## 1 Introduction
### Main Contribution  
The idea behind this work is that field inversion can be used to exploit easily measurable data to correct the entire flow field and to improve the prediction of variables that cannot be easily measured. Then it is possible to exploit these improved fields in two ways: the first approach is the classical FIML approach in which machine learning is used to identify a function of some local flow features that can provide the local value of the correction [8; 21; 22]. In this work, we propose a second approach involving the training of a U-Net model, a type of convolutional neural network (CNN) commonly used to accelerate computations in computational fluid dynamics (CFD). The U-Net model has demonstrated significant potential in various CFD applications. For instance, the authors in [23] employed the U-Net model for fast and precise prediction and optimization of wind environments around buildings. Similarly, a modified U-Net architecture has been successfully applied to predict flow field information in porous media, showcasing the ability of deep learning to efficiently handle complex fluid dynamics problems [24]. Beyond flow field prediction, U-Net architectures have also been employed in thermal-hydraulic analysis for reactor simulations. An example of this is the Unet-IMR, an intelligent mesh refinement method that improves mesh quality by predicting and refining poor mesh regions, thereby enhancing simulation accuracy in complex fluid dynamics scenarios [25]. High-fidelity CFD simulations are essential for the analysis and optimization of components like heat exchangers in thermal management systems, particularly in electric aircraft propulsion. These simulations, however, are computationally intensive, especially during sensitivity analyses and geometry optimizations. To address these challenges, a deep-learning-based surrogate model, specifically a geometry-adaptive CNN, was developed to predict RANS mean flow fields in heat exchangers with varying geometries. By integrating a physics-informed approach, this CNN model achieved high accuracy in predicting velocity, pressure, and temperature distributions while adhering to governing equations and boundary conditions [26]. In our study, we utilize the U-Net model to learn and correct the turbulent eddy viscosity field obtained from the original RANS model, thereby producing an augmented flow field provided by the field inversion procedure. Unlike approaches that seek to develop analytical corrections for the RANS model, our method focuses on providing a global correction to the entire RANS flow field. The motivation behind adopting the second strategy originates from the limitations of using a simple feed-forward neural network (FFNN). By using an FFNN, there is no guarantee that a function describing the correction field obtained through optimization will exist. This is primarily due to the complexity involved in selecting the input variables, a task that is challenging and may not yield a set of inputs that are universally optimal across different test cases. Another issue arises when integrating the model into CFD software. Since the training is based on a database containing results at convergence, the model must handle numerical transients and input values that were not encountered during training and may significantly deviate from those seen in the training data. This can lead to numerical instability, which can be difficult to mitigate, even by reducing the CFL number. The second approach, using a U-Net model, addresses these concerns more effectively because it is applied as a post-processing tool. Unlike the FFNN, the U-Net does not attempt to establish a relationship between the correction field and specific input variables. Instead, it performs a global correction based on the original RANS model fields to produce an augmented field that represents the optimal solution. This corrected field is then imposed on the CFD solver, and if it is accurately determined, the solver converges within a few iterations.