PINN for solving forward and inverse problems involving integrable two-dimensional nonlocal equations

Abstract In this paper, the physics informed neural network (PINN) deep learning method is applied to solve two-dimensional nonlocal equations, including the partial reverse space y -nonlocal Mel’nikov equation, the partial reverse space-time nonlocal Mel’nikov equation and the nonlocal two-dimensional nonlinear Schrödinger (NLS) equation. By the PINN method, we successfully derive a data-driven two soliton solution, lump solution and rogue wave solution. Numerical simulation results indicate that the error range between the data-driven solution and the exact solution is relatively small, which verifies the effectiveness of the PINN deep learning method for solving high dimensional nonlocal equations. Moreover, the parameter discovery of the partial reverse space-time nonlocal Mel’nikov equation is analysed in terms of its soliton solution for the first time.