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Volume 28, Issue 5
Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains

Ziqi Liu, Wei Cai & Zhi-Qin John Xu

DOI: 10.4208/cicp.OA-2020-0179

Commun. Comput. Phys., 28 (2020), pp. 1970-2001.

Published online: 2020-11

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  • Abstract

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.

  • Keywords

Deep neural network, Poisson-Boltzmann equation, multi-scale, frequency principle.

  • AMS Subject Headings

35Q68, 65N99, 68T07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-1970, author = {Liu , ZiqiCai , Wei and John Xu , Zhi-Qin}, title = {Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {5}, pages = {1970--2001}, abstract = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0179}, url = {http://global-sci.org/intro/article_detail/cicp/18402.html} }
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 -

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.

Ziqi Liu, Wei Cai & Zhi-Qin John Xu. (2020). Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains. Communications in Computational Physics. 28 (5). 1970-2001. doi:10.4208/cicp.OA-2020-0179
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