  The nonlinear Schrodinger equation is a classical integrable equation which
contains plenty of significant properties and occurs in many physical areas.
However, due to the difficulty of solving this equation, in particular in high
dimensions, lots of methods are proposed to effectively obtain different kinds
of solutions, such as neural networks, among others. Recently, a method where
some underlying physical laws are embeded into a conventional neural network is
proposed to uncover the equation's dynamical behaviors from spatiotemporal data
directly. Compared with traditional neural networks, this method can obtain
remarkably accurate solution with extraordinarily less data. Meanwhile, this
method also provides a better physical explanation and generalization. In this
paper, based on the above method, we present an improved deep learning method
to recover the soliton solutions, breather solution and rogue wave solutions to
the nonlinear Schrodinger equation. In particular, the dynamical behaviors and
error analysis about the one-order and two-order rogue waves of the Schrodinger
equation are revealed by the deep neural network for the first time. Moreover,
the effects of different numbers of initial points sampled, residual
collocation points sampled, network layers, neurons per hidden layer on the
one-order rogue wave dynamics of this equation have been considered with the
help of the control variable way under the same initial and boundary
conditions. Numerical experiments show that the dynamical behaviors of soliton
solutions, breather solution and rogue wave solutions of the integrable
nonlinear Schrodinger equation can be well reconstructed by utilizing this
physically-constrained deep learning method.
