fOGA: An Orthogonal Greedy Algorithm for Fractional Laplacian Problems

Abstract

In this paper, we propose a numerical method for fractional Laplace equations that combines finite difference discretization with shallow neural network approximation. The fractional Laplace operator is discretized using a directional representation of Riemann--Liouville type, which leads to a finite difference approximation of the nonlocal operator. In two dimensions, the angular integral is approximated by a quadrature rule, and auxiliary points are introduced along each direction to facilitate the evaluation of the operator. Based on the resulting discrete system, the solution is then represented by a shallow neural network constructed through the orthogonal greedy algorithm (OGA).