# Efficient LES Parametric Studies via ANN-Based Multi-Fidelity Modeling and Adaptive Sampling

Thomas Berthelon, Ali Mahdi, Guillaume Balarac

► To cite this version:

Thomas Berthelon, Ali Mahdi, Guillaume Balarac. Efficient LES Parametric Studies via ANN-Based Multi-Fidelity Modeling and Adaptive Sampling. Flow, Turbulence and Combustion, 2025, 115, pp.603-622. 10.1007/s10494-025-00669-0. hal-05111848

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Submitted on 13 Jun 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Efficient LES Parametric Studies via ANN-Based
Multi-Fidelity Modeling and Adaptive Sampling.

Thomas Berthelon¹*, Ali Mahdi¹ and Guillaume Balarac¹

¹Université Grenoble Alpes, CNRS, Grenoble-INP, LEGI, Grenoble, France.

*Corresponding author(s). E-mail(s):
thomas.berthelon@univ-grenoble-alpes.fr;

**Abstract**

Large Eddy Simulations (LES) are increasingly used in industry due to their superior accuracy compared to traditional statistical methods like Reynolds-Averaged Navier-Stokes (RANS) simulation. However, their high computational cost remains a major obstacle to performing daily parametric studies in engineering design offices. The objective of this work is to improve the efficiency of LES-based parametric studies through multi-fidelity surrogate modeling. Taking into account the computational cost of each turbulence modeling approach, multi-fidelity technic propose to combine limited number of LES results with more numerous RANS simulations. To achieve this, we use Artificial Neural Networks (ANN), which are particularly effective at capturing complex relationships between fidelity levels and handling discontinuities. To further reduce computational cost, we propose a new adaptive sampling strategy that selects high-fidelity LES points based on an estimation of interpolation error. This approach enhances the accuracy of the multi-fidelity method by efficiently allocating computational resources where they are most needed. The proposed strategy is first validated on an analytical test case before being applied to the study of the lift coefficient as a function of the angle of attack for a NACA0012 airfoil. We demonstrate that with only five LES evaluations, our method accurately captures the main features of this function, including the stall angle. This work paves the way for more efficient LES-based parametric studies.

**Keywords:** Turbulent flows, Large Eddy Simulation, Multi-fidelity, Adaptive Sampling Strategy
# 1 Introduction

Computational Fluid Dynamics (CFD) simulations are essential as an engineering tool, especially for conducting parametric studies or evaluating machine performance across various operating conditions. For example, they are used to determine the efficiency of a turbine based on the flow rate passing through it. To date, the Reynolds Averaged Navier-Stokes (RANS) approach is widely used in design offices because of its low calculation cost. However, the simplifying assumptions on which this approach is based can limit the accuracy of predictions. Because of these limitations, RANS methods are generally considered to be low-fidelity approaches. To overcome these shortcomings, an alternative method, Large Eddy Simulation (LES), is beginning to be adopted by industry. This method makes it possible to explicitly solve large scale structures while modeling the smallest scales, which significantly improves the accuracy of predictions compared with RANS methods. However, the computational cost associated with LES remains prohibitive for design purpose. Due to its high cost, the LES method can only be used to study a limited number of operating points.

In the context of parametric studies, a common approach is to use hybrid RANS/LES methods, such as DES [1]. These methods reduce the computational cost compared to LES simulations, making parametric studies more cost-effective, albeit with a trade-off in accuracy. Another approach, explored in this work, consists in combining the accurate but sparse results from LES with the more approximate yet abundant results from RANS. This combination allows for a broader exploration of the parameter space while maintaining a certain level of overall accuracy. In this work, we will refer to this type of method as multi-fidelity methods. The first research on multi-fidelity methods dates back to the work of Kennedy and O'Hagan in 2000 [2], and they have since been applied in various fields. For a full review of the different multi-fidelity methods, see Fernandez [3].

