Discovery of Quasi-Integrable Equations from traveling-wave data using the Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for analyzing nonlinear partial differential equations and identifying governing equations from observational data. In this study, we apply PINNs to investigate vortex-type solutions of quasi-integrable equations in two spatial dimensions, specifically the Zakharov-Kuznetsov (ZK) and the Regularized Long-Wave (RLW) equations. These equations are toy models for geostrophic shallow water dynamics in planetary atmospheres. We first demonstrate that PINNs can successfully solve these equations in the forward process using a mesh-free approach with automatic differentiation. However, in the inverse process, substantial misidentification occurs due to the structural similarities between the ZK and the RLW equations. To address this issue, we then introduce conservation law-enhanced PINNs, initial condition variations, and a friction-based perturbation approach to improve identification accuracy. Our results show that incorporating small perturbations while preserving conservation laws significantly enhances the resolution of equation identification. These findings may contribute to the broader goal of using deep learning techniques for discovering governing equations in complex fluid dynamical systems, such as Jupiter's Great Red Spot.