Journal
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
Volume 28, Issue 5, Pages 1970-2001Publisher
GLOBAL SCIENCE PRESS
DOI: 10.4208/cicp.OA-2020-0179
Keywords
Deep neural network; Poisson-Boltzmann equation; multi-scale; frequency principle
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Funding
- US National Science Foundation [DMS-1950471]
- National Key R&D Program of China [2019YFA0709503]
- Shanghai Sailing Program
- Natural Science Foundation of Shanghai [20ZR1429000]
- HPC of School of Mathematical Sciences at Shanghai Jiao Tong University
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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' 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.
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