A Shallow Ritz Method for Elliptic Problems with Singular Sources

TL;DR

This paper introduces a shallow Ritz neural network method for solving elliptic equations with singular sources, effectively removing delta function singularities and improving training efficiency through the use of level set features.

Contribution

The work presents a novel shallow neural network approach that naturally handles singular sources and incorporates level set functions as features, enhancing accuracy and efficiency.

Findings

Successfully removes delta singularities without regularization.

Improves training efficiency with level set feature input.

Demonstrates high accuracy in irregular and higher-dimensional domains.

Abstract

In this paper, a shallow Ritz-type neural network for solving elliptic equations with delta function singular sources on an interface is developed. There are three novel features in the present work; namely, (i) the delta function singularity is naturally removed, (ii) level set function is introduced as a feature input, (iii) it is completely shallow, comprising only one hidden layer. We first introduce the energy functional of the problem and then transform the contribution of singular sources to a regular surface integral along the interface. In such a way, the delta function singularity can be naturally removed without introducing a discrete one that is commonly used in traditional regularization methods, such as the well-known immersed boundary method. The original problem is then reformulated as a minimization problem. We propose a shallow Ritz-type neural network with one hidden layer to approximate the global minimizer of the energy functional. As a result, the network is trained by minimizing the loss function that is a discrete version of the energy. In addition, we include the level set function of the interface as a feature input of the network and find that it significantly improves the training efficiency and accuracy. We perform a series of numerical tests to show the accuracy of the present method and its capability for problems in irregular domains and higher dimensions.