## 3 Results and Discussion
Figure 8 shows the actual against predicted Aluminum crack depth for selected machine learning models. _Composite Models_: For all models (Gradient Boosting Regressor, SVR, and Random Forest Regressor), the Mean Squared Error (MSE) is used to evaluate their performance, which measures the average squared difference between the predicted and actual values. The lower the MSE, the better the performance of the model. The results show that the Gradient Boosting Regressor model has a Training MSE of 0.208 and a Validation MSE of 0.014, indicating that the model performs well on both the training and test sets and has not overfit the training data. The SVR model has a Training MSE of 0.182 and a Validation MSE of 0.094, indicating that the model's predictions are off by 0.182 and 0.094 squared units from the actual values in the training and test sets, respectively. The Random Forest Regressor model has a Training MSE of 1.649 and a Test MSE of 0.189. Although the Test MSE is lower than the baseline model, it is higher than the Test MSE of the other two models, suggesting that the model may have to overfit the training data. Overall, the Gradient Boosting Regressor and SVR models seem to be the best performers based on these results. They have the lowest Test MSE values and are less likely to overfit compared to the Random Forest Regressor model. Figure 9 shows the actual against predicted Concrete crack depth for selected machine learning models. The most important predictive features for aluminium are natural frequency, crack position, and temperature. For ABS and concrete, natural frequency and amplitude dominate,  
Figure 8: The actual against predicted Aluminum crack depth for selected machine learning models. Figure 9: The actual against predicted Concrete crack depth for selected machine learning models. while the position of the crack appears to be of little importance. This may seem counterintuitive, as theory suggests that the closer the crack is to the fixed position of the specimen, the more pronounced the effect on dynamic response. However, the effect of crack position is captured by the model under the natural frequency and amplitude terms, resulting in a small and relatively unimportant crack position coefficient. Using all of the features as predictors yields the lowest mean squared error (MSE) but removing crack location does not significantly increase the error in ABS and concrete models. Removing amplitude has a much more significant effect on the error, increasing it by approximately 60% relative to the all-features baseline. The most significant effect is seen by removing natural frequency from the feature pool, which increases the error significantly. This suggests that natural frequency is the dominant feature in the prediction of crack depth, and its effect on the model error is greater than the rest of the features combined. According to [30; 31], the formal verification techniques mentioned further down provide a general framework for assessing the correctness, reliability, and robustness of machine learning models used in predicting crack propagation under coupled load and temperature. 1. Formal Specification: The first step in applying formal verification methods is to establish a formal specification that defines the desired properties or requirements for the crack propagation prediction models. 2. Model Checking: Model checking can be used to verify if the machine learning models satisfy the specified properties. In this context, it would involve exhaustively exploring the model's behavior under various coupled load and temperature conditions to check if the predicted crack propagation adheres to known physical laws or expected behavior. 3. Sensitivity Analysis: Formal verification methods can also include sensitivity analysis, which involves analyzing the impact of changes in input parameters or model assumptions on the predicted crack propagation. This analysis can provide insights into the robustness of the models and their sensitivity to variations in load, temperature, or other factors. 4.