## 3 Results and Discussion
In the context of the study, MSE was used as an evaluation metric to assess the performance of the machine learning models in predicting crack propagation. Figure 5: The correlation between Crack depth (mm) and amplitude (mm) for (**a**)—Aluminum, (**b**)—ABS, (**c**)—Concrete. Figure 6: The correlation between natural frequency (Hz) and amplitude (mm) in for (**a**)—Aluminum, (**b**)—ABS, (**c**)—Concrete. Lower MSE values indicate that the model's predicted crack propagation values closely match the actual values, suggesting better predictive accuracy. It is worth noting that while MSE provides a useful measure of model performance, it is not the only metric available. Depending on the specific objectives of the study, other metrics, such as mean absolute error (MAE), R-squared (coefficient of determination), or domain-specific metrics, could also be considered to evaluate and compare the performance of the models. Essentially, these metrics represent the average predictive error of the models [1]. Table 2 summarizes the performance metrics for training and validation predictions for the three materials. _ABS Models:_ The results show that the K-Neighbors Regressor model has a Training MSE of 0.195, indicating that, on average, the model's predictions are off by 0.195 squared units from the actual values in the training set. The Validation MSE of the same model is 0.086, indicating that the model's predictions are off by 0.086 squared units from the actual values in the test set. These values indicate that the K Neighbors Regressor model performs well on both the training and test sets, suggesting that it has not overfit the training data. The Lasso model's Training MSE is 0.195, and the Validation MSE is 0.182. Both values are lower than the MSE of the baseline model (0.437), indicating that the Lasso model outperforms the baseline. The results for the Ridge model show that it has a Training MSE of 0.194 and a Validation MSE of 0.182. These values are also lower than the MSE of the baseline model (0.437), indicating that the Ridge model outperforms the baseline. Overall, all three models (K Neighbors Regressor, Lasso, and Ridge) outperform the baseline model, suggesting that they have learned meaningful patterns in the data and can make reasonably accurate predictions. Figure 7 shows the actual against predicted ABS crack depth for selected machine learning models. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Learning Model** & **Training MSE** & **Validation MSE** \\ \hline \multirow{4}{*}{Aluminum} & K-Neighbors & 0.189 & 0.079 \\  & Regressor & 0.180 & 0.15 \\  & Lasso & 0.170 & 0.14 \\ \hline \multirow{4}{*}{ABS} & K-Neighbors & 0.195 & 0.086 \\  & Regressor & 0.195 & 0.182 \\  & Lasso & 0.194 & 0.182 \\  & Ridge & 0.194 & 0.182 \\ \hline \multirow{4}{*}{Concrete} & Gradient Boosting & 4.100 & 0.597 \\  & Regressor & 4.401 & 3.114 \\ \cline{1-1}  & SVM Forest & 3.903 & 0.509 \\ \cline{1-1}  & Regressor & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The evaluation of training and validation for selected models. Figure 7: The actual against predicted ABS crack depth. _Aluminum Models_: The results show that the K Neighbors Regressor model has a Training MSE score of 0.189 and a Validation MSE score of 0.079, which is lower than the MSE of the baseline model (0.8907), indicating that the K Neighbors Regressor model outperforms the baseline. The lower the MSE score, the better the performance of the model. The Ridge model has a Training MSE score of 0.170 and a Validation MSE score of 0.149, both of which are lower than the MSE of the baseline model. The Lasso model has a Training MSE score of 0.180 and a Validation MSE score of 0.159, which are also lower than the MSE of the baseline model. In all cases, the Validation MSE score is lower than the MSE of the baseline model, suggesting that the models have learned meaningful patterns in the data and can make reasonably accurate predictions.