## 3 Results
### Comparison between the Proposed Models and Empirical Formulae  
To test the effectiveness of the suggested \(DNN\) model, a comparative study was performed using 12 empirical models proposed in the literature of predicting the bearing capacity, as presented in Table 7. It should be noted that no author has shared the mathematical equations of the proposed \(ML\) model to compare results with the same database. Published research was limited in presenting modeling results. Therefore, we cannot validate the proposed models using the current collected dataset. Consequently, the current study was limited to compare the proposed models based on the correlation coefficient. It is needless to say that the correlation coefficient is an important indicator when assessing the prediction precision, as the best model is represented by a prediction value close to 1. The results of the comparative study indicated that the proposed \(DNN\) model in our study is the best-performing model, with maximum accuracy (0.9996 for all data). Furthermore, our model is closely followed by the \(ANN\) model which was proposed by [PERSON] et al. [27], and it showed an acceptable accuracy as it ranked secondly. Moreover, the results revealed the poor performance of the ANN model proposed by [PERSON] et al. [25] in the bearing capacity. With respect to the performance of machine-learning models, the hierarchy follows the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(MAE\times 10^{3}\) & \(RMSE\times 10^{3}\) & \(IOS\) & \(R\) & \(R^{2}\) & \(IOA\) \\ \hline Concrete piles & & & & & & \\ \hline DNN & 0.1650 & 0.2140 & 0.0755 & 0.9977 & 0.9954 & 0.9988 \\ \hline ELM & 3.0424 & 4.2390 & 0.7737 & 0.9320 & 0.8686 & 0.9610 \\ \hline Lasso & 2.4324 & 3.5390 & 0.6637 & 0.9620 & 0.9254 & 0.9700 \\ \hline PLS & 2.5524 & 3.6390 & 0.6837 & 0.9688 & 0.9386 & 0.9700 \\ \hline RF & 1.1024 & 2.1690 & 0.3837 & 0.9880 & 0.9761 & 0.9912 \\ \hline Kridge & 2.2930 & 3.5917 & 0.6816 & 0.9433 & 0.8899 & 0.9641 \\ \hline Ridge & 2.4268 & 3.6145 & 0.6876 & 0.9409 & 0.8853 & 0.9636 \\ \hline LS & 2.3093 & 3.5867 & 0.6824 & 0.9414 & 0.8863 & 0.9656 \\ \hline Step & 2.4738 & 3.6421 & 0.6970 & 0.9352 & 0.8746 & 0.9626 \\ \hline SVR & 1.9787 & 4.0984 & 0.7734 & 0.9315 & 0.8676 & 0.9360 \\ \hline GP & 0.5966 & 0.9612 & 0.1731 & 0.9975 & 0.9951 & 0.9961 \\ \hline Steel piles & & & & & & \\ \hline DNN & 0.1870 & 0.3100 & 0.0448 & 0.9997 & 0.9994 & 0.9998 \\ \hline ELM & 3.1064 & 4.3966 & 0.9081 & 0.8478 & 0.7187 & 0.9118 \\ \hline Lasso & 2.7149 & 3.6962 & 0.7527 & 0.8990 & 0.8082 & 0.9437 \\ \hline PLS & 2.6329 & 3.6973 & 0.7763 & 0.8966 & 0.8038 & 0.9398 \\ \hline RF & 1.1213 & 2.3475 & 0.4893 & 0.9875 & 0.9751 & 0.9712 \\ \hline Kridge & 2.2482 & 3.6937 & 0.7342 & 0.8993 & 0.8088 & 0.9441 \\ \hline Ridge & 2.3820 & 3.7165 & 0.7402 & 0.8969 & 0.8044 & 0.9436 \\ \hline LS & 2.2646 & 3.6887 & 0.7350 & 0.8974 & 0.8054 & 0.9456 \\ \hline Step & 2.4291 & 3.7441 & 0.7496 & 0.8912 & 0.7943 & 0.9426 \\ \hline SVR & 1.9340 & 4.2004 & 0.8260 & 0.8875 & 0.7876 & 0.9160 \\ \hline GP & 0.5518 & 1.0632 & 0.2257 & 0.9975 & 0.9951 & 0.9965 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance indicators values of the AI models for predicting the pile-bearing capacity in both concrete and steel piles (bold: the optimal model). following order: [PERSON] et al. [27], [PERSON] et al. [32], [PERSON] et al. [28], [PERSON] et al. [34], [PERSON] et al. [30], [PERSON] [26], [PERSON] et al. [2], [PERSON] et al. [1], [PERSON] et al. [33], [PERSON] and [PERSON] [31], [PERSON] et al. [29], and [PERSON] et al. [25]. We believe that the reasonable ground standing behind the high accuracy found in our suggested model is due to deep learning (more than one hidden layer). The latter could offer the necessary flexibility for modeling complex functions in many cases.