## 2 Acquisition of Experimental Stress-strain Data
### Comparisons of Study Abilities of the ANN, LHS-SVR and GA-SVR
After Gaussian distribution analysis, the mean number of all of the relative errors (\(\upmu\)) and standard deviation (\(\upnu\)) is obtained. The \(\upmu\)-value expressed by Equation (12) is the mean number of all of the relative errors. The standard deviation (\(\upnu\)) expressed by Equation (13), as an evaluation index to measure the discrete degree of an individual in the dataset, was introduced to measure the distribution of the relative error (\(\updelta\)). Here, a small \(\upnu\) indicates that most of \(\delta\)-values are close to the \(\upmu\)-value, and vice versa. Additionally, a smaller \(\upmu\)-value indicates that more predicted stress data approach the experimental stress data. \[\upmu=\frac{1}{N}\sum_{i=1}^{N}\delta_{i} \tag{12}\]  
\[\upnu=\sqrt{\frac{1}{(N-1)}\sum_{i=1}^{N}\left(\delta_{\upnu}-\upmu\right)^{ 2}} \tag{13}\]  
where \(\delta\) is the sample of the relative error; \(\upmu\) is the average number of \(\delta\)-values; N is equal to the number of samples. Figure 10: Correlation between the experimental flow stresses and the testing predictions of the LHS-SVR and GA-SVR in (\(\upalpha+\upbeta\)) and \(\upbeta\) phase: (**a**) predictions in (\(\upalpha+\upbeta\)) phase predicted by the LHS-SVR; (**b**) predictions in \(\upbeta\) phase predicted by the LHS-SVR; (**c**) predictions in (\(\upalpha+\upbeta\)) phase predicted by the GA-SVR; (**d**) predictions in \(\upbeta\) phase predicted by the GA-SVR. From Figure 12 and [20], it can be found that the \(\delta\)-values acquired from the mathematical regression model, ANN, LHS-SVR and GA-SVR vary from \(-\)20%-20%, \(-\)3%-3%, \(-\)0.35%-0.05% and \(-\)3%-2%, respectively. Figure 12a,b shows the histogram of \(\delta\)-values of LHS-SVR and GA-SVR, respectively, which show the relative frequency of each \(\delta\)-level. The \(\mu\)-value and w-value of the mathematical regression model, ANN, LHS-SVR and GA-SVR are 1.3146 and 8.2033, \(-\)0.1246 and 0.8745, 0.025 and 0.000407 and 0.00216 and 0.44195, respectively. It can be summarized that the generalization ability of the mathematical regression model is the worst, and the generalization abilities of LHS-SVR and GA-SVR are at higher levels. Table 2 exhibits the \(R\)-values and \(AARE\)-values of the testing dataset of the mathematical regression model, ANN, LHS-SVR and GA-SVR, so as to further compare the generalization abilities of these models. The \(R\)-values and \(AARE\)-values of the testing dataset at the strain of 0.5 of the mathematical regression model, ANN, LHS-SVR and GA-SVR are 0.9902 and 6.583%, 0.9998 and 0.572%, 0.999975 and 0.074450% and 0.999923 and 0.529599%, respectively. It can be observed that the LHS-SVR and GA-SVR have larger \(R\)-values and lower \(AARE\)-values, which indicate that the LHS-SVR and GA-SVR can accurately predict the highly non-linear flow behaviors. Compared to ANN, the globally-optimal solution can be obtained by using LHS-SVR and GA-SVR, and the computational processes of LHS-SVR and GA-SVR are robust and will avoid falling into local extreme values. For a certain dataset, the same network topology and training parameters of an ANN will obtain fluctu  
Figure 11: Correlation between the experimental flow stresses and the testing predictions by the (**a**) LHS-SVR and (**b**) GA-SVR at a strain of 0.5. Figure 12: Distributions of the relative errors of the testing data by (**a**) LHS-SVR and (**b**) GA-SVR. attempts. ANN can meet network topologies and training parameters well to achieve a higher accuracy level; however, these accurate results have poor reproducibility. The significant character of SVR is that an SVR with identical training parameters will maintain training accuracy and prediction accuracy at stable levels in different attempts for a certain dataset. The generalization abilities of these models were shown as follows in ascending order: the mathematical regression model! ANN! GA-SVR! LHS-SVR.