Parameter identification of shallow water waves using the generalized equal width equation and physics-informed neural networks: a conservative approximation scheme

In this investigation, we implement a numerical approach employing Physics-Informed Neural Networks (PINN) based on a shallow water waves model described by the generalized equal width (GEW) equation, a highly nonlinear partial differential equation (PDE) as well as an extremely difficult PDE that is well-known for its stiffness. Utilizing a mesh-free technique, we achieve a continuous solution and derive a nonlinear function for the water waves solution using a reduced number of points within the problem domain. To insure the numerical procedure adheres to mass, momentum, and energy conservation, we introduce a new term in the loss function to insure the adherence to these properties and we demonstrate that it performs better compared to PINN. Furthermore, we closely monitor the conservation of mass, momentum, and energy throughout the simulation and on the other hand we estimated unknown parameters of GEW model using inverse PINN with high accuracy. To assess the effectiveness of our proposed methodology, we demonstrate its effectiveness on three classic test scenarios: the propagation of a single solitary wave, the interaction of two solitary waves, and the Maxwellian initial condition.