Convergence analysis of neural networks for solving a free boundary system

TL;DR

This paper introduces a neural network-based method for solving free boundary problems, specifically a modified Hele-Shaw problem, with theoretical existence proofs and numerical verification of symmetric and asymmetric solutions.

Contribution

It presents a novel neural network discretization approach for free boundary problems, with theoretical existence results and numerical demonstrations of complex solution structures.

Findings

The neural network method successfully computes symmetric and asymmetric solutions.

The approach is validated through bifurcation analysis and numerical experiments.

Theoretical proof of the existence of solutions with this discretization.

Abstract

Free boundary problems deal with systems of partial differential equations, where the domain boundaries are apriori unknown. Due to this special characteristic, it is challenging to solve free boundary problems either theoretically or numerically. In this paper, we develop a novel approach for solving a modified Hele-Shaw problem based on neural network discretization. The existence of the numerical solution with this discretization is established theoretically. We also numerically verify this approach by computing the symmetry-breaking solutions which are guided by the bifurcation analysis near the radially-symmetric branch. Moreover, we further verify the capability of this approach by computing some non-radially symmetric solutions which are not characterized by any theorems.