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Nonlocal periodic operators in partial differential equations (PDEs) posechallenges in constructing neural network solutions, which typically lackperiodic boundary conditions. In this paper, we introduce a novel PDEperspective on approximating these nonlocal periodic operators. Specifically,we investigate the behavior of the periodic first-order fractional Laplacianand Riesz transform when acting on nonperiodic functions, thereby initiating anew PDE theory for approximating solutions to equations with nonlocalitiesusing neural networks. Moreover, we derive quantitative Sobolev estimates andutilize them to rigorously construct neural networks that approximate solutionsto the two-dimensional periodic critically dissipative SurfaceQuasi-Geostrophic (SQG) equation.

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