## 1 Introduction
[30] performed compression tests on GCr15 cylinders in a temperature range of 750 \({}^{\circ}\)C to 1300 \({}^{\circ}\)C in 50 \({}^{\circ}\)C steps, strain rates of 0.001 s\({}^{-1}\), 0.01 s\({}^{-1}\) and 0.1 s\({}^{-1}\) and strains up-to 0.7. The results of these compression tests, plotted in Figure 1, show a decrease in flow stress \(\sigma^{y}\) with respect to an increase in the temperature \(T\) and a increase of \(\sigma^{y}\) with respect to an increase in the strain rate \(\hat{\varepsilon}\), as in most metallic materials. Figure 1: Original data extracted from the publication of [PERSON] et al. [30]. The evolution of the flow stress as a function of the plastic deformation shows the presence of a dynamic recrystallization (DRX) phenomenon within the material. This phenomenon is an additional non-linearity of this type of material compared to other materials, mainly because of high temperatures and low strain rates, which should be considered when describing the material behavior. As stated in the publication of [PERSON] et al. [14], depending on the flow model used--Johnson-Cook, modified Zerilli-Armstrong, Arrhenius or new modified Johnson-Cook--the fidelity of considering the real behavior varies widely with the complexity of the flow model, which includes between 5 parameters for the Johnson-Cook model and 16 parameters for the Arrhenius model. Thus, and for the input data provided by [PERSON] et al. [30], the two most common models, Johnson-Cook and Zerilli-Armstrong do not correctly describe the material behavior. Only the modified Johnson-Cook and Arrhenius models can describe correctly the behavior of the material during the compression process. Unfortunately, and this is not part of their study, if these last two models are satisfactory from a theoretical point of view, from a practical point of view for the user of a FE code such as Abaqus, it will be necessary to carry out a numerical implementation in a FORTRAN 77 VUMAT subroutine of the modified Johnson-Cook flow law or the Arrhenius law as carried out by a few authors [24; 27; 28] to use these laws for numerical simulation. This requires a certain expertise in the development and implementation of flow laws, which is not available to all users of the Abaqus FE code. From this observation, and from the necessity to select a flow law for a type of material, then to identify the parameters of this flow law according to experimental tests, and finally to implement this flow law as a user subroutine in FORTRAN in the Abaqus FE code, we recently proposed in [PERSON] et al. [31] an alternative approach based on the ability of artificial neural networks (ANNs) to behave as universal approximators as reported by [PERSON] et al. [32] and [PERSON] et al. [33]. In fact, artificial neural networks can solve problems that are difficult to conceptualize using traditional computational methods. Unlike a classical approach based on a regression method, an ANN does not need to know the mathematical form of the model it seeks to reproduce; hence, we do not need anymore to postulate the mathematical form of the constitutive equation to use it in a FE simulation using this kind of approach. Using a neural network instead of an analytical constitutive law can lead to a bias related to the validity of the answers according to the domain of use and the learning domain. Thus, if ANNs are efficient for the interpolation of results inside the learning domain, their behavior outside of it is not controlled. Therefore, if the input values are far from those provided during the training, the outputs can be far from the physical reality of the process. It is, of course, the same for analytical laws when modeling non-linear behavior, but if the choice of the model is made properly, based on physical considerations, they will provide results closer to reality than the ANN model. Therefore, care should be taken when using ANN-based flow laws, and the validity of the model input data should always be verified.