## 4 Conclusions and Future Work  
In this paper, a flow law based on an artificial neural network capable of predicting the flow stress of a material as a function of input data, such as the plastic strain \(\dot{\epsilon}^{p}\), the strain rate \(\dot{\epsilon}\) and the temperature \(T\), for a metallic material subjected to high thermomechanical loading is presented.  
From the equations that govern the writing of the neural network, the expressions of the derivatives of the flow stress of the model as a function of the plastic strain \(\dot{\epsilon}^{p}\), the strain rate \(\dot{\epsilon}\) and the temperature \(T\) were established. These expressions allow the transfer of the neural network behavior into a VUHARD subroutine written in FORTRAN 77 language to allow the use of this network for the computation of the material flow stresses within the Abaqus FE code.  
Figure 13: Predicted flow stress \(\sigma^{y}\) as a function of \((\dot{\epsilon}^{p},\dot{\epsilon})\) for a fixed temperature \(T=750\)\({}^{\circ}\)C using the 3-7-4-1 model.  
In a first step, data from the works of [PERSON] et al. (2018) and [PERSON] et al. (2018), allowed to train several architectures of the proposed model and to compare the results of these ANN models regarding the fidelity of reproduction of the experimental behavior. The comparison of the results obtained allowed to validate the approach and to show the superiority of the ANN model compared to the analytical models based on Johnson-Cook or Arrhenius flow laws, both in terms of the fidelity of the model and quality of the results. In a second step, after transferring the training data to the VUHARD subroutines for the Abaqus FE code, we showed the consistency and quality of the numerical results obtained during the numerical simulation of the compression of a GCr15 alloy cylinder. In the same section, we also discussed the problems of over-fitting the ANN when the number of neurons is too large compared to the training data ranges. It is therefore important to always adjust the structure and size of the neural network to the experimental data that we wish to approximate to avoid this over-fitting phenomenon.  
This work thus allowed to highlight the significant contributions of flow laws based on neural networks in numerical simulation by finite elements on a commercial FE code, such as the Abaqus software. The quality of the results obtained allows us to go further in the use of the simulation results and in particular to consider that the results of these finite element simulations can predict the phase transformations and the dynamic recrystallization within the material during the thermomechanical transformation at high temperature.  
This research received no external funding.  
**Data Availability Statement:** Source files of the numerical simulations are available from the author.  
**Conflicts of Interest:** The author declare no conflict of interest.  
**Abbreviations**  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} ANN & Artificial Neural Network \\ DRX & Dynamic Recrystallization \\ CPU & Central Processing Unit \\ FE & Finite Element \\ VUMAT & User subroutine to compute the stress tensor for Abaqus/Explicit \\ VUHARD & User subroutine to compute the flow stress for Abaqus/Explicit \\ \end{tabular}