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We systematically analyze the accuracy of Physics-Informed Neural Networks(PINNs) in approximating solutions to the critical Surface Quasi-Geostrophic(SQG) equation on two-dimensional periodic boxes. The critical SQG equationinvolves advection and diffusion described by nonlocal periodic operators,posing challenges for neural network-based methods that do not commonly exhibitperiodic boundary conditions. In this paper, we present a novel approximationof these operators using their nonperiodic analogs based on singular integralrepresentation formulas and use it to perform error estimates. This idea can begeneralized to a larger class of nonlocal partial differential equations whosesolutions satisfy prescribed boundary conditions, thereby initiating a newPINNs theory for equations with nonlocalities.

Accuracy Analysis of Physics-Informed Neural Networks for Approximating the Critical SQG Equation