A shallow Ritz method for elliptic problems with singular sources

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) singularity naturally removed, (ii) level set introduced as feature input, (iii) it completely shallow, comprising only one hidden layer. We first introduce energy functional of problem and then transform contribution to regular surface integral along interface. such way, can be removed without introducing discrete that commonly used traditional regularization methods, well-known immersed boundary method. The original reformulated minimization problem. propose layer approximate global minimizer functional. As result, trained by minimizing loss version energy. addition, we include input find significantly improves training efficiency accuracy. perform series numerical tests show accuracy method its capability problems irregular domains higher dimensions.