## 4 Results and Discussions  
To evaluate and examine the performance of the 2D-CNN, 1D-CNN, 1D-DNN, and VNE, the three neural network models in the experiment utilized identical parameter configurations and employed the same training and testing datasets. Table 2 presents the pertinent parameter values. The training approach used 100 epochs and a batch size of 512. These parameters were established through numerous trials, ensuring both training efficacy and enhanced training efficiency. The network used a two-layer design: the first layer had a 3 \(\times\) 2 convolutional kernel (time span \(\times\) real-imaginary channels), where the three-symbol coverage helped track the phase consistency between nearby symbols, and the two channels referred to the in-phase (I) and quadrature-phase (Q) parts of the QPSK signal. This setup allowed for simultaneous modeling of how the signal changed over time and the complex relationships between I/Q components. The second layer had a smaller 3 \(\times\) 1 kernel, which kept the focus on capturing important time-related features while also looking for more complex patterns. This design allowed the network to simultaneously understand how the signal changes over time and how the I/Q components were related to each other in a complex way. The second-layer kernel was reduced to 3 \(\times\) 1, which kept the ability to extract time-related features while focusing on capturing more complex abstract features. The pooling layer implemented (2, 1) subsampling to reduce feature dimensions along the time axis, hence augmenting translation invariance while maintaining complete information in the I/Q channels. An L1 regularization term (\(\lambda=10^{-6}\)) was added to the loss function to help prevent overfitting by encouraging the weight matrix to have fewer non-zero values. We included a Dropout layer at the network output with a probability of 0.2, which randomly deactivated neurons, thereby enhancing the ability to generalize. A dynamic early stopping mechanism (patience = 25) monitored validation loss and halted training when performance remained unchanged for 25 consecutive epochs, therefore preventing overfitting and optimizing computing resource use. We empirically confirmed that a batch size of 512 optimized CPU memory limitations and gradient update stability, facilitating reliable training convergence in intricate channel settings. The design of the VNE detailed in this work specifies the tap count of the LMS linear filter as 15 and the order of the Volterra nonlinear filter as 13. The settings were adjusted through testing: the 13 th-order Volterra structure found a good balance between handling nonlinearity and being efficient in calculations, as improvements levelled off and the risk of overfitting rose if we went beyond this order. The 15-tap version of the LMS filter adequately addressed the principal inter-symbol interference (ISI) range and reduced unnecessary calculations, as established by the autocorrelation analysis of the received signals. Dynamic step-size adjustment (\(u_{LMS\_init}=2\times 10^{-5},\ u_{polterra}=2\times 10^{-6}\)) and a regularization parameter (\(\lambda_{init}=10^{-6}\)) were used to make the model more stable and adaptable, helping to effectively correct for nonlinear distortions and phase noise in long-distance transmissions. Figure 4 illustrates the BER curves with optical power into PD. The figure illustrates that, under varying optical power settings, the 2D-CNN exhibited the lowest overall BER