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.
翻译:在本文中,我们提出了一个解决高维谱分光谱拉普拉西亚方程式的新方法。使用Caffarelli-Silvestre扩展法,美元维谱分光方程式被重新改制成一个正常的维度部分差异方程式,即$d+1美元。我们把扩展方程式转换成一个最小的里兹能量功能问题,并在一个特殊的深神经网络类中搜索其最小化器。此外,根据网络的近似特性,我们确定了对深里兹法错误的估计。报告的数字结果表明,拟议的解决分光谱方程式最多达到10维。