  This work aims to provide an effective deep learning framework to predict the
vector-soliton solutions of the coupled nonlinear equations and their
interactions. The method we propose here is a physics-informed neural network
(PINN) combining with the residual-based adaptive refinement (RAR-PINN)
algorithm. Different from the traditional PINN algorithm which takes points
randomly, the RAR-PINN algorithm uses an adaptive point-fetching approach to
improve the training efficiency for the solutions with steep gradients. A
series of experiment comparisons between the RAR-PINN and traditional PINN
algorithms are implemented to a coupled generalized nonlinear Schr\"{o}dinger
(CGNLS) equation as an example. The results indicate that the RAR-PINN
algorithm has faster convergence rate and better approximation ability,
especially in modeling the shape-changing vector-soliton interactions in the
coupled systems. Finally, the RAR-PINN method is applied to perform the
data-driven discovery of the CGNLS equation, which shows the dispersion and
nonlinear coefficients can be well approximated.
