Objective  In the field of nonlinear dynamic of fiber optics, various fiber optic effects can be mathematically described by differential equations such as the nonlinear Schrödinger equation (NLSE) that describes the evolution of optical signals due to many physical effects such as loss, dispersion and nonlinearity; Stimulated Raman scattering (SRS) ordinary differential equation describes the power evolution caused by stimulated Raman scattering; And the paraxial Helmholtz equation (PHE) describes the distribution and propagation of optical mode fields in fibers with various geometric structures. For a long time in the past, differential equations, including these three equations, were solved using numerical methods, most of which are based on the idea of difference and microelement, and discretize the computational domain and then obtain the approximate solution of the differential equation through iteration, such as finite difference method (FDM), finite-difference time-domain method (FDTD), finite element method (FEM), spectral method (SM), etc. However, the main problems faced by numerical methods are as follows. In complex scenes (such as high nonlinearity, large scale, high dimension, etc.), in order to obtain stable and accurate results, it is necessary to divide the grid more precisely, and the number of iterations is proportional to the scale of the desired scene. As the complexity of the scene increases, the amount of computation increases Exponential growth, which consumes a lot of computing resources and computing time; The computational resources and time consumed by numerical methods are unbearable, and there is currently no reasonable solution to these problems. Therefore, it is necessary to introduce a new equation solving tool with the properties of efficiency and low complexity to avoid the difficulties faced by numerical methods to meet the needs of accurate modeling of the dynamic process of interest physical quantity in complex scene. In recent years, in the field of computational physics, a revolutionary scheme for directly solving differential equations using neural networks, the physics-informed neural network (PINN), was proposed, which has attracted widespread attention and has been successfully validated in various fields related to differential equations. For the purpose of accurate modeling of nonlinear dynamic of fiber optics, PINNs were employed to solve NLSE, SRS ordinary differential equation and PHE to preliminarily verify PINN's feasibility in the field of fiber optics in this paper.

  Methods  The principle of PINN is firstly elucidated in this paper (Fig.1). Since PINN is real-valued, while the NLSE and PHE are actually complex equations. Thus, when solving these two equations by PINN, it is necessary to first separate the real and imaginary parts of the equation to obtain the real and imaginary part equations (Eq.5, Eq.8, Fig.5). The loss function of SRS ODE is reformed in the form of a matrix due to the coupling effect between different channels (Fig.3). Taking mean square error as accuracy evaluator, the results of NLSE, SRS ODE and PHE obtained by PINN are respectively compared with that of split-step Fourier method (SSFM), multi-step per span (MSPS) method and finite difference beam propagation method (FD-BPM) (Fig.2, Fig.4, Fig.6-8) to verify the feasibility of PINN for the modelling of nonlinear dynamic of fiber optics. Additionally, the computational complexity and running time of PINN and numerical method is quantitatively analyzed (Fig.9).

  Results and Discussions   The feasibility of PINN for solving NLSE is verified in the scenario that considers the effects of group velocity dispersion (GVD), self-phase modulation (SPM) and third order dispersion (TOD) with multiple input signals such as Gaussian pulse, first-order soliton and second-order soliton in the transmission distance of 80 km (Fig.2). The verification scenario of SRS ODE is set to the C+L-band transmission system of a transmission bandwidth from 186.1 THz to 196.1 THz, a channel bandwidth of 100 GHz, and a protection interval of 600 GHz between C and L bands, which has a total of 96 channels under full load (Fig.4). The PHE is solved respectively in step-index fiber in the geometry of straight, bended and tapered with five lowest linear polarization modes employed as fiber inputs (Fig.6-8). The above validation schemes all achieved accuracy results compared to those of numerical methods with low computational complexity and running time (Fig.9).

  Conclusions  As a revolutionary differential equation solving scheme, PINNs are introduced to the modelling of the nonlinear dynamic of fiber optics in this paper. The feasibility of PINN is verified in three typical nonlinear scenarios by solving the NLSE, the SRS ODE and the PHE. At present, the scientific computing community driven by artificial intelligence is gradually improving. Artificial intelligence algorithms that introduce physical information may provide a new idea, method, and tool for nonlinear dynamic modelling of fiber optics in the future, which may fundamentally provide a reliable technique for various fields including modeling of nonlinear dynamic of fiber optics in scientific computing, design, modeling, and other aspects.