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

TL;DR

This paper introduces a deep neural network-based Ritz method leveraging the Caffarelli-Silvestre extension to efficiently solve high-dimensional spectral fractional Laplacian equations, with proven error estimates and numerical validation up to ten dimensions.

Contribution

It presents a novel deep Ritz approach for fractional Laplacian equations using the Caffarelli-Silvestre extension, including error analysis and specialized network design for singularities.

Findings

Effective solution for high-dimensional fractional Laplacian equations.

Error estimates established for the deep Ritz method.

Numerical experiments up to ten dimensions confirm accuracy.

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 ten 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 Monte Carlo method and sinc quadrature is developed to compute the loss function with higher accuracy.