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。

1D 和 2D PT - 对称饱和非线性薛定谔方程的深度学习孤子动力学和复杂势能识别

Deep learning soliton dynamics and complex potentials recognition for 1D  and 2D PT-symmetric saturable nonlinear Schrödinger equations

In this paper we analyze crowd turbulence from both classical and quantum
perspective. We analyze various crowd waves and collisions using crowd
macroscopic wave function. In particular, we will show that nonlinear
Schr\"{o}dinger (NLS) equation is fundamental for quantum turbulence, while its
closed-form solutions include shock-waves, solitons and rogue waves, as well as
planar de Broglie's waves. We start by modeling various crowd flows using
classical fluid dynamics, based on Navier-Stokes equations. Then, we model
turbulent crowd flows using quantum turbulence in Bose-Einstein condensation,
based on modified NLS equation.
Keywords: Crowd behavior dynamics, classical and quantum turbulence, shock
waves, solitons and rogue waves

该论文分析了古典和量子视角下的人群紊流现象，从人群宏观波函数出发，研究了各种人群波和碰撞，特别地，表明非线性薛定谔方程是量子紊流的基础，其解包括冲击波、孤子和流氓波以及平面德布罗意波；其次，基于纳维 - 斯托克斯方程，建模了各种人群流，然后利用玻色 - 爱因斯坦凝聚的量子紊流模拟了紊乱的人群流。