## 1 Introduction
Implementing ANNs for plasticity in thermomechanics has been studied, and a review of the literature can be found, for example, in the work of [PERSON] et al. [34] concerning the use of recurrent neural networks, in that of [PERSON] et al. [35] concerning their application in finite element analysis of metal forming processes, or in that of [PERSON] et al. [36] concerning the applicability to meta-materials and their characterization. A distinction must be made between ANN-based flow models (the focus of this study) and ANN-based constitutive models. Both approaches have been studied by many researchers during the last thirty years. [PERSON] [37] proposed an ANN-based constitutive model for concrete under monotonic biaxial loading and cyclic uniaxial loading. They extended their work by introducing adaptive and auto-progressive networks in [38; 39], where the architecture of the network evolves during the learning phase to better learn the complex stress-strain behavior of the materials using a global load-deflection response, where the evaluation of the flow stress of the material computed by the ANN is combined with a radial return algorithm. [PERSON] et al. [40] proposed an ANN to predict the flow stress of 42 CrMo4 steel in hot compression tests on a Gleeble thermomechanical device and showed a very good correlation between the experimental results and the model predictions. [PERSON] et al. [41] compared the predictive capabilities of an ANN versus an analytical model for Johnson-Cook, Arrhenius, and strain-compensated Arrhenius laws and concluded that the neural network had better efficiency and accuracy in predicting the hot behavior of the Al-Cu-Mg-Pb alloy. The underlying idea proposed in our approach is to implement a flow law described by a trained ANN as a FORTRAN 77 subroutine in the Abaqus FE code. This ANN was previously trained from the data extracted from mechanical tests of the material and can directly define the value of the flow stress \(\sigma^{y}\) as a function of the plastic strain \(\varepsilon^{p}\), the strain rate \(\dot{\varepsilon}\) and the temperature \(T\). After a training phase based on the use of the Python library TensorFlow [42; 43], the weights and biases of the trained neural network are transcoded into a subroutine in FORTRAN 77, which is compiled and linked with the libraries of the Abaqus FE code to include the behavior of the material by allowing the computation of the flow stress \(\sigma^{y}\) as a function of \(\varepsilon^{p}\), \(\dot{\varepsilon}\) and \(T\), and of its three derivatives \(\partial\sigma^{y}/\partial\varepsilon^{p}\), \(\partial\sigma^{y}/\partial\dot{\varepsilon}\) and \(\partial\sigma^{y}/\partial T\). The structure of this paper is as follows: Section 2 addresses the presentation of a neural-network-based flow law and its training from the data proposed by [PERSON] et al. [30] and reported in Figure 1. The comparison of several neural network architectures regarding accuracy and implementation complexity will be presented and compared. In Section 3, we will present the transposition of this neural network into a FORTRAN 77 subroutine for the Abaqus FE code. Validation is based on the numerical simulation Abaqus Explicit FE code of a compression test in the same configuration as the one proposed by [PERSON] et al. [30] using four different ANN flow laws. In this Section, we will present the problems of over-fitting the neural network and its visible consequences on the results concerning numerical simulations. Finally, a conclusion and perspective section will conclude this paper.