Deep learning soliton dynamics and complex potentials recognition for 1D
  and 2D PT-symmetric saturable nonlinear Schrödinger equations
Jin Song Zhenya Yan
In this paper, we firstly extend the physics-informed neural networks (PINNs)
to learn data-driven stationary and non-stationary solitons of 1D and 2D
saturable nonlinear Schr\"odinger equations (SNLSEs) with two fundamental
PT-symmetric Scarf-II and periodic potentials in optical fibers. Secondly, the
data-driven inverse problems are studied for PT-symmetric potential functions
discovery rather than just potential parameters in the 1D and 2D SNLSEs.
Particularly, we propose a modified PINNs (mPINNs) scheme to identify directly
the PT potential functions of the 1D and 2D SNLSEs by the solution data. And
the inverse problems about 1D and 2D PT -symmetric potentials depending on
propagation distance z are also investigated using mPINNs method. We also
identify the potential functions by the PINNs applied to the stationary
equation of the SNLSE. Furthermore, two network structures are compared under
different parameter conditions such that the predicted PT potentials can
achieve the similar high accuracy. These results illustrate that the
established deep neural networks can be successfully used in 1D and 2D SNLSEs
with high accuracies. Moreover, some main factors affecting neural networks
performance are discussed in 1D and 2D PT Scarf-II and periodic potentials,
including activation functions, structures of the networks, and sizes of the
training data. In particular, twelve different nonlinear activation functions
are in detail analyzed containing the periodic and non-periodic functions such
that it is concluded that selecting activation functions according to the form
of solution and equation usually can achieve better effect.