## 3 Results and Discussion
### Flow Stress Behavior Modeling Experiments
#### 3.3.4 Models Cross-validation  
The cross-validation method was used to compare the predictability of the proposed model, ACE and ANN. Both models were evaluated by extracting the stress-strain experimental curves tested at 500 \({}^{\circ}\)C, and then the models were reconstructed without the data for 500 \({}^{\circ}\)C. The prediction of the data tested at 500 \({}^{\circ}\)C was made. In the same manner, the prediction of the data tested at 1 \(\times\) 10\({}^{-2}\) s\({}^{-1}\) was performed. The comparison between the predicted by both ACE and ANN modelled and the experimental curves are shown in Figure 12. The predicted data by both models at 500 \({}^{\circ}\)C (Figure 12a,c) and 1 \(\times\) 10\({}^{-2}\) s\({}^{-1}\) (Figure 12b,d) were in good agreement with the experimental data.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **R** & **RMSE** & **AARE (\%)** \\ \hline ACE & 0.98 & 1.42 & 6.54\% \\ \hline ANN & 0.99 & 0.27 & 0.91\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Performance indicators the correlation coefficient (R), average absolute relative error (AARE) (%), and root means square error (RMSE) values of the proposed models after fitting and approximation of the flow behavior for the alloy studied.  
Figure 11: Correlation between the experimental and modeled flow stresses obtained by (**a**) ACE and (**b**) ANN models.  
The prediction quality for the proposed models after cross-validation was quantified by R, AARE, and RMSE (Table 5). Figure 13 demonstrates the overall correlation between the experimental and the predicted flow stress via the ACE (Figure 13a,b) and ANN (Figure 13c,d) models at 500 \({}^{\circ}\)C and 0.01 s\({}^{-1}\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**R**} & \multicolumn{2}{c}{**RMSE**} & \multicolumn{2}{c}{**AARE (\%)**} \\ \cline{2-7}  & \multicolumn{2}{c}{**500 \({}^{\circ}\)C**} & \multicolumn{2}{c}{\(\mathbf{1\times 10^{-2}}\)**} & \multicolumn{2}{c}{**500 \({}^{\circ}\)C**} & \multicolumn{2}{c}{\(\mathbf{1\times 10^{-2}}\)**} & \multicolumn{2}{c}{**500 \({}^{\circ}\)C**} & \multicolumn{2}{c}{\(\mathbf{1\times 10^{-2}}\)**} \\  & \multicolumn{2}{c}{**s\({}^{-1}\)**} & \multicolumn{2}{c}{**s\({}^{-1}\)**} & \multicolumn{2}{c}{**s\({}^{-1}\)**} & \multicolumn{2}{c}{**s\({}^{-1}\)**} \\ \hline ACE & 0.98 & 0.98 & 1.92 & 1.2 & 9.5\% & 7\% \\ \hline ANN & 0.94 & 0.99 & 1.87 & 0.7 & 10.5\% & 5\% \\ \hline \hline \end{tabular}
\end{table}
Table 5: Performance indicators R, AARE (%), and RMSE values of the proposed models after prediction of the flow behavior of the investigated alloy via a cross-validation technique.  
Both ACE and ANN models demonstrated a good predictability of the flow stress behavior of the studied alloy. At 500 \({}^{\circ}\)C, the minor difference between the modeled and experimental data were achieved for the strain rates in a range of 0.002-0.01 s\({}^{-1}\) (Figure 14a,c). For higher strain rates, 0.03-0.1 s\({}^{-1}\), the ACE model demonstrated a lower deviation from the experiments than that of the ANN model (Figure 14b,d). The proposed models are required for the simulation superplastic forming process and the prediction forming parameters. The models are important instruments for the optimization superplastic forming regimes and processed high-quality complex-shape parts from the alloy studied.  
Figure 13: Correlation between the experimental and predicted flow stress obtained by (**a,b**) ACE and (**c,d**) ANN models.