## 2 Training of the ANN Flow Law
### Training of the Neural Networks
Concerning the internal accuracy \(\text{E}_{\text{MS}}\) of the network, it varies in proportions in accordance with the graphical representation of Figure 3. From the plot in Figure 3 and the results reported in Table 1, the user would normally be tempted to select the most complex model, namely 3-15-7-1, as it gives the smallest deviation between predicted and experimental values of flow stress \(\sigma^{y}\). This will be analyzed in the next section concerning the numerical simulation of the compression of a cylinder on the Abaqus Explicit software using the ANN flow law. Since, as reported in Figure 3, the 3-7-4-1 model seems to have converged after the training phase, we are going to compare it with the 'accurate' model, the 3-15-7-1 model. From a more physical point of view, Figures 4 and 5 show the correlation between the data \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**ANN** & \(n_{v}\) & \(t\) & \(\text{E}_{\text{MS}}\) & \(\text{E}_{\text{mar}}\) & \(\text{E}_{\text{RMS}}\) \\  & & **(min)** & \(\times 10^{-5}\) & **(\%)** & **(MPa)** \\ \hline
3-7-4-1 & 65 & 48 & 3.91 & 1.88 & 3.05 \\
3-9-4-1 & 81 & 48 & 3.29 & 1.70 & 2.75 \\
3-9-7-1 & 114 & 49 & 1.83 & 1.25 & 2.44 \\
3-15-7-1 & 180 & 50 & 1.01 & 0.97 & 2.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results concerning the training of the four ANN flow laws. Figure 3: Convergence of the ANN models during the training phase. predicted by the neural network and the experimental data for the 3-7-4-1 and 3-15-7-1 networks, respectively. Figure 4: Comparison of the flow stress \(\sigma^{y}\) predicted by the 3-7-4-1 ANN (continuous line) and the experimental data for the GCr15 (square markers). Figure 5: Comparison of the flow stress \(\sigma^{y}\) predicted by the 3-15-7-1 ANN (continuous line) and the experimental data for the GCr15 (square markers). Analysis of Figures 4 and 5 shows a very good correlation between the ANN results and the experimental results, which is reflected by the very low values of the \(\mathrm{E_{MAR}}\) and \(\mathrm{E_{RMS}}\) coefficients reported in Table 1. As reported by [PERSON] [48], the correlation coefficient (R) is generally not a good measure in our case of study because it only shows the correlation of the model with respect to the data and not its accuracy, which is a determining factor in the qualification of a model. Therefore, this type of coefficient is not used in this work for comparing the different models. Concerning the performance of the ANN flow laws, the correlation results for both reported models, are much better than the ones obtained by [PERSON] et al. [14] during his work on the same material with four different analytical flow laws, especially since he had to split the data into two groups according to the temperature value (one on the range \(T=750\)-\(850\)\({}^{\circ}\)C and one on the range \(T=850\)-\(1300\)\({}^{\circ}\)C) and to identify two sets of parameters for each flow law to reduce the error of his identified analytical models. This of course raises the question of the usability of those analytical laws where the temperature of the material changes from one group to the other during a thermomechanical transformation. In our approach and by using an ANN flow law, the identified law is not only valid over the whole temperature range, but it displays a \(\mathrm{E_{MAR}}\) value 5 times lower than the best flow law proposed by [PERSON] et al. [14]: the Arrhenius law with an \(\mathrm{E_{MAR}}=3.74\%\) over the range \(T=750\)-\(850\)\({}^{\circ}\)C and \(\mathrm{E_{MAR}}=5.76\%\) over the range \(T=850\)-\(1300\)\({}^{\circ}\)C, while the \(\mathrm{E_{MAR}}=0.97\%\) for the 3-15-7-1 and \(\mathrm{E_{MAR}}=1.88\%\) for the 3-7-4-1 ANN flow laws proposed here. The disadvantage of developing a flow law model based on neural networks is the number of internal variables in the network (180 in the case of the 3-15-7-1 network), which makes it difficult to translate the network into printable results.