Physics-driven deep learning methods and numerically intractable “bad” Jaulent–Miodek equation

The “bad” Jaulent–Miodek (JM) equation serves to describe the motion of non-viscous shallow water wave packets in a flat-bottomed domain subject to shear forces. The “bad” JM equation exhibits poor properties, characterized by the linear instability of nonlinear waves on the zero-plane background, rendering it challenging to solve through traditional analytical and numerical methods. In this paper, two classic physics-driven deep learning approaches, namely, Physics-Informed Neural Networks (PINN) and Physics and Equality-Constrained Artificial Neural Networks (PECANN), are combined into a two-stage “PINN+PECANN” neural network to address the nonlinear wave evolution on the zero-plane background for the “bad” JM equation. The two-stage “PINN+PECANN” neural network method employs PINN in the first stage to pre-train the neural network, followed by fine-tuning of the network parameters using PECANN in the second stage. This approach not only correctly obtains solutions to the “bad” JM equation but also enhances computational efficiency. Specifically, we present the evolutionary behavior of nonlinear waves for the common initial values of the “bad” JM equation: Gauss wave packets, sech wave packets, and rational wave packets. Furthermore, the nonlinear interactions between two Gauss, sech, rational wave packets are provided. The results in this paper validate the advantages of physics-driven deep learning methods in solving equations with poor properties and open up a new pathway for obtaining unstable solutions of nonlinear equations.