# Stochastic approach for elliptic problems in perforated domains

arXiv: https://arxiv.org/abs/2403.11385 · HTML: https://mlsanity.com/paper/2403.11385/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 37 references that are in the map.
- First posted on arXiv 18 Mar 2024.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Generalized Multiscale Finite Element Methods (GMsFEM)](https://mlsanity.com/paper/1301.2866/) (2013)
- [A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networks](https://mlsanity.com/paper/2001.06145/) (2020)
- [A Neural Network Approach for Homogenization of Multiscale Problems](https://mlsanity.com/paper/2206.02032/) (2022)
- [An analysis of the derivative-free loss method for solving PDEs](https://mlsanity.com/paper/2309.16829/) (2023)

## Abstract (arXiv)

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.