Localized waves for modified Boussinesq equation and Mikhailov–Lenells equation based on Physics-informed neural networks and Miura transformation

Finding the new localized waves of nonlinear partial differential equations via the analytical or numerical methods is an important research area. This study focuses on obtaining the new localized waves of modified Boussinesq equation and Mikhailov–Lenells equation. By embedding the Miura transformation into the Physics-informed neural networks (PINN), we simulate the solutions of the two coupled systems through taking the solution of good Boussinesq equation as the starting. Experiencing multiple trainings, we find that the initial random seed (IRS) can influence the trained result under the condition that the hyper-parameters and initial dataset keep invariable in the training. In virtue of the learning capacity of PINN and one-to-many nonlinear mapping of Miura transformation, the localized waves of the two coupled equations are discovered under the same initial dataset and hyper-parameters, and show the W-shaped, antidark soliton, antikink and the overlapping between dark soliton and antidark soliton, and two dark solitons. The two coupled equations experience an unsupervised training for they are simulated under the initial and boundary conditions of good Boussinesq equation. It means that their solutions are constrained by the physical laws and the nonlinear transformation. The localized waves are analyzed through the graphics. PINN show the ability to predict the solutions of coupled system at the same network under the nonlinear mapping with the initial data of single equation. The algorithm is expected to be applied to the unsupervised learning of more coupled nonlinear models.