A Shallow Ritz Method for elliptic problems with Singular Sources

In this paper, a shallow Ritz-type neural network for solving elliptic problems 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 feather input, (iii) it is completely shallow consisting of 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 the introduction of discrete delta function 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 and find that it significantly improves the training efficiency and accuracy. We perform a series of numerical tests to demonstrate the accuracy of the present network as well as its capability for problems in irregular domains and in higher dimensions.