Adaptive sampling physics-informed neural network method for high-order rogue waves and parameters discovery of the (2 + 1)-dimensional CHKP equation.

Rogue waves are important physical phenomena, which have wide applications in nonlinear optics, hydrodynamics, Bose-Einstein condensates, and oceanic and atmospheric dynamics. We find that when using the original PINNs to study rogue waves of high dimensional PDEs, the prediction performance will become very poor, especially for high-order rogue waves due to that the randomness of selection of sample points makes insufficient use of the physical information describing the local sharp regions of rogue waves. In this paper, we propose an adaptive sampling physics-informed neural network method (ASPINN), which renders the points in local sharp regions to be selected sufficiently by a new adaptive search algorithm to lead to a prefect prediction performance. To valid the performance of our method, the (2+1)-dimensional CHKP equation is taken as an illustrative example. Experimental results reveal that the original PINNs can hardly be able to predict dynamical behaviors of the high-order rogue waves for the CHKP equation, but the ASPINN method can not only predict dynamical behaviors of these high-order rogue waves, but also greatly improve the prediction efficiency and accuracy to four orders of magnitude. Then, the data-driven inverse problem for the CHKP equation with different levels of corrupted noise is studied to show that the ASPINN method has good robustness. Moreover, some main factors affecting the neural network performance are discussed in detail, including the size of training data, the number of layers of the neural network, and the number of neurons per layer.