Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains
Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains
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DOI：
10.4208/cicp.oa-2020-0179
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发表时间：
2020-06
期刊：
影响因子：
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通讯作者：
Ziqi Liu;Wei Cai;Zhi-Qin John Xu
中科院分区：
文献类型：
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作者：
Ziqi Liu;Wei Cai;Zhi-Qin John Xu
在本文中，我们提出了多尺度深度神经网络（MscaleDNNs）利用径向尺度的思想在频域和紧支持的激活函数。径向标度将偏微分方程解的高频内容逼近问题转化为低频函数的学习问题，紧凑的支持激活函数有助于分离待用dnn逼近的目标函数的频率内容。因此，mscalednn在多个尺度上实现了快速一致收敛。所提出的mscalednn优于传统的全连接深度神经网络，是一种在复域和奇异域上具有丰富频率内容的泊松-玻尔兹曼方程的有效无网格数值方法。
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.