Most multi-fidelity models are traditionally built using Gaussian Processes (GP) [4–6] due to their ability to quantify uncertainty and provide reliable estimates. In fluid mechanics, several studies have applied GP-based methods to Euler and RANS approaches [7–9]. The GP-based method has also been used in an optimization context. For example, Pellegrini et al. [10] applied GP-based multi-fidelity methods to optimize hydrodynamic shapes using RANS simulations and Mole et al. [11] used GP-based multi-fidelity methods in a context of optimisation of wind farm wake by combining wake models and LES simulations. In these work, they used the uncertainty associated with the GP-based multi-fidelity model to adaptively sample regions of interest and improve prediction accuracy while limiting computational cost. GP-based methods are therefore well-suited for multi-fidelity modeling in RANS/LES context, as they offer effective point selection strategies. However, GP-based methods are unable to capture discontinuities [12]. This limitation restricts the use of these methods, particularly in turbulence flow contexts where such discontinuities are likely to occur.

On the other hand, new developments in multi-fidelity modeling have incorporated Artificial Neural Networks (ANN) as an alternative to GP-based methods. These ANN-based approaches have demonstrated strong performance, particularly in handling complex functions that involve non-linearities, phase shifts, and discontinuities [13]. Nevertheless, studies utilizing ANN-based multi-fidelity approaches have not
yet incorporated an criterion for high-fidelity point selection. For instance, Zhang et
al. [14] employed ANN-based multi-fidelity methods to optimize aerodynamic shapes
using RANS simulations at varying mesh resolutions. Their approach did not include
any criterion to adaptively refine the description of the target function. Similarly, Tao
et al. [15] introduced multi-fidelity convolutional neural networks to enhance opti-
mization performance in RANS-based applications. While their method effectively
improved prediction accuracy, no adaptive sampling strategy to better discretize the
target function has been proposed, which limits the ability to target regions requiring
additional high-fidelity data. To implement a point selection strategy, Bayesian Neu-
ral Networks (BNN) can be used, as they provide uncertainty estimates. This is the
approach taken by Xiong et al. [16] in multi-fidelity frameworks for wing optimization
within a RANS context. Their method integrated uncertainty estimates from BNNs to
guide the adaptive sampling process in airfoil optimization. However, BNNs are more
complex to implement and train compared to traditional ANN, making their use more
challenging.

Multi-fidelity models combining RANS and LES approaches remain scarce. To the best of our knowledge, the work of Mole et al. [17] is the only study that explores such a combination within a multi-fidelity framework. They utilized both ANN- and GP-based multi-fidelity methods to study turbulent flow around an array of tandem wall-mounted cubes. In addition to applying multi-fidelity methods to predict integrated forces, they extended these methods to a multi-objective framework, generating surrogate models for both flow profiles and slices of the flow field. Their results showed that the multi-fidelity approach outperforms single-fidelity methods and is highly promising for describing turbulent flows with limited LES computations. However, this work did not address the selection of high-fidelity sampling locations, as the high-fidelity points were uniformly distributed. Since LES calculations are very costly, a sampling adaptation strategy is needed to get the most benefit from multi-fidelity methods. Moreover, the case treated in this study does not present any discontinuities.

In this work, we aim to demonstrate the potential of multi-fidelity methods within
the RANS-LES framework, a context that remains underexplored in the literature.
This framework can involve cases with discontinuities, necessitating the use of neu-
ral networks to effectively capture the complex relationships between fidelity levels.
To address these challenges, we propose a simple high-fidelity point selection strategy
that is independent of the specific multi-fidelity method employed. In this work, the
strategy is applied to classical ANN-based multi-fidelity methods, providing an effec-
tive and practical solution for selecting high-fidelity points and improving the overall
description of the target function. First, the procedure to select high fidelity point
is detailed. Then, this procedure is evaluated on an analytic case, characterized by a
discontinuity and a phase shift between high and low fidelity data. The procedure is
compared to GP-based approach in particular. The procedure is then used in a con-
text combining RANS results of a NACA0012 airfoil at incidence, demonstrating its
versatility and effectiveness. And finally, the procedure is used blindly in a real-life
situation combining both RANS and LES.
# 2 High-Fidelity Point Selection Strategy

## 2.1 Multi-fidelity framework

The multi-fidelity method is first briefly recalled. The method used in this work is similar to the method used by Mole et al. [17], which is similar of that proposed by Meng and Karniadakis [13]. The focus is on analyzing the evolution of a quantity of interest, y, as a function of the input parameter x. To account for varying levels of model precision, two fidelity levels are introduced: $y_H(x)$, representing the high-fidelity function, which offers greater accuracy at the expense of higher computational cost, and $y_L(x)$, the low-fidelity function, which have a lower computational cost but less precise. Multi-fidelity approaches aim to establish a relationship not only between $y_H$ and x, but also between $y_H$ and $y_L$. The goal is to approximate the potentially non-linear function f that links $y_H$ to both x and $y_L$, expressed as

$$y_H = f(x, y_L). \qquad (1)$$

This formulation enables the integration of information from multiple fidelity levels to improve predictive capability while managing computational resources.

