Deep Ritz Method for the Spectral Fractional Laplacian Equation Using the Caffarelli--Silvestre Extension
Abstract
In this paper, we propose a novel method for solving high-dimensional spectral fractional Laplacian equations. Using the Caffarelli--Silvestre extension, the $d$-dimensional spectral fractional equation is reformulated as a regular partial differential equation of dimension $d+1$. We transform the extended equation as a minimal Ritz energy functional problem and search for its minimizer in a special class of deep neural networks. Moreover, based on the approximation property of networks, we establish estimates on the error made by the deep Ritz method. Numerical results are reported to demonstrate the effectiveness of the proposed method for solving fractional Laplacian equations up to 10 dimensions. Technically, in this method, we design a special network-based structure to adapt to the singularity and exponential decaying of the true solution. Also, a hybrid integration technique combining the Monte Carlo method and sinc quadrature is developed to compute the loss function with higher accuracy.
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Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A2018 - A2036
DOI: 10.1137/21M1442516
ISSN (online): 1095-7197
Copyright
© 2022, Society for Industrial and Applied Mathematics.
History
Submitted: 26 August 2021
Accepted: 26 January 2022
Published online: 13 July 2022
Keywords
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Authors
Funding Information
Research Grants Council, University Grants Committee https://doi.org/10.13039/501100002920 : C1013-21GF, 12300218, 12300519, 17201020, 17300021
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