## 6 Results and Discussion  
The ELM, KELM, KELM-GWO, DELM, BPNN models, and the empirical equation were run to predict UCS after ensuring that the optimal involving parameters and the performance of all the six models were evaluated in terms of six statistical indices (i.e., RMSE, R\({}^{2}\), MAE, U\({}_{1}\), U\({}_{2}\), and VAF) previously mentioned in this study. The results of the performance indices in the training and testing phases of all models were represented in Table 5. As can be seen in this table, the KELM-GWO model was the best model for predicting UCS with the best performance indices in both the training (RMSE: 17.2176; R\({}^{2}\): 0.8846; MAE: 12.0577; U\({}_{1}\): 0.0798; U\({}_{2}\): 0.0259; VAF: 88.4566%) and testing phases (RMSE: 14.2176; R\({}^{2}\): 0.9152; MAE: 11.4315; U\({}_{1}\): 0.0706; U\({}_{2}\): 0.0259; VAF: 91.5207%). Nevertheless, it is important to note that not all AI models perform better than the empirical formulas. It should be noted that the DELM model is worse than the empirical formula in both training and testing phases with an RMSE of 28.4753-27.6213, R\({}^{2}\) of 0.6843-0.7019, MAE of 22.4743-22.9152, U\({}_{1}\) of 0.1306-0.1332, U\({}_{2}\) of 0.0677-0.0730, and VAF of 69.1654-70.3401%. The reason for this result is that the DELM lacks a reverse fine-tuning process compared with a traditional deep learning algorithm. Although more hidden layers are added, excessive input weights aggravate the training load and result in a low prediction performance. After the KELM-GWO model, the BPNN, KELM, and ELM have a close competition in both the training and testing phases. To further compare the predictive performance of the six models, the regression diagrams of all the models in the training and testing phases are demonstrated in Figure 9. As can be seen in this figure, the horizontal and vertical axes represent the observed values of UCS measured in laboratory experiments and the predicted values proposed by AI models and the empirical equation in this study, respectively. Particularly, the data point  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{4}{c}{**Performance (Training)**} \\ \cline{2-7}  & **RMSE** & **R\({}^{2}\)** & **MAE** & **U\({}_{1}\)** & **U\({}_{2}\)** & **VAF (\%)** \\ \hline ELM & 22.3844 & 0.8049 & 15.9951 & 0.1041 & 0.0443 & 80.4891 \\ KELM & 22.2684 & 0.8069 & 15.9380 & 0.1038 & 0.0442 & 80.7237 \\
**KELM-GWO** & **17.2176** & **0.8846** & **12.0577** & **0.0798** & **0.0259** & **88.4566** \\ DELM & 28.4753 & 0.6843 & 22.4743 & 0.1306 & 0.0677 & 69.1651 \\ BPNN & 19.2109 & 0.8563 & 13.6784 & 0.0905 & 0.0344 & 85.9230 \\ Empirical & 26.3309 & 0.7300 & 19.8876 & 0.1223 & 0.0610 & 73.0331 \\ \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{4}{c}{**Performance (Testing)**} \\ \cline{2-7}  & **RMSE** & **R\({}^{2}\)** & **MAE** & **U\({}_{1}\)** & **U\({}_{2}\)** & **VAF (\%)** \\ \hline ELM & 21.3123 & 0.8225 & 17.0308 & 0.1038 & 0.0452 & 82.4108 \\ KELM & 17.5050 & 0.8803 & 12.4699 & 0.0854 & 0.0308 & 88.1894 \\
**KELM-GWO** & **14.7327** & **0.9152** & **11.4315** & **0.0706** & **0.0259** & **  
will appear on the black diagonal line in each diagram when the predicted value is equal to the observed value. Meanwhile, other radial lines with 10% and 30% are second and third criteria to indicate deviations from the predicted values. Accordingly, the more predicted values appear within the 10% and 30% lines and the closer they are to the black diagonal line, the better the prediction performance of the model. As can be realized, the KELM-GWO model not only has the least predicted values outside the 30% line, but also has the most predicted values close to the black diagonal line in both the training and testing phases. After the KELM-GWO model, BPNN, KELM, and ELM have a similar prediction performance. Nevertheless, there are more predicted values of UCS away from the black diagonal in the empirical and DELM model. Figure 9: Regression diagrams of the models in the training and testing phases.