## 2 Methodology
### Hybrid Metaheuristic Algorithms
To do so, all four networks were tested with nine different population sizes including 10, 25, 50, 75, 100, 200, 300, 400, and 500. Each model performed 1000 repetitions to minimize the error. In this process, root-mean-square error (RMSE) was set as the objective function (OF) to measure the training error in each iteration. This function is expressed in Equation (2). Figure 3a shows the obtained RMSEs for the tested population sizes. Also, the convergence curve of the most accurate one is illustrated in this figure. \[RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\bigl{(}Y_{i_{observed}}-Y_{i_{predicted}} \bigr{)}^{2}}, \tag{2}\]

where \(N\) is the number of data, and \(Y_{i\,observed}\) and \(Y_{i\,predicted}\) stand for the observed and predicted stability values. As can be seen, all four models exhibited an acceptable error in analyzing the relationship between the stability condition and its influential parameters. In detail, the smallest error was obtained for the

Figure 3: Executed sensitivity analysis based on the population size: (**a**) obtained RMSE values, the convergence curves of (**b**) MFO-ANN, (**c**) WOA-ANN, (**d**) ACO-ANN, and (**e**) LCA-ANN. WOA-ANN with a population size of 400 (RMSE = 0.307658318), LCA-ANN with a population size of 200 (RMSE = 0.312263011), MFO-ANN with a population size of 50 (RMSE = 0.298588793), and ACO-ANN with a population size of 10 (RMSE = 0.274504799). ### Accuracy Assessment Criteria

The classification accuracy of the models was measured using a well-known criterion, namely, the area under the receiving operating characteristic curve (AUROC). Note that it was obtained by plotting the ROC diagrams, which is a good way of assessing the accuracy in diagnostic problems, such as natural hazard models [57; 58; 59; 60; 61]. Moreover, two error criteria of RMSE and mean absolute error (MAE) were used to measure the performance error of the models. Equation (3) expresses the formulation of the MAE. \[MAE=\ \frac{1}{N}\sum_{i=1}^{N}\Bigl{(}Y_{i_{observed}}\ -Y_{i_{predicted}}\ \Bigr{)}. \tag{3}\]

### Accuracy Assessment of the Predictive Models

In this part, the results of the best-fitted models (i.e., with elite population sizes) are evaluated to examine their simulation capability. As is known, the results of the training phase address the learning quality of the model, and the testing results indicate the generalization capability for unseen conditions of the problem. In the training phase, the calculated values of RMSE and MAE for the typical ANN were 0.3465 and 0.3055, respectively. Both of these indices experienced considerable decreases by applying the WOA (0.3076 and 0.2555), LCA (0.3122 and 0.2592), MFO (0.2985 and 0.2430), and ACO (0.2745 and 0.1783) optimization techniques. Also, in terms of the AUROC, the accuracy of the ANN was increased from 0.956 to 0.969, 0.964, 0.969, and 0.965, respectively. At a glance, it can be deduced that the models can improve the learning capability of the ANN. Figure 4 displays the predicted and actual stability values for the ensemble models. The output ranges were [\(-\)0.196124259, 1.163826771], [\(-\)0.285459666, 1.165811194], [\(-\)0.280854543, 1.220819059], and [\(-\)0.323683705, 1.197618633], respectively. Figure 4: _Cont._

Similar to the first phase, all the neural-metaheuristic ensembles surpassed the ANN in the testing phase which means the algorithms have performed efficiently in adjusting the computational weights and biases of this tool. In detail, the RMSE was reduced from 0.3465 to 0.3076, 0.3122, 0.2985, and 0.2745. As for the MAE, it fell from 0.3055 to 0.2555, 0.2592, 0.2430, and 0.1783. The differences between the actual and predicted stability values (labeled as error) are illustrated in Figure 5, along with the histogram of the errors. The products of the WOA-ANN, LCA-ANN, MFO-ANN, and ACO-ANN vary in the extents [\(-\)0.19260775, 1.121547514], [\(-\)0.221848351, 1.183820947], [\(-\)0.176990489, 1.103579629], and [\(-\)0.072023941, 1.206028442], respectively.