Large-scale dynamics of the oceans and the atmosphere are governed by primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is generally challenging. Neural networks have been shown to be a promising machine learning tool to tackle this challenge. In this work, we employ physics-informed neural networks (PINNs) to approximate the solutions to the PEs and study the error estimates. We first establish the higher-order regularity for the global solutions to the PEs with either full viscosity and diffusivity, or with only the horizontal ones. Such a result for the case with only the horizontal ones is new and required in the analysis under the PINNs framework. Then we prove the existence of two-layer tanh PINNs of which the corresponding training error can be arbitrarily small by taking the width of PINNs to be sufficiently wide, and the error between the true solution and its approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, all the estimates are a priori, and our analysis includes higher-order (in spatial Sobolev norm) error estimates. Numerical results on prototype systems are presented to further illustrate the advantage of using the $H^s$ norm during the training.

翻译：海洋和大气的大尺度动力学由原始方程控制。由于其非线性和非局部性，原始方程的数值研究通常具有挑战性。神经网络已被证明是应对这一挑战的有前景的机器学习工具。本文采用物理信息神经网络逼近原始方程的解，并研究其误差估计。我们首先建立了具有完全粘性和扩散性，或仅具有水平粘性和扩散性的原始方程全局解的高阶正则性。其中，仅具有水平粘性和扩散性的情况在物理信息神经网络框架下的分析中属于全新且必要的结果。然后，我们证明存在两层tanh物理信息神经网络，使得通过将网络宽度取足够宽，相应的训练误差可任意小；当训练误差足够小且样本集足够大时，真实解与其逼近之间的误差也可任意小。特别地，所有估计均为先验的，且我们的分析包含高阶（空间Sobolev范数意义下）误差估计。文中还给出了原型系统的数值结果，以进一步说明在训练过程中使用$H^s$范数的优势。