Unraveling Nonlinear Wave Dynamics in Hydraulic Engineering: Symbolic Neural Network Solutions for the (31)-Dimensional P-Type Model+

In this paper, the nonlinear wave dynamics of the (3+1)-dimensional P-type model are investigated using a symbolic neural-network-based hybrid technique. The model is considered in the context of nonlinear wave propagation, which is of practical importance in different areas. This approach combines the modified generalized Riccati equation mapping method with a neural network structure, where each neuron in the first hidden layer represents a Riccati-based activation function, which allows the construction of new physics-based trial solutions. This approach allows the systematic generation of exact traveling wave solutions in space and time. Solutions including solitary waves, and periodic and localized pulses are obtained, which provide essential insight into the influence of model parameters on wave profiles and evolution in complex media. The dynamics of the waves are illustrated by two-dimensional, three-dimensional, and contour plots, which show the behavior of propagation, localization, and stability. The results demonstrate the method is feasible, efficient, and effective for studying high-dimensional nonlinear systems. This work not only extends the Riccati-based neural network approach to the model but also provides a platform for exploring nonlinear wave behavior in shallow-water flows and other hydraulic engineering problems.