Predicting bright and dark solitons in (2+1)-dimensional Bose-Einstein Condensates by two types of neural operator networks

Abstract

This study makes predictions regarding the soliton dynamics of the Gross-Pitaevskii equation in Bose-Einstein Condensates. Based on the exact solutions for the bright and dark one-soliton obtained by the Hirota bilinear method, numerical experimental calculations are performed using two neural operator networks, namely the deep operator network (DeepONet) and the fourier neural operator (FNO). The results show that both of these two methods can successfully predict the bright solitons and dark solitons of the system, and the obtained numerical solutions have high accuracy. The FNO method is superior to DeepONet in terms of prediction accuracy, while the latter has more advantages in terms of time economy. Both methods have perfectly solved the limitations of the traditional neural network methods, such as slow prediction speed and fixed grids. The excellent results of predicting the dynamics of complex ring dark solitons (RDSs) and semi-vortex solitons also demonstrate the broad applicability of the algorithm employed. This provides a new and promising approach for deep learning methods to address the challenges of accumulable systems.