  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 equation
involves advection and diffusion described by nonlocal periodic operators,
posing challenges for neural network-based methods that do not commonly exhibit
periodic boundary conditions. In this paper, we present a novel approximation
of these operators using their nonperiodic analogs based on singular integral
representation formulas and use it to perform error estimates. This idea can be
generalized to a larger class of nonlocal partial differential equations whose
solutions satisfy prescribed boundary conditions, thereby initiating a new
PINNs theory for equations with nonlocalities.
