Data-driven solutions of the nonlinear Boussinesq and Schrödinger equations of second order in time with physics-informed neural networks

Abstract Deep learning combining the physics information is employed to solve the Boussinesq and Schrödinger equations with second-order time derivative. High prediction accuracies are achieved by adding a new initial loss term in the physics-informed neural networks along with the adaptive activation function and loss-balanced coefficients. The numerical simulations are carried out with different initial and boundary conditions, in which the relative L₂-norm errors are all around 10⁻⁴. The prediction accuracies have been improved by two orders of magnitude compared to the former results in certain simulations. The dynamic behavior of solitons and their interaction are studied in the colliding and chasing process for the Boussinesq equation. More training time is needed for the solver of the Boussinesq equation when the width of the two-soliton solutions is narrower with other parameters fixed. Simulations in large domains are conducted to explore soliton and rogue wave solutions of the Schrödinger equation and the high precisions are achieved.