fOGA: Orthogonal Greedy Algorithm for Fractional Laplace Equations

TL;DR

This paper introduces a neural network-based method combined with an orthogonal greedy algorithm to efficiently approximate solutions to fractional Laplace equations, demonstrating promising convergence in numerical experiments.

Contribution

It presents a novel approach integrating finite difference discretization with neural networks and the OGA algorithm for fractional Laplace equations.

Findings

Favorable convergence results in numerical tests

Effective approximation of fractional Laplace operator

Validation of the combined method's feasibility

Abstract

In this paper, we explore the finite difference approximation of the fractional Laplace operator in conjunction with a neural network method for solving it. We discretized the fractional Laplace operator using the Riemann-Liouville formula relevant to fractional equations. A shallow neural network was constructed to address the discrete fractional operator, coupled with the OGA algorithm. To validate the feasibility of our approach, we conducted numerical experiments, testing both the Laplace operator and the fractional Laplace operator, yielding favorable convergence results.