## 5 Use of Artificial Neural Network
The hyperparameters are adjusted according to the determinations made in the DoE, and the MAE values for the validation data are calculated. Once the model with the optimum MAE is identified, the residuals are determined to establish the prediction limits. The testing data are used to assess whether the network possesses the desired predictive capability. The columns of Table 3 display the MAE obtained for the validation and testing data for each configuration of the neural network and its replicates. Additionally, the average computation time per epoch in seconds is included, along with the number of trainable parameters, which provides an indication of the network's complexity. From the response surface model for the MAE of the validation and testing data, it is inferred that the significant factors are the size of the data series (A), the number of encoder blocks (E), the size of the multilayer perceptron (F), and the learning rate (G). Additionally, the normality of the data allows for the application of ANOVA, leading to the conclusion that the mean errors of the two data groups (validation and testing) are significantly different, which may be attributed to various underlying causes. The  
Figure 5: Identification of the best ANN model. presence of _data drift_ or _data shift_ refers to a change in the distribution of the predictors, a change in the distribution of the target variable, or a change in the relationship between the predictors and the target variable. This phenomenon leads to a progressive decline in the model's predictive accuracy [71]. An additional factor contributing to this issue is the model's tendency to overfit the validation and training data, which compromises its ability to generalize effectively to new testing datasets. Resolving this challenge is non-trivial, and [72] highlights that the root causes often lie in the use of excessively complex models with a high number of trainable parameters or the model's inadvertent assimilation of noise inherent in the data series. To further analyze the network, a design is implemented in which the discarded variables (B, C, and D) are fixed at their maximum, median, and minimum values, verifying whether the variables excluded by the DoE truly have no impact on the network's performance. Figure 6 presents the boxplot of the validation and test values, highlighting the non-significant factors and enabling the clear observation of distinct processes. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**No. **} & \multicolumn{2}{c}{**Run 1**} & \multicolumn{2}{c}{**Run 2**} & \multicolumn{2}{c}{**Run 3**} & \multirow{2}{*}{**s/Epoch**} & \multirow{2}{*}{**Trainable**} \\ \cline{2-2} \cline{5-8}  & **Val. ** & **Test** & **Val. ** & **Test** & **Val. ** & **Test** & **(Average)** & **Param. ** \\ \hline
1 & 0.1452 & 0.1930 & 0.1291 & 0.1746 &  
After analyzing the normality of the obtained MAE values, ANOVA is performed on the normally distributed data, and the non-parametric Kruskal\(-\)Wallis test is applied to the non-normal distributions (test data). In both cases, it is concluded that the most complex architecture yields the lowest MAE, although the input data series must be the shortest. Likely, the use of a larger number of data points in the input series introduces additional noise into the network, resulting in poorer predictions. Therefore, the optimal architecture corresponds to the network (A = 20, B = 192, C = 6, D = 6, E = 2, F = 64, G = 15; MAEVal = 0.06289 and MAETest = 0.09573). Figure 7 illustrates the residuals of the validation and test data, which represent the difference between the observed actual values and those estimated by the neural network. It is readily apparent that the residuals exhibit bias, specifically 0.91 MPa, indicating that the predicted values are systematically lower than the actual values. Furthermore, the distributions of these residuals are non-normal.