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.
Original language | English |
---|---|
Pages (from-to) | A2018-A2036 |
Number of pages | 19 |
Journal | SIAM Journal on Scientific Computing |
Volume | 44 |
Issue number | 4 |
DOIs | |
Publication status | Published - Aug 2022 |
Scopus Subject Areas
- Computational Mathematics
- Applied Mathematics
User-Defined Keywords
- Caffarelli-Silvestre extension
- deep learning
- fractional Laplacian
- Ritz method
- singularity