In practice, $n_L$ low-fidelity points are defined, denoted $X_L = x_{L,i}$ for $i = 1, ..., n_L$, which are evaluated using the low-fidelity method. Then, $n_L$ low-fidelity evaluations, i.e. $Y_L = y_L(x_{L,i})$ for $i = 1, ..., n_L$, are obtained. These points are used to train a low-fidelity surrogate model, denoted $\tilde{y}_L$, defined by

$$\tilde{y}_L = \mathcal{G}(X_L, Y_L)(x), \qquad (2)$$

where $\mathcal{G}$ represents the neural network calibrated using the inputs $X_L$ and the outputs $Y_L$. Next, $n_H$ high-fidelity points are defined, denoted $X_H = x_{H,i}$ for $i = 1, ..., n_H$, which are evaluated using the high-fidelity method. Then, $n_H$ high-fidelity evaluations, i.e. $Y_H = y_H(x_{H,i})$ for $i = 1, ..., n_H$, are obtained. In addition, the low-fidelity surrogate model is used to evaluate $\tilde{Y}_L = \tilde{y}_L(X_H)$. These points are used to train a multi-fidelity surrogate model, denoted $\tilde{y}_{MF}$, defined by

$$\tilde{y}_{MF} = \mathcal{H}(X_H, \tilde{Y}_L, Y_H)(x, \tilde{y}_L(x)), \qquad (3)$$

where $\mathcal{H}$ represents the neural network calibrated using the inputs $X_H$, $\tilde{Y}_L$ and the outputs $Y_H$.

For both networks, the Mean Squared Error (MSE) cost function is minimized using the L-BFGS algorithm [18]. In order to choose certain parameters that are difficult to intuït, a hyperparameter tuning phase is carried out. Given the small amount of data in the cases evaluated, the Leave One Out Cross Validation (LOOCV) method is used [19]. The hyperparameter tuning is used to select the activation functions (hyperbolic tangent or logistic function) and the intensity of the regularization.

In order to take into account a possible phase shift between the two fidelity levels, an hyperparameter, $\tau$, is added when $\tilde{Y}_L$ is built, i.e. $\tilde{Y}_L = \tilde{y}_L(X_H - \tau)$, and the
multi-fidelity surrogate model is now defined by

$$ \tilde{y}_{MF} = \mathcal{H}_{(\mathbf{X}_H, \tilde{\mathbf{Y}}_L, \mathbf{Y}_H)}(x, \tilde{y}_L(x - \tau)). \quad (4) $$

The user chooses different values of $\tau$, in practice equidistant between a minimum value and a maximum value of phase shift. Each value is evaluated using the LOOCV method. Finally, the value with the lowest MSE is chosen to make the prediction. Note that this method assumes that there is a single phase shift between the two fidelity levels. In order to take into account more complex phase shift relationships, it is possible to use the embedding theory [13, 20]. Note that this integration of phase shift is compatible with both ANN-based and GP-based multi-fidelity methods. The entire multi-fidelity method was developed in Python using the scikit-learn library [21].

## 2.2 Description of the strategy

In this section the proposed high-fidelity point selection strategy is detailed. This is a key ingredient to minimize the computational cost by adequately selecting the needed high-fidelity points to evaluate. While GP-based and BNN-based multi-fidelity methods both provide uncertainty quantification, which enables point selection strategies, to our knowledge, no equivalent approach has been proposed for ANN-based multi-fidelity methods. An effective and practical solution is then provided in this work for selecting high-fidelity points. Our strategy belongs to the Query-by-Committee adaptive sampling methods [22], which involve comparing the predictions of multiple surrogate models to select the next evaluation point, located where the discrepancy between models is the greatest. In this work, we choose to compare the ANN-based multi-fidelity to a simple piece-wise linear prediction. This choice is mainly motivated by the fact that piecewise linear prediction can handle discontinuities, unlike high-order interpolations.

