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Volume 13, Issue 1
Neural Network Method for Integral Fractional Laplace Equations

Zhaopeng Hao, Moongyu Park & Zhiqiang Cai

DOI: 10.4208/eajam.010122.210722

East Asian J. Appl. Math., 13 (2023), pp. 95-118.

Published online: 2023-01

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  • Abstract

A neural network method for fractional order diffusion equations with integral fractional Laplacian is studied. We employ the Ritz formulation for the corresponding fractional equation and then derive an approximate solution of an optimization problem in the function class of neural network sets. Connecting the neural network sets with weighted Sobolev spaces, we prove the convergence and establish error estimates of the neural network method in the energy norm. To verify the theoretical results, we carry out numerical experiments and report their outcome.

  • Keywords

Deep Ritz method, neural network, fractional elliptic PDE, ReLU.

  • AMS Subject Headings

35B65, 65N35, 65N12, 41A25, 26B40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-13-95, author = {Hao , ZhaopengPark , Moongyu and Cai , Zhiqiang}, title = {Neural Network Method for Integral Fractional Laplace Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2023}, volume = {13}, number = {1}, pages = {95--118}, abstract = {

A neural network method for fractional order diffusion equations with integral fractional Laplacian is studied. We employ the Ritz formulation for the corresponding fractional equation and then derive an approximate solution of an optimization problem in the function class of neural network sets. Connecting the neural network sets with weighted Sobolev spaces, we prove the convergence and establish error estimates of the neural network method in the energy norm. To verify the theoretical results, we carry out numerical experiments and report their outcome.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.010122.210722}, url = {http://global-sci.org/intro/article_detail/eajam/21304.html} }
TY - JOUR T1 - Neural Network Method for Integral Fractional Laplace Equations AU - Hao , Zhaopeng AU - Park , Moongyu AU - Cai , Zhiqiang JO - East Asian Journal on Applied Mathematics VL - 1 SP - 95 EP - 118 PY - 2023 DA - 2023/01 SN - 13 DO - http://doi.org/10.4208/eajam.010122.210722 UR - https://global-sci.org/intro/article_detail/eajam/21304.html KW - Deep Ritz method, neural network, fractional elliptic PDE, ReLU. AB -

A neural network method for fractional order diffusion equations with integral fractional Laplacian is studied. We employ the Ritz formulation for the corresponding fractional equation and then derive an approximate solution of an optimization problem in the function class of neural network sets. Connecting the neural network sets with weighted Sobolev spaces, we prove the convergence and establish error estimates of the neural network method in the energy norm. To verify the theoretical results, we carry out numerical experiments and report their outcome.

Zhaopeng Hao, Moongyu Park & Zhiqiang Cai. (2023). Neural Network Method for Integral Fractional Laplace Equations. East Asian Journal on Applied Mathematics. 13 (1). 95-118. doi:10.4208/eajam.010122.210722
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