论文标题
通过深层神经网络的分数拉普拉斯的新方法
A new approach for the fractional Laplacian via deep neural networks
论文作者
论文摘要
在过去的几十年中，对小节拉普拉斯人进行了深入研究。在本文中，我们使用最新的深度学习技术为相关的Dirichlet问题提供了一种不同的方法。实际上，已经通过神经网络来理解具有随机表示的强度PDE，从而克服了所谓的维度诅咒。在这些方程式中，可以在$ \ mathbb {r}^d $中找到抛物线词，而有限域中的椭圆形$ d \ subset \ mathbb {r}^d $。在本文中，我们考虑了针对指数$α\ in（1,2）$的分数拉普拉斯的Dirichlet问题。我们表明，它以随机方式代表的解决方案可以使用深层神经网络近似。我们还检查了这种近似值不会遭受维数的诅咒。
The fractional Laplacian has been strongly studied during past decades. In this paper we present a different approach for the associated Dirichlet problem, using recent deep learning techniques. In fact, intensively PDEs with a stochastic representation have been understood via neural networks, overcoming the so-called curse of dimensionality. Among these equations one can find parabolic ones in $\mathbb{R}^d$ and elliptic in a bounded domain $D \subset \mathbb{R}^d$. In this paper we consider the Dirichlet problem for the fractional Laplacian with exponent $α\in (1,2)$. We show that its solution, represented in a stochastic fashion can be approximated using deep neural networks. We also check that this approximation does not suffer from the curse of dimensionality.
