Volume 28, Issue 5
A Multi-Scale DNN Algorithm for Nonlinear Elliptic Equations with Multiple Scales

Xi-An Li, Zhi-Qin John Xu & Lei Zhang

Commun. Comput. Phys., 28 (2020), pp. 1886-1906.

Published online: 2020-11

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  • Abstract

Algorithms based on deep neural networks (DNNs) have attracted increasing attention from the scientific computing community. DNN based algorithms are easy to implement, natural for nonlinear problems, and have shown great potential to overcome the curse of dimensionality. In this work, we utilize the multi-scale DNN-based algorithm (MscaleDNN) proposed by Liu, Cai and Xu (2020) to solve multi-scale elliptic problems with possible nonlinearity, for example, the p-Laplacian problem. We improve the MscaleDNN algorithm by a smooth and localized activation function. Several numerical examples of multi-scale elliptic problems with separable or non-separable scales in low-dimensional and high-dimensional Euclidean spaces are used to demonstrate the effectiveness and accuracy of the MscaleDNN numerical scheme.

  • Keywords

Multi-scale elliptic problem, p-Laplacian equation, deep neural network (DNN), variational formulation, activation function.

  • AMS Subject Headings

65N30, 35J66, 41A46, 68T07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-1886, author = {Li , Xi-An and John Xu , Zhi-Qin and Zhang , Lei}, title = {A Multi-Scale DNN Algorithm for Nonlinear Elliptic Equations with Multiple Scales}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {5}, pages = {1886--1906}, abstract = {

Algorithms based on deep neural networks (DNNs) have attracted increasing attention from the scientific computing community. DNN based algorithms are easy to implement, natural for nonlinear problems, and have shown great potential to overcome the curse of dimensionality. In this work, we utilize the multi-scale DNN-based algorithm (MscaleDNN) proposed by Liu, Cai and Xu (2020) to solve multi-scale elliptic problems with possible nonlinearity, for example, the p-Laplacian problem. We improve the MscaleDNN algorithm by a smooth and localized activation function. Several numerical examples of multi-scale elliptic problems with separable or non-separable scales in low-dimensional and high-dimensional Euclidean spaces are used to demonstrate the effectiveness and accuracy of the MscaleDNN numerical scheme.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0187}, url = {http://global-sci.org/intro/article_detail/cicp/18399.html} }
TY - JOUR T1 - A Multi-Scale DNN Algorithm for Nonlinear Elliptic Equations with Multiple Scales AU - Li , Xi-An AU - John Xu , Zhi-Qin AU - Zhang , Lei JO - Communications in Computational Physics VL - 5 SP - 1886 EP - 1906 PY - 2020 DA - 2020/11 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2020-0187 UR - https://global-sci.org/intro/article_detail/cicp/18399.html KW - Multi-scale elliptic problem, p-Laplacian equation, deep neural network (DNN), variational formulation, activation function. AB -

Algorithms based on deep neural networks (DNNs) have attracted increasing attention from the scientific computing community. DNN based algorithms are easy to implement, natural for nonlinear problems, and have shown great potential to overcome the curse of dimensionality. In this work, we utilize the multi-scale DNN-based algorithm (MscaleDNN) proposed by Liu, Cai and Xu (2020) to solve multi-scale elliptic problems with possible nonlinearity, for example, the p-Laplacian problem. We improve the MscaleDNN algorithm by a smooth and localized activation function. Several numerical examples of multi-scale elliptic problems with separable or non-separable scales in low-dimensional and high-dimensional Euclidean spaces are used to demonstrate the effectiveness and accuracy of the MscaleDNN numerical scheme.

Xi-An Li, Zhi-Qin John Xu & Lei Zhang. (2020). A Multi-Scale DNN Algorithm for Nonlinear Elliptic Equations with Multiple Scales. Communications in Computational Physics. 28 (5). 1886-1906. doi:10.4208/cicp.OA-2020-0187
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