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

研究成果: 雜誌貢獻期刊論文同行評審

5 引文 斯高帕斯(Scopus)

摘要

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.

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文章編號111547
期刊Journal of Computational Physics
469
DOIs
出版狀態已出版 - 15 11月 2022

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