TY - JOUR
T1 - Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains
AU - Liu , Ziqi
AU - Cai , Wei
AU - John Xu , Zhi-Qin
JO - Communications in Computational Physics
VL - 5
SP - 1970
EP - 2001
PY - 2020
DA - 2020/11
SN - 28
DO - http://doi.org/10.4208/cicp.OA-2020-0179
UR - https://global-sci.org/intro/article_detail/cicp/18402.html
KW - Deep neural network, Poisson-Boltzmann equation, multi-scale, frequency principle.
AB - <p style="text-align: justify;">In this paper, we propose multi-scale deep neural networks (MscaleDNNs)
using the idea of radial scaling in frequency domain and activation functions with
compact support. The radial scaling converts the problem of approximation of high
frequency contents of PDEs&#39; solutions to a problem of learning about lower frequency
functions, 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 connected
DNNs and be an effective mesh-less numerical method for Poisson-Boltzmann equations with ample frequency contents over complex and singular domains.</p>