## 4 Conclusions  
In this work, the potential of field inversion for improving the prediction of the flow field around airfoils at low Reynolds number was investigated and critically discussed. First of all, a discussion on the choice of the goal function which drives field inversion was performed: the minimization of the error on a physical distribution (for example wall static pressure) provided better results with respect to the error on an integral quantity. Two different strategies for exploiting the results of field inversion are then discussed. The first considered approach is the classical local FIML paradigm which was originally proposed by [13]. This approach is promising but the results obtained on the considered test cases enlightened some criticalities. First of all, the regression step of the FIML paradigm requires us to assume a list of input variables for the FFNN: this choice is not trivial and it is possible to find problems where the correction is not a function of the chosen input variables because it is possible to find in the database different values of the correction for the same values of the input variables. Furthermore, even if the chosen set of input variables provides a suitable regression, this function can lead to numerical instabilities when integrated into a RANS solver and used for predictive CFD simulations. This behavior can be explained by the fact that the FFNN is trained on a set of steady solutions and so, during the numerical transient of a predictive simulation, it can encounter input values that are far from the training range. In the present work, it was observed that a significant source of instability is related to the development of negative corrections during the transients: in order to solve this problem, a mapping function on the correction is introduced during the field inversion optimization in order to guarantee the positivity of the correction. This significantly improved the numerical behavior of the corrected model during the predictions but it limited its correction capability. This last limitation was not observed in the global U-Net approach. This approach has already demonstrated promising results. For instance, [PERSON] et al. in [55] utilized a U-Net model as a surrogate, referring to it as a physics-informed CNN (PI-CNN) because they enhanced the loss function with physical constraints. They applied this model to predict the turbulent mixing layer over time at various Reynolds numbers. In contrast, our work employs a different strategy. Rather than providing an augmented RANS model like the FIML paradigm, our method offers a post-processing procedure that globally adjusts the flow field predicted by the baseline RANS model. Since the corrected eddy viscosity is directly imposed as a frozen field in the CFD simulation, the numerical instabilities, which represent a problem for FIML, are not observed in the global U-Net approach. In the  
Figure 26: Lift and drag coefficients predicted with U-Net-augmented Spalart–Allmaras model for the NACA0021 test case at various angles of attack. present implementation, the U-Net correction acts on a Cartesian grid and so it is necessary to interpolate the CFD solution provided by the baseline RANS model from the original unstructured mesh to a Cartesian mesh and back. Interpolating data onto a Cartesian mesh for U-Net can lead to significant errors if the mesh is too coarse, particularly in the boundary layer region. This issue is also highlighted in [56], where the authors opted to use a simple FFNN to circumvent the problem. In our study, we integrated the FFNN within the CFD software, but this introduced instability, unlike in their work where the FFNN served as a surrogate model to predict flow quantities based on airfoil shape, Mach number, and angle of attack. The interpolation issue can be mitigated by using a sufficiently fine Cartesian mesh, limited to the region of interest (typically near the solid wall where the goal function is computed), or by employing U-Nets that operate on unstructured meshes.