## 4 Construction of the Cascaded Model and Training
### Cascaded Model and Training  
As illustrated in Figure 9, a cascaded Transformer and DNN model is utilized in this work to accurately predict the COM value and its corresponding equalizer coefficients. A total of 325 channel features have been collected as a dataset, and a DNN model is used to predict the spectral feature for each channel, specifically the IL and RL at the Nyquist frequency points. These predicted spectral data, along with the channel features, are then fed into a Transformer model to derive the pre-equalization SBR. Utilizing the pre-equalization SBR, the corresponding equalizer parameters under the COM configuration are calculated by the DNN model. In order to compute the post-equalization SBR, equalizer parameters combined with the channel features and frequency-domain features are then sent into another Transformer model. Finally, the sampled post-equalization SBR, channel features, and spectral features are used to compute the final corresponding COM value in the DNN model. The input vector of physical characteristics for each channel can be defined as follows:  
\[\mathbf{X}=\left\{x_{ij}\right\}_{j=1:N_{C},j=1:N_{p}} \tag{8}\]  
Figure 8: The structure of our Transformer for regression tasks. Figure 9: The structure of the cascaded model. where \(N_{C}\) represents the number of channels in the dataset, \(N_{P}\) is the total number of channel features, and \(\mathbf{X}\) is an \(N_{C}\times N_{P}\) matrix. The element \(x_{ij}\) denotes the quantified value of a specific physical feature of a channel: for instance, \(x_{11}\) corresponds to the length feature value of first channel. The DNN used in the cascaded model can be represented by Equation (9):  
\[ \begin{array}{ll}\mathbf{Y}&=F_{DNN}(\mathbf{Z},\mathbf{\theta})\\ &=\mathbf{W}^{(l)}f_{\text{ReLU}}(\cdots(\mathbf{W}^{(2)}f_{\text{ReLU}}(\mathbf{W}^{(1)} \mathbf{X}+\theta^{(1)})\\ &\hskip 14.226378 pt+\theta^{(2)})\cdots)+\theta^{(l)}\end{array} \, \tag{9}\]  
where \(\mathbf{Z}\) represents the linearly weighted output of input matrix \(\mathbf{X}\) and model weights \(\mathbf{W}\), as described in Equation (3), and \(\theta\) represents the offset. The index \(l\) denotes the layer of the DNN model, which includes the input layer, hidden layers, and output layer. Our network employs a total of four different DNN models with varying hyperparameters: \(\mathbf{Y}^{l}\) for predicting spectral features, \(\mathbf{Y}^{c}\) for predicting equalizer parameters, \(\mathbf{Y}^{c}\) for predicting the eye diagram, and \(\mathbf{Y}^{c}\) for predicting the COM. The augmented matrix \(\mathbf{X}_{f}\) is obtained by horizontally concatenating the input matrix \(\mathbf{X}\) of the first DNN model with its output \(\mathbf{Y}^{l}\):  
\[ \begin{array}{llll}\mathbf{X}^{f}&=[\mathbf{X}]\mathbf{Y}^{l}]\\ &=\left[\begin{array}{ccccc}x_{11}&x_{12}&\cdots&x_{1N_{P}}&y_{1I}&y_{1R}\\ x_{21}&x_{22}&\cdots&x_{2N_{P}}&y_{2I}&y_{2R}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ x_{N_{C}1}&x_{N_{C}2}&\cdots&x_{N_{C}N_{P}}&y_{N_{C}I}&y_{N_{C}R}\end{array} \right]\,\end{array}  \tag{10}\]  
where the column vector \(\mathbf{y}_{l}=(y_{1I},y_{2I},\cdots,y_{N_{C}I})^{T}\) represents the predicted channel IL and \(\mathbf{y}_{R}=(y_{1R},y_{2R},\cdots,y_{N_{R}R})^{T}\) represents the predicted channel RL. To enhance the accuracy of the predicted COM value, the pulse response is extracted from the pre- and post-equalization SBR in the COM tool. The peak corresponding time point is taken as the main cursor, and the amplitudes of four pre-cursors and eight post-cursors determined by the COM sampling interval are selected as the required SBR. Then, the pulse of each channel is flattened and the corresponding time labels are inserted, forming a sequential feature vector. Equation (11) represents the sequence of amplitude values. After this processing, the Transformer model can be employed to accurately predict the SBR. \[\left\{