## 2 Artificial Neural Network Flow Law
### Artificial Neural Network Equations
#### 2.1.4 Derivatives of the Neural Network
In order to test the extreme limits of the proposed networks, Figure 7 shows the comparison of predicted values according to the nature of the network for conditions globally outside the learning limits. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Activation** & **CPU** & \begin{tabular}{c} \(\mathrm{E_{MS}}\) \\ \(\times\)**10\({}^{-6}\)** \\ \end{tabular} & \begin{tabular}{c} \(\mathrm{E_{RMS}}\) \\ (MPa) \\ \end{tabular} & \begin{tabular}{c} \(\mathrm{\Delta E_{RMS}}\) \\ (\%) \\ \end{tabular} & \begin{tabular}{c} \(\mathrm{E_{MAR}}\) \\ \end{tabular} & \begin{tabular}{c} \(\mathrm{\Delta E_{MAR}}\) \\ \end{tabular} & \
\begin{tabular}{c} \(\mathrm{\Delta E}\) \\ \end{tabular} &
\begin{tabular}{c} \(\mathrm{Rank}\) \\ \end{tabular} \\ \hline Sigmoid & 1:04 & 13.853 & 0.604 & 1.007 & 1.412 & 1.201 & 1.536 & 2 \\ Tanh & 1:03 & 12.890 & 0.621 & 1.035 & 1.634 & 1.390 & 1.748 & 5 \\ ReLU & 1:03 & 31.537 & 0.860 & 1.434 & 2.750 & 2.339 & 2.881 & 6 \\ Softplus & 1:04 & 13.968 & 0.600 & / & 1.617 & 1.375 & 1.724 & 4 \\ Swish & 1:04 & 12.434 & 0.619 & 1.417 & 0.720 & 1.205 & 1.546 & 3 \\ Exp & 1:03 & 12.843 & 0.688 & 1.147 & 1.176 & / & 1.362 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the models’ error depending on the activation function used. We selected the worst case, multiplied the strain range by 2 (up to \(\epsilon=1.4\)), multiplied the strain rate by 2 (\(\dot{\epsilon}=10\)) and lowered the temperature to \(T=1000\)\({}^{\circ}\)C. Figure 7 shows the evolution of the flow stress predicted by the six models. In Figure 7, top left, when deformation alone is extended, we can see that all six models correctly predict the flow stress evolution over the interval \(\epsilon=[0,0.7]\), whereas they diverge beyond a deformation of \(\epsilon=0.7\). The behavior of the different models is highly variable, and overall, only the Sigmoid and Tanh models show a physically consistent trend. The model with an exponential activation function behaves catastrophically outside the learning range, due to the very nature of the exponential function. When deformation and temperature are out of range (top right), behavior is consistent below \(\epsilon=0.7\) and divergent beyond. Again, only the Sigmoid and Tanh models show a physically consistent trend above \(\epsilon=0.7\). When strain and strain rate are out of range (bottom left in Figure 7), the behavior is consistent below \(\epsilon=0.7\), while diverging above \(\epsilon=0.7\). The values given by the exponential model are out of range for all strain values. Finally, when all inputs are out of range (bottom right), the behavior is consistent and identical, except for the ReLU model below \(\epsilon=0.7\). It is divergent above \(\epsilon=0.7\), and consistent only for the Sigmoid and Tanh models. Figure 5: Comparison of experimental (dots) and the flow stress \(\sigma\) predicted by the ANN (continuous line) for \(\dot{\epsilon}=1\) s\({}^{-1}\). We can therefore conclude from this extrapolation study that it is important to remain as far as possible within the limits of the neural network's learning domain if the results are to be physically admissible. Furthermore, from these analyses, it appears that only the Sigmoid and Tanh models are capable of physically admissible prediction of the flow stress values outside the learning domain. This is due in particular to the double saturation of the tanh and sigmoid functions, as illustrated in Figure 3, when the input values are outside the usual limits. Figure 6: Comparison of experimental (dots) and ANN predicted flow stresses (continuous line) using the Exponential activation function.