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TITLE: [HG_AP]Representation of the solution of fractional Schrödinger equations with shallow neural network

ABSTRACT: Barron spaces have gained significant attention in the machine learning community for their ability to provide dimension-free convergence rates in two-layer neural networks. This work extends Barron norm estimates for the whole-space static Schrödinger equation to incorporate the nonlocal term. We show that if both the source term and the potential function of the equation belong to the Barron space, and the potential function has a nonnegative lower bound, the solution remains within the Barron space. As a consequence, we prove that the two-layer network can approximate solutions of the fractional Schrödinger equation with a rate determined by the constant quantified by the given data. Furthermore, we will discuss interesting future direction in this topic.