Firstly, the parameter range, consisting of $x_{min}$ and $x_{max}$, is chosen by the user. The low-fidelity function is then evaluated within this range with relatively high density, i.e. $n_L$ must be large enough to well discretize the function. Since the cost associated with the low-fidelity method is assumed to be low, this step does not require any real optimization. In order to perform the first multi-fidelity estimation, it is necessary to obtain a 3rd high-fidelity point.

We have chosen to base this on the maximum of the second derivative of the low-fidelity function, evaluated using a centered finite-difference scheme. The motivation is to locate the third point in region where the low-fidelity estimation exhibits the strongest variation of behavior. The underlying assumption is that strong variations in the low-fidelity function occur close to strong variations in the high-fidelity function. However, this choice is part of the method’s initialization and is not expected to have a significant influence once convergence is reached. The point $x_H$ is then defined by,

$$ x_H = \arg\max_x \left( \left| \frac{\partial^2 \tilde{y}_L(x)}{\partial^2 x} \right| \right). \quad (5) $$
An initial multi-fidelity prediction $\tilde{y}_{MF}(x)$ is then made using these 3 points. This prediction is compared to a piece-wise linear prediction $\tilde{y}_{lin}$, defined by,

$$ \tilde{y}_{lin}(x) = y_{H,i} + \frac{y_{H,i+1} - y_{H,i}}{x_{H,i+1} - x_{H,i}} \cdot (x - x_{H,i}), \quad \text{for } x_{H,i} \le x \le x_{H,i+1}. \quad (6) $$

The multi-fidelity and the linear prediction are compared in order to compute an interpolation error, denoted $\tilde{\epsilon}$ and defined by,

$$ \tilde{\epsilon}(x) = |\tilde{y}_{MF}(x) - \tilde{y}_{lin}(x)|. \quad (7) $$

The aim of the procedure is then to select the points that adequately describe the multi-fidelity prediction using piecewise linear interpolation. To do this, the next point, $x_H$, is then chosen where this interpolation error is highest,

$$ x_H = \arg \max_x (\tilde{\epsilon}(x)). \quad (8) $$

This operation is repeated until the L2-norm of the interpolation error, $\tilde{\epsilon}_{L2}$, reaches a user defined level, $\epsilon_{target}$. The flowchart of the procedure is represented in Figure 1.

## 3 Validation on analytical case

The proposed procedure is first tested on an analytical test case, based on the "Forrester" function including a jump [12] and in which a phase shift $\tau = 0.1$ has been added. The functions $y_H$ and $y_L$ are defined by,

$$ y_L(x) = \begin{cases} 0.5(6x - 2)^2 \sin(12x - 4) + 10(x - 0.5) - 5, & x \le 0.5, \\ 3 + 0.5(6x - 2)^2 \sin(12x - 4) + 10(x - 0.5) - 5, & x > 0.5, \end{cases} \quad (9) $$

$$ y_H(x) = \begin{cases} 2f_L(x - \tau) - 20(x - \tau) + 20, & x - \tau \le 0.5, \\ 4 + 2f_L(x - \tau) - 20(x - \tau) + 20, & x - \tau > 0.5. \end{cases} \quad (10) $$

The low-fidelity function is evaluated on 26 points, including a high density close to the discontinuity. Concerning the multi-fidelity prediction, the network is composed of 2 layers of 30 neurons for all the procedure. The training is carried out for 11 values of $\tau$ ranging from 0 to 0.2 evaluated with LOOCV.

Figure 2 illustrates the first 6 iterations of the procedure. For each iteration, the high-fidelity function, the low-fidelity function, the multi-fidelity prediction and the piece-wise linear prediction are shown. The interpolation error $\tilde{\epsilon}$ is also shown. This interpolation error is used to select the next high-fidelity point, which can be identified by the vertical line on the figures. Figure 2a shows the first iteration of the procedure. The third point, determined by Eq. (5), is $x_H = 0.5$. The shift is over-predicted for this first iteration. Then, the interpolation error $\tilde{\epsilon}$ is maximum at $x = 0.71$, which is the fourth point chosen for high-fidelity estimation (Fig. 2b). The procedure is repeated and adds points close to the discontinuity, which makes it possible to predict the shift. It can be seen from the last iteration in Fig. 2f that the strategy leads to a high
<figure><img src="image_2.png" /><figcaption>Fig. 1: Flowchart of proposed multi-fidelity approach.</figcaption></figure>

density close to the discontinuity and a lower density elsewhere, which is the expected behavior.

