## 2 Acquisition of Experimental Stress-strain Data
### The Prediction Model of Flow Behaviors Based on SVR and Latin Hypercube Sampling
#### 3.3.2 The Establishment of the Prediction Model LHS-SVR for the Flow Behaviors  
In this work, 1520 input-output pairs were selected from the stress-strain curves to train and test the LHS-SVR. Among the 1520 input-output pairs, 21.05% (320) of stress points at the strain range of 0.1-0.8 with a distance of 0.1 were not utilized for training, but for testing the generalization ability of the developed LHS-SVR, and the remaining datasets were adopted to train the LHS-SVR. The cross-validation method, as an effective means of evaluating the accuracy of machine learning, was used in this investigation to evaluate the accuracy of the established LHS-SVR. In the cross-validation method, the original data are divided into \(N\) datasets. Each separate dataset is alternately retained as validation data, and the other (\(N\)-1) datasets are used to train the LHS-SVR. The performance of the LHS-SVR is evaluated by the average number of an evaluation index in an \(N\) validation process. Here, the number \(N\) was set as five. Additionally, an evaluation index mean square error (\(MSE\)) expressed by Equation (7) between training stress data and validation stress data was introduced to evaluate the performance of the LHS-SVR. \[MSE=\frac{1}{N}\sum_{i=1}^{N}\left[f(x_{i})-y_{i}\right]^{2} \tag{7}\]  
where \(f(x_{i})\) is the predicted stress data; \(y_{i}\) is the experimental stress data. Additionally, \(N\) is equal to the number of stress-strain samples. In addition, another evaluation index correlation coefficient (\(R\)) expressed as Equation (8) was used to estimate the degree of correlation between the experimental flow stresses and predicted flow stresses [28]. A larger \(R\)-value near one demonstrates a good correlation between the two variables, and vice versa. \[R=\frac{\sum_{i=1}^{N}\left(E_{i}-\overline{E}\right)(P_{i}-\overline{P})}{ \sqrt{\sum_{i=1}^{N}\left(E_{i}-\overline{E}\right)^{2}}{\sum_{i=1}^{N}\left( P_{i}-\overline{P}\right)^{2}}} \tag{8}\]  
Figure 3: The sketch maps of: (**a**) all sampling points; (**b**) the sampling points designed by the orthogonal test; and (**c**) the sampling points generated by Latin hypercube sampling (LHS). where \(E\) is the sample of experimental stress-strain values; \(P\) is the sample of predicted stress-strain values; \(N\) is equal to the number of samples. Assuming that there is an \(n\)-dimension sampling space; \(x_{i}\) is the \(i\)-th dimension variable, \(i\ =\ 1,2,...,n\). \(x_{i}\in[l_{i},u_{i}]\): \(l_{i}\) and \(u_{i}\) are the boundary values of the \(i\)-th dimension variable. The sampling process of LHS-SVR, which selects \(n_{0}\) sampling points, is as follows:  
* Determine the searching sampling space and sample size \(n_{0}\). The domain of definition \([l_{i},u_{i}]\) of \(x_{i}\) was uniformly divided into \(n_{0}\) intervals. Then, the \(n\)-dimension space was divided into \(n_{0}\)\({}^{n}\) hypercubes. Generate an \(n_{0}\times n\) matrix, and each column of the matrix is a random permutation of {1, 2, 3,..., \(n_{0}\)}. Each row of the matrix represents a selected hypercube. The central points of all selected hypercubes are the sampling points. * The \(MSE\)-values of all sample points were calculated by the cross-validation method. * Search the optimal parameter combination. * Reset the searching sampling space; increase the searching sample size. * Search the optimal parameter combination again. * Output the optimal parameter combination. The specific flowchart of the LHS-SVR is illustrated in Figure 4.  
In the preliminary search, 500 sample points were preset in the intervals: \(C\in[0,100]\), \(\gamma\in[0,100]\), \(\zeta\in[0,100]\); and the \(MSE\)-values and \(R\)-values of all samples were calculated. Each point represents a parameter combination. In the preliminary search, 50 points with larger \(R\)-values are illustrated in Figure 5a, so as to display the distribution of solutions in the search space.