Physics-informed neural networks method in high-dimensional integrable systems

In this paper, the physics-informed neural networks (PINNs) are applied to high-dimensional system to solve the (N + 1)-dimensional initial-boundary value problem with 2N + 1 hyperplane boundaries. This method is used to solve the most classic (2+1)-dimensional integrable Kadomtsev-Petviashvili (KP) equation and (3+1)-dimensional reduced KP equation. The dynamics of (2+1)-dimensional local waves such as solitons, breathers, lump and resonance rogue are reproduced. Numerical results display that the magnitude of the error is much smaller than the wave height itself, so it is considered that the classical solutions in these integrable systems are well obtained based on the data-driven mechanism.