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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 area 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$$H^s$$$H^s$norm during the training.

We study the stochastic effect on the three-dimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale solutions and pathwise uniqueness. By adding vertical viscosity,i.e., considering the hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and allow the transport noise in the vertical direction. We establish the local existence of martingale solutions and pathwise uniqueness, and show that the solutions become analytic in the vertical variable instantaneously as$$t>0$$$t>0$and the vertical analytic radius increases as long as the solutions exist.