TITLE:
A Novel Method for Solving Nonlinear Schrödinger Equation with a Potential by Deep Learning
AUTHORS:
Chaojun Zhang, Yuexing Bai
KEYWORDS:
Physics-Informed Neural Networks, Improved Physics-Informed Neural Net-works, Defocusing NLS Equation, Rogue Wave Solution
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.10 No.10,
October
27,
2022
ABSTRACT: The improved physical information neural network algorithm has been proven to be used to study integrable systems. In this paper, the improved physical information neural network algorithm is used to study the defocusing nonlinear Schrödinger (NLS) equation with time-varying potential, and the rogue wave solution of the equation is obtained. At the same time, the influence of the number of network layers, neurons and the number of sampling points on the network performance is studied. Experiments show that the number of hidden layers and the number of neurons in each hidden layer affect the relative L2-norm error. With fixed configuration points, the relative norm error does not decrease with the increase in the number of boundary data points, which indicates that in this case, the number of boundary data points has no obvious influence on the error. Through the experiment, the rogue wave solution of the defocusing NLS equation is successfully captured by IPINN method for the first time. The experimental results of this paper are also compared with the results obtained by the physical information neural network method and show that the improved algorithm has higher accuracy. The results of this paper will be contributed to the generalization of deep learning algorithms for solving defocusing NLS equations with time-varying potential.