions of the Poisson equations in the perforated domains with Dirichlet (outer boundary) and Neumann (boundaries of holes) boundary conditions.
Highlight: handling boundary conditions without imposing explicit loss functions for boundary conditions
Non-intrusive capturing the homogenized solution
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