A Shallow Ritz Method for Elliptic Problems with Singular Sources

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.

References in corpus (5)

• On Empirical Comparisons of Optimizers for Deep Learning
• PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries
• Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs
• A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems
• Deep unfitted Nitsche method for elliptic interface problems

Cited by in corpus (4)

• A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems
• A cusp-capturing PINN for elliptic interface problems
• A shallow physics-informed neural network for solving partial differential equations on surfaces
• An efficient neural-network and finite-difference hybrid method for elliptic interface problems with applications