Boussinesq equation solved by the physics-informed neural networks

Physics-informed neural networks (PINNs) model is utilized to achieve the first- and second-order rogue wave solvers of the Boussinesq equation with different initial and boundary conditions. A new generalized loss term handling the initial first-order derivate is included in the simulations to guarantee the high prediction accuracies in the adaptive PINNs (APINNs) and the gradient-optimized PINNs (GPINNs) models, with a new regularization parameter being considered in the latter case. Learned results with high precision are fulfilled in the large domain simulations by applying more collocation points and more weight parameters in the neural network architecture. The APINNs model currently can be made use of in more situations with high prediction accuracies, while the GPINNs model is more robust in the current research where the initial condition is distributed in the localized sharp areas. Parallel computing is carried out to get the mean relative L-2-norm errors efficiently in the GPINNs model due to the random choosing of the simulation points during the training iterations.