A numerical approach for the fractional Laplacian via deep neural networks

TL;DR

This paper introduces a deep neural network-based numerical method to solve fractional elliptic problems with Dirichlet boundary conditions, demonstrating efficiency across various dimensions and fractional orders.

Contribution

It presents a novel stochastic gradient descent algorithm leveraging deep neural networks to approximate solutions of fractional Laplacian problems.

Findings

The method effectively approximates solutions for multiple fractional orders.

Numerical examples confirm the efficiency and accuracy of the approach.

The approach scales well with higher dimensions.

Abstract

We consider the fractional elliptic problem with Dirichlet boundary conditions on a bounded and convex domain $D$ of $\mathbb{R}^d$, with $d \geq 2$. In this paper, we perform a stochastic gradient descent algorithm that approximates the solution of the fractional problem via Deep Neural Networks. Additionally, we provide four numerical examples to test the efficiency of the algorithm, and each example will be studied for many values of $\alpha \in (1,2)$ and $d \geq 2$.