## 4 Conclusions
This latter approach aligns with the method proposed by [57], in which the authors used a graph convolutional network (GCN), a variant of the convolutional neural network, as described in [58]. This extension is left for future work. In this study, we evaluated the proposed methodology only in 2D problems because in this way it was possible to perform several tests due to the low cost of the 2D simulations. However, there are no theoretical limitations to the extension of the method in 3D since it would be sufficient to modify the architecture of the U-Net model. The two strategies were investigated on two test cases characterized by transition and separation (NACA0021 and SDJ7003 airfoils): these flow features are not correctly captured by the chosen baseline Spalart-Allmaras model which was developed for fully turbulent flows. However, the field inversion shows its potential by providing a correction field which can significantly improve the prediction in both cases. The correction field was then exploited with both the local FIML and the global U-Net approaches: both strategies are able to improve the prediction of the original RANS model but the FIML showed some instability problems which limited the exploitation of field inversion results. On the contrary, the global U-Net approach avoided these instability problems and provided significant improvements in the predicted flow field. Furthermore, the global U-Net approach is less intrusive since does not require us to implement the correction to the production terms in the turbulent eddy viscosity equation inside the CFD solver. Conceptualization, L.M., A.F. and F.L. ; methodology, L.M. and A.F. ; software, L.M., M.C. and A.I. ; supervision, A.F. and F.L. ; visualization, L.M. ; writing--original draft preparation, L.M. ; writing--review and editing, L.M. and A.F. All authors have read and agreed to the published version of the manuscript. This research was funded by the Italian Ministry of University and Research through the PRIN research project 2022B2X937, 'NextGenSProDesT: Next Generation Space Propulsion Design Techniques', and supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Actions, grant agreement 872442 (ARIA). The data presented in this study are available upon request from the corresponding author. The full dataset is not publicly available due to the extensive number of simulations performed for this work; however, specific data will be provided upon request. The authors would like to thank OPTIMAD Srl for providing the CFD software, immerFlow, and the necessary computational resources that made all the simulations possible. Additionally, we extend our gratitude to NUREA for their assistance in understanding the U-Net model applied in this work. Special thanks to [PERSON] for sharing her extensive knowledge of machine learning models and for all her valuable suggestions. We also wish to thank [PERSON] and [PERSON] for their insightful discussions on the various applications of machine learning models in the field of CFD. Authors [PERSON] and [PERSON] were employed by the company Optimad Srl Via Agostino da Montefeltro, Torino. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.