A new approach for the fractional Laplacian via deep neural networks

Autor:	Valenzuela, Nicolás
Rok vydání:	2022
Předmět:	Mathematics - Analysis of PDEs Mathematics - Numerical Analysis Mathematics - Probability
Druh dokumentu:	Working Paper
Popis:	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 $\alpha \in (1,2)$. 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. Comment: 55 pages, 1 figure
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2205.05229 Zobrazit plný text záznamu View this record from Arxiv