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

• URL: http://arxiv.org/abs/2410.19014v2
• Date: Mon, 28 Oct 2024 22:59:26 GMT
• Title: Discovery of Quasi-Integrable Equations from traveling-wave data using the Physics-Informed Neural Networks
• Authors: A. Nakamula, N. Sawado, K. Shimasaki, Y. Shimazaki, Y. Suzuki, K. Toda,
• Abstract summary: PINNs are used to study vortex solutions in 2+1 dimensional nonlinear partial differential equations. We consider PINNs with conservation laws (referred to as cPINNs), deformations of the initial profiles, and a friction approach to improve the identification's resolution.
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• Abstract: Physics-Informed Neural Networks (PINNs) are used to study vortex solutions in the 2+1 dimensional nonlinear partial differential equations. These solutions include the regularized long-wave (RLW) equation and the Zakharov-Kuznetsov (ZK) equation, which are toy models of the geostrophic shallow water model in the planetary atmosphere. PINNs successfully solve these equations in the forward process and the solutions are obtained using the mesh-free approach and automatic differentiation while accounting for conservation laws. In the inverse process, the proper equations can be successfully derived from a given training data. Since these equations have a lot in common, there are situations when substantial misidentification arises during the inverse analysis. We consider PINNs with conservation laws (referred to as cPINNs), deformations of the initial profiles, and a friction approach that provides excellent discrimination of the equations to improve the identification's resolution.

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