Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains

Abstract

In this paper, we propose multi-scale deep neural networks (MscaleDNNs) using\nthe idea of radial scaling in frequency domain and activation functions with\ncompact support. The radial scaling converts the problem of approximation of\nhigh frequency contents of PDEs' solutions to a problem of learning about lower\nfrequency functions, and the compact support activation functions facilitate\nthe separation of frequency contents of the target function to be approximated\nby corresponding DNNs. As a result, the MscaleDNNs achieve fast uniform\nconvergence over multiple scales. The proposed MscaleDNNs are shown to be\nsuperior to traditional fully connected DNNs and be an effective mesh-less\nnumerical method for Poisson-Boltzmann equations with ample frequency contents\nover complex and singular domains.\n