Solving nonlinear Boussinesq equation of second-order time derivatives with physics-informed neural networks

Abstract Deep learning combining the physics information is employed to solve the Boussinesq equation 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 $L_2$-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 processes for the Boussinesq equation.
More training time is needed for the solver of the Boussinesq equation when the width of the two-soliton solutions becomes narrower with other parameters fixed.