Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains (Communications in Computational Physics, 2020)
In this paper, we propose multi-scale deep neural networks (MscaleDNNs)using the idea of radial scaling in frequency domain and activation functions withcompact support. The radial scaling converts the problem of approximation of highfrequency contents of PDEs' solutions to a problem of learning about lower frequencyfunctions, and the compact support activation functions facilitate the separation of frequency contents of the target function to be approximated by corresponding DNNs.As a result, the MscaleDNNs achieve fast uniform convergence over multiple scales.The proposed MscaleDNNs are shown to be superior to traditional fully connectedDNNs and be an effective mesh-less numerical method for Poisson-Boltzmann equations with ample frequency contents over complex and singular domains.
Keywords: Deep neural network; Poisson-Boltzmann equation; multi-scale; frequency principle.