## Materials and Methods
### Statistical Performance Indicators  
The estimation precision of the suggested models was assessed through several statistical performance indicators and by utilizing graphical presentation. The statistical performance indicators are mean absolute error (\(MAE\)), root mean square error (\(RMSE\)), index of scattering (\(IOS\)), coefficient of determination (\(R^{2}\)), Pearson correlation coefficient (\(R\)), and index of agreement (\(IOA\)). They are expressed as follows [54; 55]:  
1. Mean absolute error (\(MAE\)): \[ \begin{array}{c}MEA=\frac{1}{N}\sum_{i=1}^{N}\lvert Y_{tar,i}-Y_{out,i}\rvert \ \ \ (0<MAE<\infty)\end{array} \] (1)
2. Root mean square error (\(RMSE\)): \[ \begin{array}{c}RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(Y_{tar,i}-Y_{out,i})^{2} }\ \ \ (0<RMSE<\infty)\end{array} \] (2)
3. Index of scattering (\(IOS\)): \[ \begin{array}{c}IOS=\frac{\sqrt{\frac{1}{N}\sum_{i=1}^{N}(Y_{tar,i}-Y_{out,i} )^{2}}}{\overline{Y_{tar}}}\ \ \ (0<RMSE<\infty)\end{array} \] (3)
4. Coefficient of determination (\(R^{2}\)): \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & **Algorithm Parameters** & **Value** \\ \hline \multirow{4}{*}{_ELM_} & Hidden layers & \(H=1\) \\  & Hidden neurons & \(N=12\) \\  & Activation function & \({}^{\prime}\)linear\({}^{\prime}\) \\  & Regulation parameter & C = 0.02 \\ \hline \multirow{4}{*}{_DNN_} & Hidden layers & \(H=2\) \\  & Hidden neurons in the first layer & \(N1=\) [1–20] \\  & Hidden neurons in the second layer & \(N2=\) [1–20] \\  & Activation function in the first layer & \({}^{\prime}\)Tansig\({}^{\prime}\) \\  & Activation function in the second layer & \({}^{\prime}\)Tansig\({}^{\prime}\) \\ \hline \multirow{4}{*}{_SVR_} & Regulation parameter \(C\) & Series of \(C\) \\  & Regulation parameter lambda & Series of lambda \\  & Kernel function & \({}^{\prime}\)bf\({}^{\prime}\) \\ \hline \multirow{2}{*}{_RF_} & nTrees & nTrees = 100 \\  & mTrees & mTrees = 26 \\ \hline \multirow{2}{*}{_LASSO_} & Lambda & series of lambda \\  & PLS components & NumComp = 3 for PSO \\  & & NumComp = 4 for GT and FS \\ \hline \multirow{2}{*}{_Ridge_} & Regularization parameter lambda & lambda = 1 \\  & & \\ \hline \multirow{4}{*}{_KRidge_} & Regularization parameter lambda & lambda = 1 \\  & Kernel function & \({}^{\prime}\)linear\({}^{\prime}\) \\  & Parameter for kernel & sigma = 2 \(\times\) 10\({}^{-7}\) \\ \hline \multirow{4}{*}{_GP_} & Function set & +, -, \(\times\), \(\div\), power, In, sqrt, sin, cos, tan \\  & Population size & 100 up to 500 \\ \cline{1-1}  & Number of generations & 1000 \\ \cline{1-1}  & Genetic operators & Reproduction, crossover, mutation \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial parameter settings for the algorithms. \[R^{2}=1-\frac{\sum_{i=1}^{N}(Y_{tar,i}-Y_{out,i})^{2}}{\sum_{i=1}^{N}(Y_{tar,i}- \overline{Y_{tar}})^{2}}\ \ \ (-\infty<NSE<1) \tag{4}\]
5. Pearson correlation coefficient (\(R\)): \[R=\frac{\sum_{i=1}^{N}(\big{(}Y_{tar,i}-\overline{Y_{tar}}\big{)}\big{(}Y_{ out,i}-\overline{Y_{out}}\big{)})}{\sqrt{\sum_{i=1}^{N}\big{(}\big{(}Y_{tar,i}- \overline{Y_{tar}}\big{)}^{2}\big{(}Y_{out,i}-\overline{Y_{out}}\big{)}^{2} \big{)}}}\ \ \ (-1<R<1)\] (5)
6. Index of agreement (\(IOA\)): \[IOA=1-\frac{\sum_{i=1}^{N}(Y_{tar,i}-Y_{out,i})^{2}}{\sum_{i=1}^{N}\Big{(} \sum_{i=1}^{N}\big{|}Y_{out,i}-\overline{Y_{tar}}\big{|}+\sum_{i=1}^{N}\big{|} Y_{tar,i}-\overline{Y_{tar}}\big{|}\Big{)}^{2}}\ \ \ (0<IOA<1)\] (6)  
where \(Y_{tar,i}\), \(Y_{out,i}\), \(\overline{Y_{tar}}\), and \(\overline{Y_{out}}\) characterize the target, output, mean of the target, and mean of output pile-bearing capacity values for \(N\) data samples, respectively. Moreover, the suggested machine-learning model possessed the minimum value of \(RMSE\), \(IOS\), and \(MAE\), and the peak value of \(IOA\), \(R^{2}\), and \(R\) presents the optimal one and the closest to the experimental values.