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：海洋和大气的大规模动态由原始方程（PEs）所调节。由于非线性和非局部性质，数值研究PEs通常具有挑战性。神经网络已被证明是一种有希望的机器学习工具，用于解决这一挑战。在这项工作中，我们使用基于物理的神经网络（PINNs）近似PEs的解并研究误差估计。我们首先确定具有完全粘性和扩散的PEs的全局解的高阶正则性，或具有仅水平粘度和扩散的PEs的高阶正则性。对于只有水平粘度和扩散的情况下的这样的结果是新的，并且在PINNs框架下的分析中是必需的。然后我们证明了存在两层tanh PINNs，其相应的训练误差可以通过将PINNs的宽度取得足够宽来任意小，并且在训练误差足够小且样本集足够大的情况下，真实解和其近似解之间的误差可以任意小。特别地，所有估计都是先验的，我们的分析包含高阶的（在空间Sobolev范数中）误差估计。我们还介绍了原型系统的数值结果，以进一步说明在训练期间使用$H^s$范数的优势。