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In this paper, we firstly extend the physics-informed neural networks (PINNs)to learn data-driven stationary and non-stationary solitons of 1D and 2Dsaturable nonlinear Schr\"odinger equations (SNLSEs) with two fundamentalPT-symmetric Scarf-II and periodic potentials in optical fibers. Secondly, thedata-driven inverse problems are studied for PT-symmetric potential functionsdiscovery rather than just potential parameters in the 1D and 2D SNLSEs.Particularly, we propose a modified PINNs (mPINNs) scheme to identify directlythe PT potential functions of the 1D and 2D SNLSEs by the solution data. Andthe inverse problems about 1D and 2D PT -symmetric potentials depending onpropagation distance z are also investigated using mPINNs method. We alsoidentify the potential functions by the PINNs applied to the stationaryequation of the SNLSE. Furthermore, two network structures are compared underdifferent parameter conditions such that the predicted PT potentials canachieve the similar high accuracy. These results illustrate that theestablished deep neural networks can be successfully used in 1D and 2D SNLSEswith high accuracies. Moreover, some main factors affecting neural networksperformance are discussed in 1D and 2D PT Scarf-II and periodic potentials,including activation functions, structures of the networks, and sizes of thetraining data. In particular, twelve different nonlinear activation functionsare in detail analyzed containing the periodic and non-periodic functions suchthat it is concluded that selecting activation functions according to the formof solution and equation usually can achieve better effect.

Deep learning soliton dynamics and complex potentials recognition for 1D and 2D PT-symmetric saturable nonlinear Schr\"odinger equations