In this analytic case, we can define the error between multi-fidelity prediction and the true function by $\epsilon = \tilde{y}_{MF} - y_H$. The $L_2$-norm and the $L_\infty$-norm of this error can be evaluated by,

$$|\epsilon|_2 = \sqrt{\frac{1}{n} \sum_{i}^{n} (\tilde{y}_{MF,i} - y_{H,i})^2}, \quad |\epsilon|_{\infty} = \max_i |\tilde{y}_{MF,i} - y_{H,i}|, \quad (11)$$

Where $n$ represents the number of evenly spaced points chosen for evaluation, which is set to 101 in this case. The $L_2$-norm and the $L_\infty$-norm of this error are illustrated in function of the number of points in Fig. 3. In this figure, the proposed strategy is compared to two others approaches. First, it is compared with the classical GP-based multi-fidelity method widely used in the literature. In this strategy, the next point
<figure><img src="image_3.png" /><figcaption>(a) Iteration 1 (3 points).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(b) Iteration 2 (4 points).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(c) Iteration 3 (5 points).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(d) Iteration 4 (6 points).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(e) Iteration 5 (7 points).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(f) Iteration 6 (8 points).</figcaption></figure>

**Fig. 2:** Iterations of the procedure apply to the discontinuity-shifted test case, Eq. (9) and (10). At the top of each subfigure : The *dashed red* line corresponds to the true low-fidelity function $y_L$. The *dashed blue* line corresponds to the true high-fidelity function $y_H$. The *blue* symbols correspond to high-fidelity data $Y_H$. The *dotted black* line corresponds to the piece-wise linear interpolation $\tilde{y}_{lin}$. The *black* line corresponds to multi-fidelity prediction $\tilde{y}_{MF}$. At the bottom : interpolation error function $\tilde{\epsilon}$.
<figure><img src="image_9.png" /><figcaption>**Fig. 3:** Comparison of error convergence as a function of the number of points for the discontinuity-shifted test case, Eqs. (9) and (10). The solid lines represent the $L_2$-norm, while the dotted lines represent the $L_\infty$-norm. The *blue* lines correspond to the proposed strategy using ANN, the *red* lines correspond to the classical GP strategy, and the *black* lines correspond to uniformly distributed points using ANN.</figcaption></figure>

is selected where the uncertainty provided by the GP is the highest [10, 11, 23]. In this work, GP-based multi-fidelity method used is the one detailed in [17]. It is also compared with a naive approach with ANN-based multi-fidelity method where high-fidelity points are distributed uniformly. As shown on Fig. 3, the proposed strategy results in lower errors compared to the two other methods. This is true for the $L_2$-norm of the error, but especially for the $L_\infty$-norm of the error, which drops drastically after five points. This sharp decrease is not observed with the two other methods, where the $L_\infty$-norm of the error remains relatively high. Figure 4 illustrates the predictions obtained with the two other methods using 8 points. For the GP-based strategy, Figure 4a highlights the difficulty of these approaches in capturing discontinuities [12]. Specifically, the low-fidelity prediction fails to capture the discontinuity accurately. As a result, the multi-fidelity prediction exhibits significant discrepancies near the discontinuity, which explains why the $L_\infty$-norm remains high despite the increased number of points. Furthermore, this strategy tends to distribute points uniformly without refining the point density near the discontinuity. This behavior is markedly different from that observed with the proposed strategy (Fig. 2f). The ANN-based multi-fidelity prediction with uniform distribution of 8 points is shown in Figure 4b. The use of neural networks enables the low-fidelity prediction to adequately capture the discontinuity. However, although the discontinuity is also reflected in the multi-fidelity prediction, a slight misalignment with the high-fidelity function remains. The absence of points near the discontinuity prevents fine capture of the phase shift between the low- and high-fidelity functions. This misalignment explains why the $L_\infty$-norm of the error remains high.

