Data-driven localized waves of a nonlinear partial differential equation via transformation and physics-informed neural network

The data-driven localized waves are discussed for a nonlinear partial differential equation, which can be connected with the modified Korteweg de Vries (KdV) equation by the Bäcklund transformation. The solutions of analyzed equation can be converted into the ones of the modified KdV equation through the Bäcklund transformation, which is an irreversible transformation. The numerical solutions of two equations are obtained via the physics-informed neural network (PINN) and the initial and boundary conditions of the modified KdV equation at the same network. The localized wave solutions of the analyzed equation are generated under an unsupervised training for they are only supervised by the governing equations residual and the Bäcklund transformation in the neural network. As the result of the choices of initial and boundary conditions of the modified KdV equation, different types of localized waves for analyzed equation are predicted, such as the types of flat-top soliton and stair, which are analyzed through the graphics. The evolutions of initial Gaussian wave for two equations are analyzed to verify the capability of PINN.