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

TL;DR

This paper develops higher-order error estimates for physics-informed neural networks (PINNs) applied to primitive equations, providing theoretical guarantees and numerical validation for their accuracy in modeling oceanic and atmospheric dynamics.

Contribution

The work introduces new higher-order regularity results for primitive equations and establishes a priori error bounds for two-layer tanh PINNs, including cases with partial viscosity.

Findings

Higher-order regularity for primitive equations with horizontal viscosity is established.

PINNs can achieve arbitrarily small training and approximation errors with sufficient width and data.

Numerical results demonstrate the effectiveness of $H^s$ norm in training PINNs.

Abstract

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.