Higher-Order Error estimates for physics-informed neural networks approximating the primitive equations

Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is in general a hard task. Neural network has been shown a promising machine learning tool to tackle this challenge. In this work, we employ the 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 higher-order regularity results are 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 the approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, our analysis includes higher-order (in spatial Sobolev norm) error estimates, and improves existing results in PINNs' literature which concerns only the $L^2$ error. Numerical results on prototype systems are presented for further illustrating the advantage of using $H^s$ norm during the training.