## 4 Dimensional Deviations Study
### Data Processing
At the end of the training process, the \"Train Network\" window displays the values of the correlation coefficient R\({}^{2}\) and the mean squared error (MSE), as shown in Table 6.  
The results presented in Table 6 indicate a strong correlation between the predicted and actual values across all datasets, as reflected by the high R\({}^{2}\) values (above 0.96). The mean squared error (MSE) remains at a very low magnitude, suggesting that the network has achieved a high level of accuracy in training, validation, and testing. Notably, the validation error (MSE = 2.84401 \(\times\) 10\({}^{-7}\)) is slightly higher than the training error (MSE = 2.16272 \(\times\) 10\({}^{-7}\)), which is expected as the model generalizes to unseen data. Figure 3: The neural network architecture. However, the increase in MSE for the test set (4.87121 \(\times\) 10\({}^{-7}\)) suggests a slight overfitting, meaning the model might perform marginally better on training data than on entirely new inputs. Nonetheless, the overall performance remains strong, indicating a well-trained network with good generalization capabilities. Figure 4 illustrates the learning curve of the neural network by displaying the mean squared error (MSE) for the training, validation, and test datasets over eight epochs. The blue, green, and red curves represent the training, validation, and test errors, respectively. The model achieves its best validation performance at epoch 7, with an MSE of 2.844 \(\times\) 10\({}^{-7}\), as indicated by the green dotted line and the marked point on the graph. The downward trend of all three curves suggests that the model is learning effectively, with the errors decreasing as training progresses. However, after epoch 7, the validation and test errors stabilize, while the training error continues to decrease slightly. This behavior suggests that further training might lead to overfitting, where the model becomes too specialized to the training data and loses generalization capability. Therefore, stopping the training at epoch 7 ensures the best trade-off between accuracy and generalization. The training process was validated when an overall R\({}^{2}\) value (0.957) was obtained. Figure 5 presents the values of R\({}^{2}\) for training, testing, validation, and the value of the overall R\({}^{2}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset** & **Number of Samples** & \
\begin{tabular}{c} **MSE (Mean** \\ **Squared Error**) \\ \end{tabular} &
\begin{tabular}{c} **R2 (Correlation** \\ **Coefficient)** \\ \end{tabular} \\ \hline Training & 19 & 2.16272 \(\times\) 10\({}^{-7}\) & 0.962024 \\ Validation & 4 & 2.84401 \(\times\) 10\({}^{-7}\) & 0.987041 \\ Testing & 4 & 4.87121 \(\times\) 10\({}^{-7}\) & 0.968548 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The experimental plan. Figure 4: Training, validation, and test error evolution over epochs. The values of the specific gamultiobj multi-objective optimization using genetic algorithm parameters were the following: population size = 10, crossover, crossover ratio = 0.8, reproduction, and crossover fraction = 0.8. After 103 iterations, in the case corresponding to diameter \(\mathcal{O}\)d1, \(\mathcal{O}\)D1, the optimized conditions were obtained, as shown in Figure 6.  
Figure 5: The values of R\({}^{2}\) for training, testing, and validation processes. Figure 6: Optimization using a genetic algorithm for \(\mathcal{O}\)d1, \(\mathcal{O}\)D1 diameters. In this case, an identical architecture to the first network was used for a second neural network, which was trained, tested, and validated: three input neurons, corresponding to the input parameters; two output neurons, corresponding to the output parameters; and ten neurons on the hidden layer, empirically chosen. For this process, the overall R\({}^{2}\) is 0.947.