## 2 Principles of the Proposed ANN Equalizer
### Advanced Training Scheme Algorithm  
The flow diagram summarizing the training algorithm of the proposed ANN-based equalizer, utilizing the introduced advanced training scheme, is shown in Figure 3. First, the ANN weights are assigned to random initial values, and the stopping criteria of the training algorithm are defined (e.g., minimum required mean square error (MSE)). The maximum allowed number of training epochs and/or improvement of validation accuracy is determined. The advanced training scheme is then utilized to generate the ANN training input-output dataset, as illustrated in Figure 1(b). Using the current weight values, the corresponding ANN equalizer outputs are evaluated, and the corresponding MSE between current equalizer outputs and desired outputs is calculated (see Equation (2)). The algorithm then checks whether the predefined stopping criteria are satisfied or not.  
Figure 2: Schematic diagram of the training data preparation for an \(M\)-ary modulation format in the case of **(a)** extensive training scheme [21] and **(b)** advanced training scheme. \(K\) is the number of interfering symbols and \(N=2K+1\) is the ISI span length, i.e., the required number of equalizer taps. The arrows refer to the original data used during the training of the ANN equalizer, after discarding the redundant guard data.  
In other words, it determines whether the current ANN model is well trained or not. If all stopping criteria are satisfied, the algorithm terminates, and the best-achieved ANN model is saved for testing. Otherwise, the Adam optimization algorithm is employed to update the current ANN weights such that the current MSE is reduced. The new ANN model (i.e., ANN with new weight parameters) is considered as the current model. The current equalizer outputs and MSE are re-calculated, and the cycle is repeated till the algorithm terminates.  
Once the training of the ANN equalizer is completed, actual unknown data can then be transmitted through the channel, and the trained ANN equalizer is utilized at the DSP unit of the receiver to compensate for the system distortions, recovering the transmitted data. In this case, the input vector of the ANN equalizer at time instant \(i\) is given as  
\[\mathbf{Y}_{i}=\left[y_{-K+i,}\ \ y_{-K+i+1}\ \ldots\ \ y_{i}\ldots\ y_{K+i-1,}\ \ y_{K+i}\right]^{T},\ \ \ i=K,\ K+1,\ K+2,\ \ldots, \tag{3}\]  
where \(y_{i}\) is the received sampled symbol at time instant \(i\), and \(K\) is the number of interfering symbols. For each input vector \(\mathbf{Y}_{i}\), the ANN equalizer produces an equalized signal sample \(\hat{X}_{i}\), which is assumed to be a reliable estimate of the actual transmitted signal sample \(\hat{X}_{i}\).  
Notice that during the transmission of actual data, the nominal parameters of the fiber channel (e.g., dispersion coefficient \(\beta_{2}\)) might be subjected to fluctuations due to variations in environmental conditions. Although these fluctuations occur at a much slower rate than the transmission data rate, it can slightly deteriorate the performance achieved by the trained ANN equalizer. However, the proposed ANN equalizer can be simply modified to adaptively re-adjust its weights and trace these channel fluctuations using a small variation in the weights' values. In practice, only one optimization step is needed [2]. Since we have no information about the transmitted signal, in this case, the single optimization step is performed to minimize the MSE between the final decided symbol value \(\hat{a}_{i}\) and the equalizer output \(\hat{X}_{i}\), i.e., the objective function of Equation (2) becomes \(E(\mathbf{W})=\left[\hat{X}_{i}(\mathbf{W},\mathbf{Y}_{i})-\hat{a}_{i}\right] ^{2}\).