## 5 ANN-Based Prediction Models
#### 5.1.2 \(\varepsilon_{cu}\)  
A similar procedure to that used for \(f_{cc}\) was performed to find the best ANN architecture for predicting \(\varepsilon_{cu}\). The obtained outcome for independent variables variations and the number of layers is presented in Table 7. Based on the table, the combination of variables with the Bayesian regularization algorithm offered the highest accuracy compared to the other algorithms. Equation (6) presents the best composition of input variables to obtain the highest accuracy for the prediction of \(\varepsilon_{cu}\). \[\varepsilon_{cu}=f\left(f_{co},\ \frac{K_{l}}{f_{co}},\ \varepsilon_{fu},\ k_{2},\ E_{ f},\ t_{f},\ f_{f},\ \varepsilon_{co},\ D\right), \tag{6}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & **Number of** & **Number of Beurons** & **Training Function** & **Transfer Function** & **Learning Rate** & **Objective Function** \\  & **Hidden Layers** & **First** & **Second** & **Function** & **Rate** & **Function** \\ \hline \(f^{r}_{cc}\) & 2 & 15 & 10 & Bayesian regularization & tansig & 0.005 & MSE \\ \(\varepsilon_{cu}\) & 2 & 6 & 8 & Levenberg–Marquardt & tansig & 0.001 & MSE \\ \(\varepsilon_{h,rup}\) & 2 & 10 & 15 & Bayesian regularization & tansig & 0.005 & MSE \\ \(f^{r}_{c1}\) & 1 & 25 & - & Bayesian regularization & tansig & 0.005 & MSE \\ \(\varepsilon_{c1}\) & 2 & 25 & 20 & Levenberg–Marquardt & tansig & 0.001 & MSE \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Final ANN parameters for prediction models. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Transfer Function** & **AAE (\%)** & **M (\%)** & **RMSE (MPa)** \\ \hline tansig-tansig & 6.4 & 100.4 & 7.4 \\ \hline logsig-logsig & 8.4 & 101.2 & 9.9 \\ \hline logsig-tansig & 10.2 & 105.0 & 11.2 \\ \hline pureline-pureline & 14.0 & 103.8 & 14.1 \\ \hline tansig-pureline & 8.5 & 101.1 & 9.1 \\ \hline elliotsig-elliotsig & 9.9 & 101.8 & 10.8 \\ \hline tribas-tribas & 20.1 & 105.8 & 22.9 \\ \hline purelin-logsig & 11.2 & 105.0 & 12.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Summary of different studied cases for transfer function in predicting \(f_{cc}\). where \(k_{2}\) is strain enhancement coefficient proposed by [PERSON] et al. [4]. As shown in Table 2, simultaneous use of \(f^{\prime}{}_{co}\) and \(\varepsilon_{co}\) could lead to multicollinearity in \(\varepsilon_{cu}\) model. However, the obtained results of \(\varepsilon_{cu}\) showed that using both parameters in predicting the model resulted in a more accurate prediction. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{10}{c}{**Leverberg–Marquardt**} \\ \cline{2-13}  & \multicolumn{3}{c}{**10 Neurons**} & \multicolumn{3}{c}{**15 Neurons**} & \multicolumn{3}{c}{**20 Neurons**} & \multicolumn{3}{c}{**25 Neurons**} & \multicolumn{3}{c}{**30 Neurons**} \\ \hline
**Input** & **AAE** & **M** & **RMSE** & **AAE** & **M** & **RMSE** & **AAE** & **M** & **RMSE** & **AAE** & **M** & **RMSE** & **AAE** & **M** & **RMSE** \\
**Variables** & **(\%)** &  
ANN parameters, which offered more accurate model for predicting \(\varepsilon_{cu}\), are presented in Table 6. According to the table, the Levenberg-Marquardt algorithm and the tansig transfer function with two hidden layers of six and eight neurons offered the most accurate and simple ANN model for predicting \(\varepsilon_{cu}\). Cross-validation analysis was carried out on the final ANN map. The developed model performance for \(\varepsilon_{cu}\) is shown in Figure 5. Based on Figure 5a, the predicted \(\varepsilon_{cu}\) values were close to experimental \(\varepsilon_{cu}\) values. Figure 5b exhibits that the considered goal as limit for BP neural network analysis of \(\varepsilon_{cu}\) reached at epoch of 95. Additionally, it is shown in Figure 5c that, similar to \(f^{\prime}_{cx}\), the selection of the dataset exhibited a major influence on the \(\varepsilon_{cu}\) model accuracy.