## 2 Methodology
### Hybrid Metaheuristic Algorithms
Moreover, the ROC curves for the prediction of ensemble models are shown in Figure 6. The calculated areas under the curves indicate more than 90% accuracy for all five models. However, the AUROCs of the hybrid ensembles were significantly higher than the unreinforced ANN (AUROC = 0.930). Figure 4: The results obtained for (**a**) WOA-ANN, (**b**) LCA-ANN, (**c**) MFO-ANN, and (**d**) ACO-ANN predictions in the training phase. Figure 5: The results obtained for (**a,b**) whale optimization algorithm (WOA)–artificial neural network (ANN), (**c,d**) league champion optimization algorithm (LCA)–ANN, (**e,f**) moth–flame optimization (MPO)–ANN, and (**g,h**) ant colony optimization (ACO)–ANN predictions for the testing samples. Until now, all used criteria confirmed that the metaheuristic algorithms can develop a more powerful ANN compared to the BP learning method. The results of the WOA-ANN, LCA-ANN, MFO-ANN, and ACO-ANN tools are evaluated in this section to compare the efficiency of the algorithms. A score-based system was developed to rank the models and determine the most accurate one. As Table 2; Table 3 denote, each model received three scores based on the calculated RMSE, MAE, and AUROC. Then, the summation of these scores determined the model producing the most consistent outputs in each phase. According to Table 3, the ACO-based model grasped the highest scores in terms of all accuracy criteria except for the training AUROC (0.965) which was second to the WOA and MFO (0.969). Therefore, it grasped the highest overall scores in both training and testing phases, followed by the MFO, which closely surpassed the WOA, while the LCA featured the lowest rank. Figure 6: The receiver operating characteristic (ROC) curves of (**a**) WOA–ANN, (**b**) LCA–ANN, (**c**) MFO–ANN, and (**d**) ACO–ANN predictions in the testing phase. Moreover, in comparison with the HHO and DA applied to the same data by [PERSON] et al. [23], it was deduced that the methods of the current study present a more accurate analysis and approximation of bearing capacity. In detail, the RMSE and MAE obtained for the DA-MLP (superior to the HHO-MLP) were 0.3421 and 0.2904, which are larger than the results obtained for our WOA-ANN, MFO-ANN, and ACO-ANN. Also, the best AUROC of this study was higher than that for both models in the mentioned reference (0.944 vs. 0.942). ### Presenting the Neural Predictive Formula

In this section, due to the largest accuracy obtained for the ACO-ANN, the neural relationship of this model was extracted and presented in the form of Equation (4) to predict the stability value using the considered effective parameters. Note that this formula was developed by the optimized parameters of the MLP output



was the second efficient ensemble. Based on the findings of this study, the combination of neural computing and metaheuristic techniques (especially the ACO-ANN) provides fast and inexpensive yet accurate models for analyzing the stability of the footings over two-layered soils. This is in contrast to the use of traditional methods (e.g., laboratory studies and finite element techniques), which are time-consuming and entail implementing costly and destructive tests. Furthermore, a comparison with conventional methods (MLP neural network) showed that using metaheuristic techniques can be an advantageous way of enhancing their performance (i.e., around 95% accuracy of prediction). In fact, this research suggests the use of powerful inspirations from real-world phenomena for optimizing the computational parameters of the ANN. [PERSON] and [PERSON] wrote the manuscript, and discussed and analyzed the data. [PERSON] and [PERSON] edited, restructured, and professionally optimized the manuscript. This research received no external funding. The authors declare no conflict of interest. ## References

* _[PERSON] et al. (2015)_ [PERSON]; [PERSON]; [PERSON]; [PERSON] Bearing capacity of precast thin-walled foundation in sand. _Proc. Inst. Civ. Eng. Geotech. Eng._**2015**, _168_, 539-550. [CrossRef]
* _[PERSON] et al.