## 3 ANN Flow Law Implementation in FE Software
### Numerical Simulations and Benchmarks Tests  
To validate the proposed approach and to compare the different neural network architectures proposed in Section 2.3, we propose here to simulate on the Abaqus Explicit FE code the high-temperature compression of a cylinder on a Gleeble-type thermomechanical device. We consider a cylinder in compression with an initial diameter \(d_{0}=8\) mm and an initial height \(h_{0}=12\) mm made of GCr15 material, for which the final height after compression is \(h=6\) mm, which is a reduction of 50% of its total height. The compression of the sample is done in 10 s so that the strain rate \(\hat{\epsilon}\) is in the corresponding range of the  
Figure 7: General flowchart of the radial return algorithm to compute the final stress tensor \(\mathcal{O}_{1}\). characterization of the material behavior defined by [PERSON] et al. [30]. Concerning the flow laws, the 4 models presented in Section 2.3 will be used and compared between them. Unfortunately, it is not possible here to compare these results with the experimental results, even if the initial shape of the specimen is the same, as these are not available in the references of the work of [PERSON] et al. [14] or [PERSON] et al. [30]. The mesh of the sample is made with 850 axis-symmetric quadrilateral finite elements with 4 nodes and reduced integration (named CAX4R in the Abaqus software) with 50 elements in the vertical direction and 17 elements in the radial direction, respectively. The cylinder is between two rigid surfaces, and the Coulomb friction law with a friction coefficient at the contact surfaces was set to \(\mu=0.15\). The simulation time being fixed at 10 s in order to reduce the simulation time, considering that an explicit integration scheme is used, a global mass scaling is used for all simulations. The VUHARD subroutine is compiled using the GNU gfortran 11.3.0 and linked to the main Abaqus Explicit executable. All benchmarks tests were solved using Abaqus Explicit 2022 on a Dell XPS 13 laptop running Ubuntu 20.04 64 bits with 16 GiB of RAM and one 4 core i/-10510U Intel Processor. All computations were performed using the double precision option of Abaqus, with parallel threads execution on two cores. Figure 8 shows the plastic strain field \(\varepsilon^{p}\) contourplot within the structure at the end of the simulation for both the 3-7-4-1 (left side) and the 3-15-7-1 (right side) flow laws, while Figure 9 shows the temperature field \(T\) contourplot for the same models. Both sides of the figures look more or less the same with some visible differences from the left to the right concerning the shape of the isovalues zones and the maximum value, but in fact, the two models with the lowest and the highest number of neurons give  
Figure 8: Equivalent plastic strain \(\varepsilon^{p}\) contourplot for the compression of a cylinder using the 3-7-4-1 (**left side**) and the 3-15-7-1 (**right side**) ANN flow laws. Figure 9: Temperature \(T\) contourplot for the compression of a cylinder using the 3-7-4-1 (**left side**) and the 3-15-7-1 (**right side**) ANN flow laws. coherent results concerning for the plastic strain \(\varepsilon^{p}\) and temperature \(T\) contourplots. The maximum plastic strains are concentrated in the center of the specimen with a maximum value of \(\varepsilon^{p}=0.89\) for the 3-7-4-1 model and \(\varepsilon^{p}=0.95\) for the 3-15-7-1 model, which is slightly beyond the limit set by the training data, which varies from 0 to 0.7. As shown by [PERSON] et al. (2018), the flow laws defined by neural networks are able to correctly extrapolate the flow stresses \(\sigma^{y}\) when the plastic deformations are higher than at least 150% of the maximum plastic strain used during training. Concerning the temperature \(T\), the maximum value is around \(T=795\)\({}^{\circ}\)C and \(T=799\)\({}^{\circ}\)C, which is very close and in accordance to the experiments used for the learning phase.