The analysis of this analytical case highlights the relevance of the proposed strategy for handling signals featuring both a discontinuity and a phase shift between the two fidelity levels. In the next section, the strategy is applied with real data coming from RANS and LES results of a classic aerodynamic case.
<figure><img src="image_10.png" /><figcaption>**Fig. 4:** Multi-fidelity prediction with 8 points for classical strategies applied to the discontinuity-shifted test case, Eqs. (9) and (10). The red dots represent the low-fidelity data $\mathbf{Y}_L$, and the red line represents the low-fidelity prediction $\tilde{y}_L$. The blue dots represent the high-fidelity data $\mathbf{Y}_H$, and the dashed blue line represents the high-fidelity function $y_H$. The black line represents the multi-fidelity prediction $\tilde{y}_{MF}$.</figcaption></figure>

# 4 Application on turbulent flow

## 4.1 Numerical method

Simulations are performed with YALES2 incompressible fractional-step solver [24] using finite-volume formulation and precise fourth order space centered numerical schemes. The YALES2 solver is able to deal with unstructured meshes composed of tetrahedral elements allowing to perform direct, large eddy or statistical simulations (DNS, LES or RANS) of complex geometries. For LES, a time advancement using a fourth-order modified Runge-Kutta scheme [25] is used, whereas a linearized implicit time integration [26] is used for both momentum and turbulent quantities for RANS simulations. In this work, the Spalart-Allmaras model is used [27] for statistical simulations, whereas LES are performed using the $\sigma$-model [28]. Both RANS and LES simulations of NACA 0012 airfoil are simulated in this work. Concerning the RANS configuration, simulation is done on a 2D wide rectangular domain of size $25c \times 20c$, where c is the chord length of the airfoil. Concerning the LES configuration, simulation is done on a 3D wide rectangular domain of size $30c \times 30c \times c$. The Reynolds number of the configuration is defined by $Re = \frac{Uc}{\nu}$ with U the inlet velocity and $\nu$ the kinematic viscosity. A constant and homogeneous velocity U with an angle of attack of $\alpha$ is imposed as inlet condition, whereas a convective outflow boundary condition is used as outlet. Periodic conditions are used on others boundaries. The mesh used for each case is generated by an automatic mesh convergence strategy [29–31].

In this work, the aim of the proposed strategy is to provide an accurate description of the mean lift coefficient $C_L$ as a function of angle of attack $\alpha$, with a minimum of high-fidelity sampling data, to limit the computational cost. For an airfoil subjected to flow with incidence, two distinct flow regimes can be observed: attached flow, typical at relatively low angles of attack, and separated flow, characteristic of higher angles of attack. Figure 5 presents the numerical simulation results for these two regimes, at angles of attack of 8° and 16°, using two turbulence modeling approaches
<figure><img src="image_11.png" /><figcaption>**Fig. 5:** Illustration of numerical computation around NACA0012 airfoil at two angles of attack. Left: $\alpha = 8^{\circ}$, right: $\alpha = 16^{\circ}$. Top: Q-criterion and instantaneous velocity magnitude of LES, second line: instantaneous vorticity magnitude of LES, third line: mean vorticity magnitude of LES, bottom: mean vorticity magnitude of RANS.</figcaption></figure>

(LES and RANS). In the upper part of the figure, the Q-criterion highlights the resolved three-dimensional structures obtained with LES approach. The magnitude of the instantaneous velocity is also shown on a 2D plane, clearly illustrating the large recirculation zone present at $\alpha = 16^{\circ}$. Figure 5 also shows a 2D view of the instantaneous and the mean vorticity from LES simulations for both angles of attack. These results are compared to the mean vorticity obtained from RANS simulations on a 2D
mesh. Although RANS methods do not capture fine details of the instantaneous turbulent structures, the mean vorticity topology is close to the one obtained with LES. Specifically, a highly separated flow is observed at $\alpha = 16^{\circ}$, while the high magnitude of mean vorticity remains close to the airfoil at $\alpha = 8^{\circ}$. Finally, note that the CPU cost of a LES computation is around 10,000 times the CPU cost of a RANS computation. This very high ratio is due to the possibility to perform 2D computation with RANS modeling, but also because the convergence of temporal statistic is very time-consuming in LES computation.

