## 2 MRP Models and Algorithm Description
### Optimization and Training Procedure  
In this part, we will mainly explain the optimization methods for various parameters of the MRP model. We use supervised learning to update the learnable parameters in the model. Our model processes both x-pol and y-pol optical signal data, so the model can be simply expressed as:  
\[ \begin{split}\left[U_{x}(z,t),U_{y}(z,t)\right]=MRP\left[U_{x}(0, t),U_{y}(0,t)\right]\end{split}  \tag{11}\]  
We use a randomly uniformly distributed signal of length, \(S_{0}\), which is filtered by a pulse shaping filter and is upsampled to two samples per symbol to generate the input sequence (i.e., sampling length of \(S=2\times S_{0}\)). The label uses the SSFM to generate sequences of the same length. The SSFM can be represented by:  
\[\begin{split}&\begin{split} SSFM_{\delta}\left[U_{x,y}(0,t) \right)\right]=U_{x,y}(\delta,t)\\ =&\begin{split}\mathbf{\mathcal{F}}^{-1}\left[\omega^{ (\frac{\delta}{2}\alpha^{2}+\frac{\delta}{2}\alpha^{3})\frac{\delta}{2}}\mathbf{ \mathcal{F}}\left[e^{\frac{\delta}{2}\gamma\delta\delta_{eff}\cdot\left[-af (\frac{\delta}{2})\right]\left\{\left\{U_{x,y}(\frac{\delta}{2},t)\right\}^{2 }+\left\lvert U_{y,x}(\frac{\delta}{2},t)\right\rvert^{2}\right\}}U_{x,y}^{(0 )}(\frac{\delta}{2},t)\right]\right]\\ & \begin{split} SSFM_{L}\left[U_{x,y}(0,t)\right) \right]=U_{x,y}(L,t)=\underbrace{SSFM_{\delta}\left[SSFM_{\delta}\left\{U_{x,y}(0,t)\right\}\right]]}_{iterate\times L/\delta}\end{split}  \tag{12}\]  
The meanings of each symbol here are the same as those of the MRP model, with the subscripts \(\delta\) and \(L\) is the distance of optical signal transmission. The \(U_{x,y}^{(0)}\) here is an operator rather than a simple function, which will not cause ambiguity when defining the MRP model, but special clarification is needed here. After generating labels, due to the obvious power increment of the RP model and its derivative models, we use a simple mean normalization to align the power of the model's outputs and the labels. \[ \begin{split} Y_{x,y}(L,sT)=\frac{\left\lvert U_{x,y}(L,sT) \right\rvert}{\text{mean}(\left\lvert U_{x,y}(L,sT)\right\rvert)},\;\;s=1,2, \cdots,S\end{split}  \tag{13}\]  
Here, \(Y\) is the MRP output sequence or the SSFM label and \(T\) is the sampling period. To minimize the error between the labels and the outputs of the MRP model, we update the learnable parameters in the model using the batch gradient descent method. The loss function is defined as the mean squared error (MSE) loss:  
\[ \begin{split} L=\underset{\delta,K}{\text{min}}\frac{1}{2B} \left[\sum_{s=1}^{S}\left\lvert\hat{Y}_{x}-Y_{x}\right\rvert^{2}+\left\lvert \hat{Y}_{y}-Y_{y}\right\rvert^{2}\right]\end{split}  \tag{14}\]  
Here, \(B\) represents the amount of the data input to the model to calculate the gradient at a time, i.e., batch size. In addition, we choose the Adam algorithm for parameter updates to minimize the loss function, and the specific learning rate and batch size will be explained in the next section. Due to the form of the loss function, we do not include ASE noise in either the generated labels or the training inputs, as the addition of ASE noise can affect the model's convergence, especially at lower launch power levels. However, in the comparison of results in the next section, we compared the accuracy of the model in both cases without and with ASE noise. For learnable parameters, \(\delta\), representing step size distribution, we made a slight modification on the basis of the RP model and used the step size of the logarithmic distribution to \(\delta\). The results in [44; 45] indicate that using a logarithmic step equalization model for channel estimation has better accuracy, as it increases the step calculation density for the majority of power in the fiber optic channel, that is, the proportion of large-power components in the total computational complexity. This has a certain effect on improving the accuracy of channel modeling from integral form to summation form.