## 1 Introduction  
As the most widely used optical signal carrier, accurate modeling of standard single-mode fibers is an indispensable step in constructing optical communication systems. The existing digital signal processing methods are already able to effectively compensate the linear distortion of signals in fiber optic transmission, such as attenuation, dispersion, and polarization mode dispersion [1]. Therefore, nonlinear effects will become the main challenge for large launch power and long-distance fiber optic signal transmission. Due to the popularity of deep learning and end-to-end optimization methods, the accuracy of the fiber nonlinearity modeling will largely determine the effectiveness of transceiver compensation for nonlinearity [2; 3; 4; 5; 6]. The most widely used method for fiber optic modeling is the standard split-step Fourier method (SSFM) and its improved methods [7; 8; 9]. It obtains the numerical solution of the optical field by solving the nonlinear Schrodinger equation (NLSE) in several steps, and each step achieves an approximate analytical solution of the optical signal by considering only dispersion or nonlinearity in a series of alternating short length fibers. SSFM can achieve high accuracy in situations of high complexity and is a commonly used fiber modeling method in commercial software. However, due to the fact that the SSFM is a serial method, there are difficulties in backpropagation and risks of gradient vanishing and exploding when combined with deep learning-based transceivers [10; 11]. The Gaussian noise (GN) method is a simplified method based on describing nonlinear noise in the frequency domain [12; 13; 14; 15]. It directly adds nonlinear noise to the transmission signal by treating it as additive Gaussian noise, while ignoring that the fundamental source of nonlinear noise is the interaction between signals. Compared to the SSFM, although its computational complexity is obviously decreased, the GN model is not effective under conditions of low dispersion accumulation and large launch power [16] and is difficult to combine with advanced equalization algorithms [10]. The RP model is another commonly used simplified fiber modeling method [17], which obtains the first-order nonlinear solution of the nonlinear Schrodinger equation [18] through series matching to simplify the SSFM. Although it sacrifices some accuracy, the RP model is a parallel method that can improve the accuracy of the model by increasing the number of branches, making up for the difficulty of backpropagation in the SSFM and facilitating parameter updates in deep learning. In the past 20 years, multiple derivative algorithms have emerged for the RP method to model various waveforms, nonlinear interferences (such as SPM, IXPM, IFWM, XPM, FWM), and ASE noise interactions [19; 20; 21; 22; 23; 24; 25]. In a recent study, learnable filters were introduced and combined with machine learning to achieve high accuracy. This approach takes into account the nonlinear interactions of adjacent symbols, but the complexity increases rapidly as the number of symbols considered expand [26]. The RP model has also been applied in end-to-end learning [27] and fiber longitudinal power distribution estimation [28]. Meanwhile, due to the fact that perturbations are additive effects relative to signals, they are easily transformed into modeling and equalization, as well as combined with deep learning equalization methods [29; 30; 31; 32; 33]. In addition, the RP model can flexibly introduce high-order solutions to tackling the problem of insufficient accuracy of the first-order perturbation models for large-power signal input [34], but this will significantly increase the complexity of the algorithm.