A Shallow Ritz Method for Elliptic Problems with Singular Sources
This work addresses a specific computational challenge in numerical analysis for researchers in applied mathematics and engineering, presenting an incremental improvement over traditional regularization methods.
The paper tackles solving elliptic equations with singular delta function sources by developing a shallow neural network method that removes the singularity naturally and improves training efficiency using a level set function, achieving accurate results in numerical tests for irregular domains and higher dimensions.
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.