## 5 Numerical Results  
### Example 1  
The proposed neural constitutive network with the architecture presented in Figure 5 has been trained with different numbers of epochs, layers and neurons. The set of data for training and testing from the representative volume elements has in total 9261 couples of input-output values, i.e., for \(\epsilon\) and \(\sigma\). For the \(\epsilon\), we have  
\[\varepsilon_{ij}^{m} = \varepsilon_{ij}^{m-1}+h,\;\;h=0.001,\] \[\varepsilon_{ij}^{i} = -0.01,\;\;m=2,3,...M,\;\;M=21. \tag{13}\]  
In this example, neural networks, which consist of four hidden layers with [15; 20; 15; 20] neurons, have been considered. The batch size is 64, and from the data set, \(M=168\) values of the stress tensor (84 for the training and 84 for the test validation) are used to construct the neural constitutive models with computations of the partial derivatives for the stress tensor. The batch size, which is an important key in machine learning and deep learning, refers to the number of training samples utilized in one iteration of model training. It can influence the time of training, the efficiency and the model performance. The batch size can be chosen while training the model, e.g., in the Keras library by the module \"keras.models.Model.fit(xtrain, ytrain, validationdata= (xtest,ytest), epochs=epochs, batchsize=64, verbose=2,callbacks=callback)\".  
Here, Formula (5) is used for the loss function; i.e., each neural model is trained separately. The results with 4000 training iterations are shown in Figure 7.  
Figure 7 shows a match between the data set and the NN stress components. In Table 1, the error loss function  
\[\text{MSE}\sigma=\text{max}\{\text{MSE}\sigma_{xx},\text{MSE}\sigma_{yy}, \text{MSE}\sigma_{xy}\}\]  
is given for different numbers of layers, neurons and epochs.  
The time of computations can be decreased by using the three surrogate neural networks with the residual neural model, which includes the residual (4). The loss error function includes all three components of the stress function (Figure 4). This approach is less expensive, but it is effective only for computations of the stress tensor. Figure 8 shows the loss function (4) while training the neural network to calculate the components of the stress tensor. The number of epochs is 4000. The neural network as before has four hidden layers with [15; 20; 15; 20] neurons, respectively. The batch size is 64, and from the data set, we have taken the \(M=82\) (42 for the training and 42 for the test validation) values of the stress tensor.  
Figure 7: The components of the stress tensor after training of the neural network (Figure 5) and the data set (\(M=82\)), obtained from the representative volume elements.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number of** & **Neurons** & **Epochs** & **Training Samples** & **MSE\(\sigma\)** & **Time (min.)** \\ \hline
2 & [40,40] & 2000 & 42 & \(1.4\times 10^{-5}\) & 3 \\ \hline
2 & [40,40] & 4000 & 42 & \(1.2\times 10^{-6}\) & 6 \\ \hline
3 & [15,30,40] & 2000 & 42 & \(1.2\times 10^{-5}\) & 5 \\ \hline
4 & [15,20,15,20] & 4000 & 84 & \(4.7\times 10^{-6}\) & 8 \\ \hline
4 & [15,20,15,20] & 8000 & 84 & \(1.6\times 10^{-6}\) & 15.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The model loss error for different values of the training parameters.