A shallow Ritz method for elliptic problems with singular sources

Ming Chih Lai, Che Chia Chang, Wei Syuan Lin, Wei Fan Hu, Te Sheng Lin

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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.

Original languageEnglish
Article number111547
JournalJournal of Computational Physics
Volume469
DOIs
StatePublished - 15 Nov 2022

Keywords

  • Deep Ritz method
  • Elliptic problems
  • Immersed boundary method
  • Level set function
  • Shallow neural network
  • Singular source

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