## 5 Result and Discussion
### Machine Learning Model  
In this study, several regression models were employed, including linear regression (LR), ridge regression (RID), lasso regression (Lasso), random forest regressor (RFR), gradient boosting decision trees regressor (GBDT), AdaBoost regressor (ABR), and XGBoost regressor (XGB). The optimal hyperparameters for each model were determined using Bayesian optimization, and prediction errors on the test set were used to evaluate their performance [68]. To reduce randomness, five-fold cross-validation was performed, and the average MSE, MAE, and R\({}^{2}\) values are shown in Figure 13. As illustrated, XGBoost regression achieved the lowest errors (MSE and MAE) and the highest R\({}^{2}\). Table 12 lists the optimal hyperparameters for each model. Figure 11: Damage patterns of the original structural configuration: (**a**) after the first thermal shock, (**b**) after the second thermal shock, and (**c**) after the third thermal shock. Figure 12: Stress nephograms of the original structural configuration: (**a**) after the first thermal shock, (**b**) after the second thermal shock, and (**c**) after the third thermal shock. Figure 13: _Cont._  
XGBoost regression showed the best performance on the dataset. To further demonstrate its predictive ability, it was used to predict all data points from both the training and test sets. As shown in Figure 14, the data points closely align with the 45\({}^{\circ}\) diagonal, confirming a strong fit. The results indicate that XGBoost can accurately predict the thermal shock resistance of the coating. Feature importance analysis based on XGBoost is presented in Figure 15a. Although regression tree-based ensemble machine learning models often yield high accuracy, they are frequently considered \"black-box\" models. To address this, the SHAP method was used to interpret model prediction [69]. SHAP values act as sensitivity coefficients, identifying key features and their contributions to predictions [70]. Positive SHAP values indicate that a feature increases the prediction, while negative values indicate the opposite. These relationships are illustrated in Figure 15b. \begin{table}
\begin{tabular}{l l} \hline \hline
**Models** & **Parameter Combination** \\ \hline Ridge Regression & alpha = 0.690 \\ Lasso Regression & alpha = 0.500, max\_iter = 100 \\ Random Forest Regressor & n\_estimators = 10, max\_depth = 15, max\_features = 3 \\  & learning\_rate = 0.268, max\_depth = 7, max\_features = 0.900, \\ Gradient Boosted Decision Trees & min\_samples\_leaf = 4, min\_samples\_split = 4, \\  & n\_estimators = 220, subsample = 0.748 \\ AdaBoost Regressor & learning\_rate = 0.410, n\_estimators = 25 \\ XGBoost Regressor & n\_estimators = 500, max\_depth = 7, learning\_rate = 0.150, \\  & subsample = 0.620, colsample\_bytes = 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Optimal hyperparameters for machine learning models. Figure 13: Performance of machine learning models with the optimal hyperparameter settings on the test data: (**a**) MSE, (**b**) MAE, and (**c**) R\({}^{2}\). As shown, fiber content had the greatest influence on thermal shock resistance, with it being significantly higher than the other two features, while coating thickness had the least impact. These results are consistent with the orthogonal simulation findings from the literature. Figure 14: Prediction results using XGBoost regression. The training and testing sets are represented by blue dots and red squares, respectively. Figure 15: (**a**) Feature importance ranking for the target output; (**b**) SHAP values of each feature for each sample based on XGBoost regression. the previous section: The primary reason for this influence is that higher fiber content increases the toughness of the coating, allowing it to better withstand thermal stress. During thermal shock, the presence of fibers alters or deflects the crack propagation path, effectively delaying or preventing rapid crack growth.