A new approach for the fractional Laplacian via deep neural networks

TL;DR

This paper introduces a novel deep learning approach to approximate solutions of the fractional Laplacian Dirichlet problem, effectively handling high-dimensional cases without suffering from the curse of dimensionality.

Contribution

It presents a new neural network-based method for solving the fractional Laplacian Dirichlet problem, demonstrating its effectiveness in high dimensions.

Findings

Neural networks can approximate solutions to the fractional Laplacian Dirichlet problem.

The method overcomes the curse of dimensionality in high-dimensional PDEs.

The approach is applicable for fractional Laplacian with exponent between 1 and 2.

Abstract

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.