A PDE Perspective on Approximating Nonlocal Periodic Operators with Applications on Neural Networks for Critical SQG Equations

Abstract: Nonlocal periodic operators in partial differential equations (PDEs) pose challenges in constructing neural network solutions, which typically lack periodic boundary conditions. In this paper, we introduce a novel PDE perspective on approximating these nonlocal periodic operators. Specifically, we investigate the behavior of the periodic first-order fractional Laplacian and Riesz transform when acting on nonperiodic functions, thereby initiating a new PDE theory for approximating solutions to equations with nonlocalities using neural networks. Moreover, we derive quantitative Sobolev estimates and utilize them to rigorously construct neural networks that approximate solutions to the two-dimensional periodic critically dissipative Surface Quasi-Geostrophic (SQG) equation.