## 3 ANN model
### Evaluation of ANN Model  
To synthetically estimate the predictability of ANN model, two commonly used statistical indicators of correlation coefficient (\(R\)) and average absolute relative error (\(AARE\)) [24], which were expressed as Equations (5) and (6), were introduced. A high \(R\)-value close to 1 illustrates that the predicted values conform to the experimental ones well, meanwhile, a low \(AARE\)-value close to 0 indicates that the sum of the errors between the predicted and experimental values tends to be 0. Thereby, such \(R\) and \(AARE\) are expected. Figure 6: The influences of the hidden layer number and neuron number in each hidden layer on the generalization performance of the neural network. \[R=\frac{\sum_{i=1}^{N}{(E_{i}-\overline{E})(P_{i}-\overline{P})}}{\sqrt{\sum_{i=1}^{ N}{(E_{i}-\overline{E})^{2}}{\sum_{i=1}^{N}{(P_{i}-\overline{P})}^{2}}}} \tag{5}\]  
\[AARE(\%)=\frac{1}{N}{\sum_{i=1}^{N}}\bigg{|}\frac{P_{i}-E_{i}}{E_{i}}\bigg{|} \times 100\% \tag{6}\]  
in which \(E\) and \(P\) are respectively the experimental value and predicted value of true stress; \(\overline{E}\) and \(\overline{P}\) are the mean values of \(E\) and \(P\) respectively; \(N\) is the number of predicted points. Using the well-trained ANN model, the true stress values under experimental conditions which include the deformation conditions corresponding to the previous training points and test points were predicted. After that, the correlation relationships between the experimental and predicted true stress were expressed in Figure 7. Each point in Figure 7 takes experimental stress and predicted stress as the horizontal and vertical axis respectively. The straight line at 45 degrees from the axes is the best linear fit line. If the experimental stress is exactly equal to the predicted stress, the corresponding point would lie on this line. Besides, the lines corresponding to 5% error were presented in Figure 7 as well. If a point is within the two 5% error lines, the prediction error corresponding to the deformation condition of this point is lower than 5%. It is clearly observed that all the errors of predicted stress values were lower than 5%. The points in Figure 7, especially the ones in Figure 7, lie very close to the ideal line, indicating well consistency between the experimental and predicted results. In addition, the \(R\)-values for the training part and test part are respectively 0.9991 and 0.9978. As stated above, such high values of correlation coefficient \(R\) suggest that the predicted stress values conform very well to the experimental ones in an alternative way. Besides, the \(AARE\)s were calculated as well. The \(AARE\)-value deriving from the test part is 1.5948%, while that for the training part is merely 0.6314%. Such minor errors signal the high accuracy exhibited in the training and test work by ANN. The relative error (\(\delta\)) [17] in Equation (7) represents the percentage error of each predicted stress-strain value relative to the homologous experimental value, in order to further and more detailedly evaluate the ANN model, it was introduced. In the training part, the relative errors range from \(-\)2.45% to 3.78%, and for the test part their distribution range is from \(-\)3.27% to 3.82%. Figure 8 expresses the columnar distribution maps of the relative errors in the training and test part. In terms of Figure 8, it is not so difficult to find that the relative errors, no matter in training or test part, are within \(\pm\)4%. However, it is more noteworthy that the most of the \(\delta\)-values are miraculously concentrated in the vicinity of the ideal value 0: in the training part, the \(\delta\)-values of 92.8% points are within the interval of [\(-\)1%, 1%], and for the test part, 75.5% are concentrated in [\(-\)1%, 1%]. These results arisen from unbiased statistical data provide direct evidence that high precise prediction both in the training and test stage was achieved by the ANN model.