## 3 Results and Discussion
### The Discussion of the MRP Model  
Firstly, we will analyze the impact of different branches numbers, \(N_{step}\), on the accuracy of the model, which guides us to further validate the required step size standards and also determines the computational complexity of the model. We fixed a large launch power \(P_{0}=13\) dBm and traversed the performance of different \(N_{step}\) at a fiber length of \(L=120\) km, as shown in Figure 5. We note that, unless otherwise stated, the pulse shaping filter uses a root raised cosine (RRC) filter with the roll-off factor of 0.01. As shown in Figure 5, it can be observed that there is a limit to the fitting accuracy for both the RP model and the MRP model. Increasing the number of branches beyond this limit will not further improve system performance. \(N_{step}\) directly affects the structure and the complexity of the model; therefore, we chose \(N_{step}=10\), \(20\), and \(30\) as the branches of the MRP model and the RP model for comparison. \begin{table}
\begin{tabular}{c c} \hline \hline Parameter Initialization & \(k_{n}=1+0j\) \\ \(\delta_{i}=\) RHS of Equation (15) \\ Loss function & MSE \\ Optimizer & Adam \\ Learning Rate & \(\alpha_{epoch}=a_{0}\times 0.9^{\frac{\left|\frac{\alpha_{epoch}}{20} \right|}}\) \\ Epoch number & 1500 \\ Batch size & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Training parameters. Figure 6: Constellation diagrams comparison between SSFM and (**a**) MRP or (**b**) RP models. Figure 7: Comparison of waveforms and MSE between SSFM and RP or MRP models. for calculating RMSE. It is obvious that the fitting error between the MRP model and SSFM is smaller in the large launch power range than standard RP. Taking the results at 14 dBm as an example, the RMSE of the 30-step RP model is 11.82%, while the RMSE of the 30-step MRP model is only 2.77%, reducing the error by more than 75%, and the computational complexity of the two models is completely the same. As we have already gone through the standard DSP process and the same normalization process before calculating RMSE, the error here is almost entirely composed of nonlinear differences. If factors such as phase rotation are considered, the performance gap between the RP model and the MRP model will further increase. In practical systems, both the signal and amplifier are not ideal. Due to the presence of ASE noise and arbitrary waveform generator AWG sampling noise, the signal at the transmitting end is modeled as the sum of ideal signal and Gaussian noise. Nonlinear effects will occur between the noise and signal, resulting in the Gordon-Mollenauer Effect [46]. To verify the accuracy of the model's estimation of non-ideal signals, we set the initial signal-to-noise ratio of the signal to 37 dB, and we consider an additive Gaussian noise of a constant power of 0.01 mW before entering the fiber to simulate sampling noise and ASE noise separately. Note that we use multiple independent Gaussian noise sources to ensure that the sum of the noise remains Gaussian distributed. Figure 9 shows the accuracy of the model in estimating the received signals' EVM without and with ASE noise. When the launch power is large, the MRP model is significantly better than the RP model. In addition, we found that both the RP model and MRP model tend to underestimate the EVM of the signal, and the underestimation degree of the MRP model is significantly lighter than that of the RP model. At a launch power of 13 dBm with ASE noise, the EVM difference between the 20-step MRP and SSFM is only 0.13%, which is much lower than the 2.51% corresponding to the 20-step RP model. Moreover, this difference will further expand with the increase in fiber launch power. It is worth noting that the joint analysis of Figures 8 and 9 shows that as the launch power increases, the growth trend of the EVM of the RP model slows down significantly, while the MRP model can continue to maintain its approximation with the SSFM, but the RMSE of both increases exponentially.