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.

翻译：本文系统分析了物理信息神经网络（PINNs）在二维周期域上逼近临界表面准地转（SQG）方程解的精度。临界SQG方程涉及由非局部周期算子描述的对流和扩散过程，这对通常不显式满足周期边界条件的神经网络方法构成了挑战。本文基于奇异积分表示公式，提出利用这些算子的非周期模拟进行新颖的近似，并据此开展误差估计。该思想可推广至解满足指定边界条件的一类更广泛的非局部偏微分方程，从而开创了非局部方程物理信息神经网络理论的新方向。