## 4.2 RANS with various Reynolds number values

The ultimate goal is to address a case combining RANS and LES. However, the computational cost of LES does not allow for a detailed description of the high-fidelity curve required to assess the method. In order to apply and validate the strategy using data from aerodynamic calculations, a multi-fidelity case is emulated based only on RANS modeling. For this purpose, two Reynolds numbers are evaluated. The low-fidelity data correspond to $Re = 50,000$, while the high-fidelity data correspond to $Re = 100,000$. Given the relatively low computational cost of the RANS method, it is possible to generate both low- and high-fidelity data across the entire range of evaluated data. The mean lift coefficient was thus evaluated for $n_{\alpha} = 30$ angles of attack, ranging from 5° to 20°. Knowledge of these high-fidelity points enables the calculation of the $L_2$-norm and the $L_{\infty}$-norm of the error as,

$$|\epsilon|_2 = \sqrt{\frac{1}{n_{\alpha}} \sum_{i}^{n_{\alpha}} (\tilde{y}_{MF,i} - y_{H,i})^2}, \quad |\epsilon|_{\infty} = \max_i |\tilde{y}_{MF,i} - y_{H,i}|. \quad (12)$$

Figure 6 illustrates the mean lift coefficient evaluated for both levels of fidelity, i.e. for both Reynolds numbers. For both Reynolds number, the dependence of the mean lift coefficient with the angle of attack is similar. For low angles of attack, the $C_L$ is almost a linear function of the angle of attack. Then, the $C_L$ reaches a maximum before falling sharply. This is known as the stall phenomenon. In this context of highly separated flows (see right on Fig. 5), the RANS methods are known to be inaccurate [1, 32]. Finally, at the highest angles, the $C_L$ is nearly constant. The variation in Reynolds number causes a shift in the stall angle and in the maximum $C_L$ reached.

Figure 7 illustrates the proposed high-fidelity point selection strategy applied to this case. The hyperparameter $\tau$ is chosen between 40 values from -4 to 4. The Figure 7 shows the first six iterations of the selection point strategy. The multi-fidelity prediction method demonstrates rapid convergence towards the reference data, confirming its efficiency. As in the analytical case, the proposed strategy is compared to the two reference strategies (GP and naive ANN) through the evolution of the $L_2$ and $L_{\infty}$ norms of the error as a function of the number of points, as illustrated in Figure 8. Although the errors are comparable between the three strategies for small numbers of points, the proposed strategy shows a drastic drop in the $L_{\infty}$-norm for 6 points and
<figure><img src="image_12.png" /><figcaption>**Fig. 6:** Lift coefficient in function of angle of attack for two Reynolds number evaluated in RANS. the *red* line corresponds to the low fidelity (*Re* = 50000), the *blue* line corresponds to high fidelity (*Re* = 100000).</figcaption></figure>

beyond. In this case, the GP-based strategy performs better than the uniform distribution strategy. However, this method is unable to refine the point distribution near the discontinuity, which causes the error to plateau.

In addition, a sensitivity study of multi-fidelity predictions to the size of the neural network is performed on this case. Three different network sizes were evaluated. The first network is composed of 3 layers of 10 neurons, noted (10, 10, 10), the second is composed of 3 layers of 20 neurons, noted (20, 20, 20), and the last one is composed of 4 layers of 20 neurons, noted (20, 20, 20, 20). Figure 9 shows the evolution of the $L_2$-norm and the $L_\infty$-norm of the error (see Eq. 12) during the procedure for different network sizes. Although a real disparity exists for a few points, where the convergence is not monotonic, all the curves have a similar behavior and end up with the same level of error. The proposed strategy is therefore robust to the choice of network size at convergence.

## 4.3 Multi-fidelity strategy based on RANS and LES data

The strategy is now applied to a real conditions where the low fidelity corresponds to the RANS and the high fidelity corresponds to the LES for a Reynolds number of 50,000. The evolution of the lift coefficient as a function of the angle of attack using the LES is too expensive to be fully described as was done in the previous section for the RANS. The strategy is therefore applied here blindly, with no a priori knowledge of the high-fidelity curve. As in the previous section, the $\tau$ hyperparameter expressing the shift between the two fidelity levels is chosen with the LOOCV method from 40 values ranging from -4 to 4 degrees. The network is composed of 3 layers of 12 neurons.

