## 3 Experimental Validation
### AlSi10 Mg Validation
#### 3.1.1 Probability ML Validation  
In this Section, the Probability ML approach is validated. Figure 6 plots the loss function in function of the k-fold considered for the validation. For example, k = 3 means that the third subset has been considered as the validation dataset. According to the k-fold validation scheme, the data available in each validation fold is at least 24, thus providing a reliable validation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Orientation** & **Power** & **Speed** & **Hatch Distance** & **Layer Thickness** & **Average Powder Size** & **Risk Volume** \\ \hline  & [W] & [mm/s] & [\(\upmu\)m] & [\(\upmu\)m] & [\(\upmu\)m] & [mm\({}^{3}\)] \\ \hline
[0; 90] & [220; 380] & [600; 1650] & [130; 190] & [30; 60] & [30; 41.5] & [250; 2300] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Range of process parameters for the datasets on AlSi10 Mg considered for the validation of the developed ML algorithms. Figure 5: AlSi10 Mg datasets considered for the validation of the proposed ML algorithms. According to Figure 6, the trend is almost constant and flat, with the loss function below 0.2 for all folds, except for the first one, characterized by a loss function above 2.5. This means that the developed ML algorithm works well for the investigated dataset, but for one fold, it has a low predicting capability. Figure 7 shows the Gumbel plot for the validation (Figure 7a) fold and a training fold (Figure 7b) for k = 1, i.e., the validation fold for which the loss function reaches its maximum value and where the Probability ML shows, therefore, its worst performance. According to Figure 7a, the predicted probabilities are not conservative, with the Probability ML algorithm providing larger probabilities and with the predicted data above the experimental data. Indeed, the ML-predicted probability of a defect smaller than \(\sqrt{a_{c}}\) is larger (with therefore a smaller probability of defects with a size larger than \(\sqrt{a_{c}}\)), being theML predictions less-safe. The percent error, i.e., the difference between the predicted probability and the experimental one, normalized by the experimental probability has also been adopted to quantify the error in the prediction. The percentage error has been found in the range [5.2, 1310]%, with a median value of 94%. As expected, the percentage error is very large, especially for small defect sizes where it reaches its maximum. Indeed, the difference range reduces to [5.2, 605]%, with the median value equal to 81%, without considering the smallest defect. However, it must be noted that this validation dataset is the one showing anomalous behavior, i.e., the one with defects significantly larger than those found in other datasets (Figure 5). This could be the reason for the large differences in Figure 7a and the predicting capability can be enhanced by increasing the number of training datasets covering the range of \(\sqrt{a_{c}}\) above 400 \(\mu\)m. On the other hand, the ML and  
Figure 6: k-fold cross-validation: loss function with respect to the k-fold considered for the validation. Figure 7: Gumbel plot for k = 1; (**a**) validation dataset; (**b**) training data. the experimental probabilities for the training dataset in Figure 7b overlap, with negligible differences (percent difference in the range [0.6, 15.6]%, with a median value equal to 2.4). Figure 8 shows the Gumbel plots for the validation dataset for k = 4, i.e., one of the validation folds characterized by a loss function below 0.2. Figure 8b shows the Probability-Probability plot, with the abscissa axis reporting the experimental probability, \(P_{\text{exp}}\), and the ordinate axis reporting the probability estimated with the ML algorithm, \(P_{\text{est}}\), for k = 4 and by considering the training and the validation data. According to Figure 8a, the ML probabilities are in good agreement with the experimental data for k = 4, with limited differences. The percent difference is in the range [2.5, 212.5]%, with a median value of 64%.