## 4 Conclusions and Major Remarks  
In this paper, several ANN-based flow laws for thermomechanical simulation of the behavior of a P20 medium-alloy steel have been identified. These six laws exhibit distinctions solely in the choice of their activation functions, while maintaining a uniform architectural framework characterized by consistent specifications regarding the quantity of hidden layers and the number of neurons present on each of these hidden layers. In addition to the five classic activation functions (Sigmoid, Tanh, ReLU, Softplus and Swift), in this paper, we proposed the use of the Exp (exponential) function as an activation function, although this is almost never used in neural network formulations. The expressions of the activation functions and their derivatives were used in the neural network writing formalism to calculate the derivatives of \(\sigma\) with respect to \(\varepsilon\), \(\hat{\varepsilon}\) and \(T\).  
Comparison of the ANNs results (in terms of flow stress \(\sigma\)) with experiments have shown that all five activation functions, with the exception of the ReLU function, give very good results, far superior to those obtained conventionally using formalisms based on analytical flow laws from the literature, such as the Johnson-Cook, Arrhenius or Zerilli-Armstrong models [14]. To improve the extrapolation ability of the models, it is recommended to use the Sigmoid and Tanh activation functions. These functions can effectively squash out of bounds values, giving the artificial neural network a more realistic behavior beyond the training bounds.  
Based on the equations describing the mathematical formulation of an ANN with two hidden layers, and depending on the nature of these activation functions, we have implemented these constitutive laws in the form of a Fortran 77 subroutine for Abaqus explicit. The same approach can also be used to write a UHARD routine enabling the same flow laws to be used in Abaqus standard.  
Numerical results obtained from a compression test on a metal cylinder using the Abaqus explicit code have shown that neural network behavior models give very satisfactory results, in line with experimental tests. The Exp activation function, which is rarely used in the formulation of artificial neural networks, showed very good results (in agreement with more complex models such as Tanh), while enabling the code user to benefit from the efficiency and ease of implementation of an exponential function. These results are satisfactory insofar as the inputs remain entirely within the model's learning domain, since the extrapolation capabilities of the network based on the exponential function are very limited. We then obtain results equivalent in terms of solution quality to sigmoid or tanh-type formulations, while having computation times comparable to a ReLU function.  
Overall, this study concludes by recommending the use of the sigmoid activation function for the development of flow laws, since it gives very good results in the identification domain, allows us to leave the learning domain with a behavior that is certainly biased, but physically admissible, and offers very good performance in terms of simulation time when implemented in a finite element code. This study emphasizes the valuable impact of neural network-derived flow laws for numerical finite element simulation executed with a commercial FE code like Abaqus.  
This research received no external funding.  
**Data Availability Statement:** Source files of the numerical simulations and the ANN flow laws are available from [36].  
**Acknowledgments:** The author thanks the team of [PERSON] from the Ecole Technique Superieure de Montreal, Canada, for providing the experimental data used in Section 1.2.  
**Conflicts of Interest:** The author declares no conflict of interest.