## 5 Results and Discussion  
The prediction results in both training and testing phases are presented in this section. To this end, the products of the used models were compared with target values by using the mentioned accuracy criteria. As explained, the quality of the training results indicates the learning capability of the model. Figure 6 shows the results of this phase, in terms of the regular error (measured shear strength - predicted shear strength). As can be seen, the results of both reinforced ANNs were more accurate than the formal version. In detail, the RMSE experienced 11.33% (from 11.74 to 10.41) and 20.44% (from 11.74 to 9.34) reductions as the result of synthesizing the ANN with SFLA and WDO, respectively. The calculated MAEs confirmed these observations, as it was reduced by 14.25% (from 8.14 to 6.98) and 28.25% (from 8.14 to 5.84). In addition, referring to the obtained values of \(R^{2}\) (0.85, 0.88, and 0.90), the superiority of the ensemble models was deduced. Therefore, both SFLA and WDO performed efficiently in enhancing the learning capability of the ANN. Figure 6: The training results of the implemented (**a**) typical ANN, (**b**) SFLA–ANN, and (**c**) WDO–ANN. The same criteria were applied to the outputs of the testing phase. In this phase, the RMSE of the ANN dropped from 15.11 to 12.20 (19.25%) and, more considerably, 11.13 (26.34%), by incorporation with the SFLA and WDO. As for the MAE, the decrease of this criterion from 11.26 to 8.38 (25.57%) and 6.84 (39.25%) indicated that the generalization accuracy of the ANN was increased. The comparison between the actual and predicted shear strengths is also shown in Figure 7. According to this figure, the hybrid models grasped a more reliable approximation of shear strength in comparison with ANN, especially for minimum and maximum values. These results are broadly discussed below. The correlation of the testing results is depicted in Figure 8. Based on these figures, the products of the ANNs trained by the SFLA and WDO were better correlated with target shear strengths, compared  
Figure 7: The testing results of the implemented (**a**) typical ANN, (**b**) SFLA–ANN, and (**c**) WDO–ANN. to the ANN trained by the back-propagation learning method. This claim can be supported by the obtained values of \(R^{2}\) of 0.8286, 0.8998, and 0.9258 (for all data), and 0.8732, 0.9187, and 0.9547 (for data inside 95% confidence limit boundaries), respectively, for the ANN, SFLA-ANN, and WDO-ANN. Notably, the observed shear strength ranged from 10.92 to 114.25, while the predicted values were in the ranges [3.40, 78.73], [7.09, 81.25], and [11.43, 86.41]. Due to the better performance (i.e., the higher accuracy) of the hybrid models, it was demonstrated that applying both SFLA and WDO is an effective way to increase the reliability of the ANN. From the comparison point of view, the results of the WDO algorithm were more promising than the SFLA. For more detail, Table 2 denotes the relative error (Equation (17)) calculated for five maximum and five minimum values of the shear strength dataset. In terms of minimum values, except for the lowest one (8.27), the relative errors of the WDO-ANN were lower than those of the SFLA-ANN. The most appreciable difference was related to the shear strength of 10.27. This value was predicted as 11.51 (around 12% error) by the WDO-ANN, while the SFLA-ANN had around 50% error in this sense. Evaluating the maximum shear strengths showed that all values were underestimated by both methods. Also, it was deduced that the relative errors of the WDO-based ensemble were lower than those of the SFLA-based ensemble. This points to the higher efficiency of the WDO for estimating the peak values of shear strength. \[\text{Relative error }=(\frac{Z_{i\text{ }predicted}-Z_{i\text{ }observed}}{Z_{i\text{ }observed}})\times 100. \tag{17}\]  
Figure 8: The correlation between the measured and predicted shear strengths in the testing phases; (a) R\({}^{2}\) = 0.8286, (b) R\({}^{2}\) = 0.8998, (c) R\({}^{2}\) = 0.9258.