## 4 Prediction of Mechanical Properties of UHPC Based on AOA-SVR
### Comparative Analysis of Prediction Results  
#### 4.4.1 Model Prediction Performance and Verification Analysis  
The performance indicators for the model prediction and verification analysis are illustrated in Figure 4a,b, representing the performance statistics of the training and test datasets, respectively. They clearly demonstrate that the AOA-SVR model achieves the best performance indicators across all models. Specifically, the AOA-SVR model generates higher R\({}^{2}\) values, while RMSE, MAE, and MAPE values are lower compared to other models. Upon combining the training phase and the test phase, the SVR model also exhibits strong performance, outperforming the other two models significantly. This indicates that, following the AOA-SVR model, the SVR model is the most reliable in predicting the compressive strength of UHPC.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Indicator** & **Equation** & **Ideal Value** \\ \hline R\({}^{2}\) & R\({}^{2}=\frac{\sum_{i=1}^{n}(\chi_{i}-\gamma_{i})^{2}-\sum_{i=1}^{n}(\chi_{i}- \gamma_{i})^{2}}{\sum_{i=1}^{n}(\chi_{i}-\mu_{i})^{2}}\) & 1 \\ RMSE & RMSE \(=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(\chi_{i}-\gamma_{i})^{2}}\) & 0 \\ MAE & MAE \(=\frac{1}{n}\  
ANN model (Figure 5a) exhibits the largest deviation from the perfect agreement line, indicating the least accurate predictions among the models. The SVM model (Figure 5b) shows moderate prediction performance, with greater dispersion around the y = x line and more points falling within the 10% and 20% deviation boundaries. The SVR model (Figure 5c) shows moderate prediction performance, with greater dispersion around the y = x line and more points falling within the 10% and 20% deviation boundaries. Finally, the AOA-SVR model (Figure 5d) shows the highest R\({}^{2}\) value, with most points lying close to the y = x line, indicating near-perfect prediction accuracy. In terms of performance, the AOA-SVR model is followed by the SVR, SVM, and ANN models.  
While achieving high performance in the training phase is beneficial for model prediction, it does not necessarily guarantee the model's performance during the test phase. In other words, a model that performs well during training may not replicate its results during testing; therefore, their performance must be evaluated on the test data.  
Figure 6 presents the regression diagrams for the models during the test phase. The AOA-SVR model demonstrates superior performance, as evidenced by the highest R\({}^{2}\) value and the compact clustering of data points along the black line (nearly 100% consistent line). The R\({}^{2}\) values for the models following the AOA-SVR model are in the following order: SVR, SVM, and ANNs.  
From Figure 6, it can be seen that the model results on the test set are consistent with those in Figure 5, indicating that using the same models (ANN, SVM, SVR, AOA-SVR) for both the training set and the test set ensures that the performance of the models is evaluated  
Figure 6: Regression diagrams of (**a**) ANNs, (**b**) SVM, (**c**) SVR, and (**d**) AOA–SVR models during the test phase.  
under the same conditions, making the conclusions in the training and testing stages consistent and comparable, ensuring consistency and stability of the model performance. By comparing the results of the training set in Figure 5 with those of the test set in Figure 6, the same conclusion is drawn, indicating that the same model learns data features on the training set and verifies its generalization ability on the test set, avoiding overfitting.