## 2 Training of the ANN Flow Law
### Training of the Neural Networks  
In neural network learning, it is necessary to define the objective function to be minimized and the evaluation of the model error. In this study, the error evaluation is based on the mean square error (\(\mathrm{E}_{\mathrm{MS}}\)) and the root mean square error (\(\mathrm{E}_{\mathrm{RMS}}\)) given by the following equation:  
\[\mathrm{E}_{\mathrm{RMS}}(\mathrm{MPa})=\sqrt{\mathrm{E}_{\mathrm{MS}}}=\sqrt {\frac{1}{N}\sum_{i=1}^{N}\left(\square_{i}^{e}-\square_{i}^{y}\right)^{2}}, \tag{16}\]  
where \(N\) is the total number of numerical training data used, \(\square_{i}^{y}\) is the \(i\)th value predicted by the neural network, and \(\square_{i}^{e}\) is the corresponding experimental value coming from the experimental tests. The accuracy and predictive ability of the models is assessed by the mean absolute relative error (\(\mathrm{E}_{\mathrm{MAR}}\)) defined by Equation (17):  
\[\mathrm{E}_{\mathrm{MAR}}(\%)=\frac{1}{N}\sum_{i=1}^{N}\left|\frac{\square_{i} ^{y}-\square_{i}^{e}}{\square_{i}^{e}}\right|\times 100 \tag{17}\]  
The numerical implementation of the learning phase of the neural network was done in Python language, using the TensorFlow library [42; 43]. The minimization procedure of the objective function is based on the use of the adaptive moment estimation (ADAM) solver proposed by [PERSON] et al. [47]. The training data used in this section are taken from the publication of [PERSON] et al. [30]. Thus, data for which compression tests were performed for the 3 strain rates \(\hat{\epsilon}=0.001\) s\({}^{-1}\), \(\hat{\epsilon}=0.01\) s\({}^{-1}\) and \(\hat{\epsilon}=0.1\) s\({}^{-1}\), and the 12 temperature values between 750 \({}^{\circ}\)C and 1300 \({}^{\circ}\)C in 50 \({}^{\circ}\)C steps are used. For each pair of data (\(\hat{\epsilon}\), \(T\)), we have a record of 71 values of flow stress \(\sigma^{e}\) corresponding to values of deformation between 0 and 0.7, regularly spaced of 0.01. We use a database of 2556 quadruplets of values (\(\varepsilon^{p}\), \(\hat{\epsilon}\), \(T\), \(\sigma^{e}\)) for the training of the ANNs. The set of these data is used as training data for the neural network. Several neural network architectures have been studied in this work; they differ from each other by the number of neurons present in the two hidden layers. Among them, we selected 4 different architectures named 3-7-4-1, 3-9-4-1, 3-9-7-1 and 3-15-7-1 for which the name 3-_n-m-1_translates an ANN with 2 hidden layers, having \(n\) neurons on the first layer and \(m\) neurons on the second layer. All models have been trained for the same number of iterations (50,000 iterations), and around 50 min of training on a Dell XPS-13 7390 laptop running Ubuntu 22.04 LTS 64 bits with 16 GB of RAM and an Intel 4-core i7-10510U processor allow obtaining the converged parameters of the ANN models. Figure 3 shows the evolution of the training error defined by the \(\log_{10}\) of the mean square error (\(\log_{10}[\text{E}_{\text{MS}}]\)) during the training phase. As we can see on this figure, after 50,000 iterations, we can consider that we have reached a stationary state of the model learning and that it is useless to continue the learning phase. As expected, the more neurons the model contains, the more it can follow the non-linear evolution of the material's behavior and therefore the more the mean square error (\(\text{E}_{\text{MS}}\)) during the learning phase decreases. Table 1 shows the main results of the training of these neural networks. It can be noted that the number of internal variables \(n_{v}\) of the networks varies from 65 to 180 for the most complex and the most powerful one, but that this complexity has no real influence on the learning time \(t\) which oscillates around a value of 50 min, regardless of the architecture chosen.