  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.
