A Multiple Transferable Neural Network Method with Domain Decomposition for Elliptic Interface Problems

TL;DR

This paper introduces Multi-TransNet, a novel neural network method combining domain decomposition and interface conditions to effectively solve elliptic interface problems with discontinuities, demonstrating high accuracy and efficiency.

Contribution

The paper develops a Multi-TransNet approach integrating domain decomposition with transferable neural networks, including an empirical formula for neuron shape and a normalization method for loss weighting.

Findings

Achieves superior accuracy in elliptic interface problems

Demonstrates robustness across different contrast diffusion coefficients

Reduces parameter tuning cost significantly

Abstract

The transferable neural network (TransNet) is a two-layer shallow neural network with pre-determined and uniformly distributed neurons in the hidden layer, and the least-squares solvers can be particularly used to compute the parameters of its output layer when applied to the solution of partial differential equations. In this paper, we integrate the TransNet technique with the nonoverlapping domain decomposition and the interface conditions to develop a novel multiple transferable neural network (Multi-TransNet) method for solving elliptic interface problems, which typically contain discontinuities in both solutions and their derivatives across interfaces. We first propose an empirical formula for the TransNet to characterize the relationship between the radius of the domain-covering ball, the number of hidden-layer neurons, and the optimal neuron shape. In the Multi-TransNet method, we assign each subdomain one distinct TransNet with an adaptively determined number of hidden-layer neurons to maintain the globally uniform neuron distribution across the entire computational domain, and then unite all the subdomain TransNets together by incorporating the interface condition terms into the loss function. The empirical formula is also extended to the Multi-TransNet and further employed to estimate appropriate neuron shapes for the subdomain TransNets, greatly reducing the parameter tuning cost. Additionally, we propose a normalization approach to adaptively select the weighting parameters for the terms in the loss function. Ablation studies and extensive experiments with comparison tests on different types of elliptic interface problems with low to high contrast diffusion coefficients in two and three dimensions are carried out to numerically demonstrate the superior accuracy, efficiency, and robustness of the proposed Multi-TransNet method.