## 2 Methods
### Field Inversion  
Recently, there has been a notable shift in turbulence modeling towards data-driven approaches, leveraging machine learning techniques to enhance the accuracy of current models. Several reviews have explored these advancements within the computational fluid dynamics (CFD) community, highlighting how these new algorithms can reduce costs and improve prediction accuracy [29; 30; 31]. By incorporating data-driven techniques, such as Artificial Neural Networks (ANNs), Support Vector Machines (SVMs), Gaussian Process Regression (GPR), and Genetic Expression Programming (GEP), turbulence models can potentially better predict turbulent flows and augment RANS models with additional experimental or simulation data. According to [32], there are three primary approaches to implementing data-driven algorithms in turbulence modeling. Firstly, these algorithms are used to estimate parameters within turbulence models, often fine-tuned based on experimental data, particularly focusing on the Reynolds stress tensor and the Boussinesq hypothesis [33]. This approach includes methods that enhance neural networks with Galilean invariance, thereby increasing flexibility in handling fluid flow complexities [15; 34; 35; 36]. Secondly, traditional closure models are replaced with fully connected neural networks (FCNN) or networks employing gated recurrent units (GRU) to address stability issues over longer time frames [37; 38; 39]. Finally, a third approach substitutes the entire PDE-based model with autoencoders or Physical Informed Neural Networks (PINNs) [40; 41; 42; 43]. One particularly promising data-driven approach is the field inversion and machine learning paradigm [13; 20; 22], which is based on the solution of an optimization problem in which high fidelity data are used to find a correction field to the production term of a baseline RANS model. This correction field is then analyzed through machine learning tools to express the local correction as a function of local flow variables. FIML has shown significant potential in improving the accuracy of RANS models by integrating additional experimental or simulated data. Various studies have demonstrated its effectiveness across different applications. For example, [PERSON] [44] used FIML in gradient-based shape optimization by replacing RANS models, while [PERSON] and [PERSON] [45] enhanced a four-equation transition model. Additionally, [PERSON] et al. [8; 21] applied FIML to improve RANS modeling in turbomachinery flows, and [PERSON] et al. [46] utilized it in incompressible turbulent flow to directly adjust turbulent viscosity. In this work, we apply the field inversion approach to enhance the SA model. The classical field inversion approach modifies the production term of the SA model to improve its transition prediction capabilities. Specifically, the production term \(P\) is adjusted using a correction field \(\beta\) which is inferred through an optimization process. This process involves solving an inverse problem to minimize the discrepancy between model predictions and experimental data. The modified SA model is expressed as:  
\[\frac{\partial\varrho\bar{\vartheta}}{\partial t}+\
abla\cdot(\rho\mathbf{u} \bar{\
u})=\rho[h(\beta)P-D]+\frac{1}{\sigma}\
abla\cdot(\rho(\
u+\bar{\
u}) \
abla\bar{\
u})+\frac{cb_{2}}{\sigma}\rho(\
abla\bar{\
u})^{2}-\frac{1}{ \sigma}(\
u+\bar{\
u})\
abla\rho\cdot\
abla\bar{\
u} \tag{21}\]  
where \(h(\beta)\) is a function mapping the correction field. The mapping function ensures that \(\beta\) remains non-negative, following the approach outlined in [46]. The equations defining this mapping function are provided below:  
\[h(\beta)= \begin{cases}\beta+1&:\beta\geq 0\\ \exp{(\beta)}&:\beta<0\end{cases}  \tag{22}\]  
The correction field \(\beta\) is determined through the optimization problem:  
\[\beta^{*}=\operatorname*{arg\,min}_{\beta}G(\bar{u},u_{exp})\text{ s.t. } \tag{23}\]  
\[R(u(x,\beta(x)),\beta(x))=0 \tag{24}\]  
where \(G\) denotes the objective function and \(R\) represents the residuals of the equations.