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In this paper, we propose a novel method for solving high-dimensionalspectral fractional Laplacian equations. Using the Caffarelli-Silvestreextension, the $d$-dimensional spectral fractional equation is reformulated asa regular partial differential equation of dimension $d+1$. We transform theextended equation as a minimal Ritz energy functional problem and search forits minimizer in a special class of deep neural networks. Moreover, based onthe approximation property of networks, we establish estimates on the errormade by the deep Ritz method. Numerical results are reported to demonstrate theeffectiveness of the proposed method for solving fractional Laplacian equationsup to ten dimensions. Technically, in this method, we design a specialnetwork-based structure to adapt to the singularity and exponential decaying ofthe true solution. Also, A hybrid integration technique combining Monte Carlomethod and sinc quadrature is developed to compute the loss function withhigher accuracy.

Deep Ritz method for the spectral fractional Laplacian equation using the Caffarelli-Silvestre extension