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.

本文首先将物理信息神经网络（PINNs）扩展到学习一维和二维可饱和非线性薛定谔方程（SNLSE）中具有两个基本的 PT - 对称 Scarf-II 和周期势的数据驱动的平稳和非平稳孤子。其次，研究了 PT - 对称潜力函数的数据驱动逆问题，而不仅仅是在一维和二维 SNLSE 中的潜力参数。尤其是，我们提出了一种修改后的 PINNs（mPINNs）方案，通过解决数据来直接识别一维和二维 SNLSE 的 PT 潜力函数。同时，使用 mPINNs 方法研究了关于依赖于传播距离 z 的一维和二维 PT - 对称势。我们还通过将 PINNs 应用于 SNLSE 的稳态方程来识别潜力函数。此外，在不同参数条件下比较了两种网络结构，使得预测的 PT 潜力可以达到相似的高准确性。这些结果说明建立的深度神经网络可以在一维和二维 SNLSE 中成功地使用，并且具有较高的准确性。此外，在一维和二维 PT Scarf-II 和周期势中讨论了影响神经网络性能的主要因素，包括激活函数、网络结构和训练数据的大小。特别地，对十二种不同的非线性激活函数进行了详细分析，包括周期和非周期函数，得出了根据解和方程的形式选择激活函数通常可以获得更好效果的结论。