Figure 10 illustrates the first 3 iterations of the procedure. The error estimate drop sharply during these 3 iterations. This enables the description of the main characteristics of the relationship between the mean lift coefficient and the angle of attack. In fact, the stall zone, characteristic of this type of configuration, is clearly identified with only 5 high fidelity calculations. Other arbitrarily chosen points were simulated as test points in order to assess the quality of the final prediction. The comparison in shown on Fig. 11. The predicted multi-fidelity curve stay close to the test points,
<figure><img src="image_13.png" /><figcaption>(a) Iteration 1 (3 points).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(b) Iteration 2 (4 points).</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(c) Iteration 3 (5 points).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(d) Iteration 4 (6 points).</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(e) Iteration 5 (7 points).</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(f) Iteration 6 (8 points).</figcaption></figure>

**Fig. 7:** Iterations of the procedure apply to the RANS case with 2 Reynolds numbers. At the top of each subfigure : The dashed red line corresponds to the true low-fidelity function $y_L$. The dashed blue line corresponds to the true high-fidelity function $y_H$. The blue symbols correspond to high-fidelity data $Y_H$. The dotted black line corresponds to the piece-wise linear interpolation $\tilde{y}_{lin}$. The black line corresponds to multi-fidelity prediction $\tilde{y}_{MF}$. At the bottom : interpolation error function $\tilde{c}$.
<figure><img src="image_19.png" /><figcaption>**Fig. 8:** Comparison of error convergence as a function of the number of points for the RANS case with 2 Reynolds numbers. The solid lines represent the $L_2$-norm, while the dotted lines represent the $L_\infty$-norm. The *blue* lines correspond to the proposed strategy using ANN, the *red* lines correspond to the classical GP strategy, and the *black* lines correspond to uniformly distributed points using ANN.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 9:** Influence study of the size of the network for the RANS case with 2 Reynolds numbers.</figcaption></figure>

which have not been used during the learning process. This indicates the quality of the multi-fidelity prediction.

To demonstrate the crucial contribution of the adaptive strategy, we performed a multi-fidelity prediction using five equally spaced points. This prediction, presented in Appendix A, shows that five points are not sufficient to accurately determine the position of the discontinuity.

It is important to note that the multi-fidelity prediction does not necessarily pass through all the high-fidelity training points. This can be due to the small size of the network. In this work, the choice was made to keep the same network size during
<figure><img src="image_21.png" /><figcaption>**Fig. 10:** Iterations of the procedure apply to the LES case. At the top of each subfigure : The dashed red line corresponds to the low-fidelity interpolation $\tilde{y}_L$ (RANS). The blue symbols correspond to high-fidelity data $\mathbf{Y}_H$ (LES). The dotted black line corresponds to the linear interpolation $\tilde{y}_{\text{lin}}$. The black line corresponds to multi-fidelity prediction $\tilde{y}_{\text{MF}}$. At the bottom : interpolation error function $\tilde{\epsilon}$.</figcaption></figure>

the high-fidelity selection point strategy, but in order to improve prediction, the network size could be automatically adapted to the number of point. This may also be due to the uncertainties associated with the convergence of the statistics from the LES calculations. In order to take this component into account, the incorporation of uncertainties into the multi-fidelity model [10] could be considered as a perspective for this work. Nevertheless, this application illustrates the potential of multi-fidelity approaches for fluid mechanics. Indeed, despite the fact that the RANS approach overestimates the stall angle in this configuration, it holds relevant information that multi-fidelity methods can extract in order to target the points of interest to be evaluated in LES.
<figure><img src="image_22.png" /><figcaption>**Fig. 11:** Lift coefficient in function of angle of attack. The red symbols correspond to low fidelity data $\mathbf{Y}_L$ (RANS). The black line corresponds to multi-fidelity prediction $\tilde{y}_{MF}$. The blue symbols correspond to high-fidelity $\mathbf{Y}_H$ (LES) data used for the multi-fidelity prediction. The orange symbols correspond to high-fidelity (LES) data for validation. The black crosses correspond to DNS data [33].</figcaption></figure>

# 5 Conclusion

The objective of this work was to develop an efficient method to accurately describe a quantity of interest in turbulent flow simulations, while minimizing the number of costly LES evaluations. To achieve this, an automated high-fidelity points selection strategy was embedded in a multi-fidelity method that combines limited but precise data from a high-fidelity method with abundant but less accurate data from a low-fidelity method. The chosen multi-fidelity method relies on neural networks, which are particularly suitable for capturing complex relationships between different fidelity levels.

The proposed strategy was first evaluated on an analytical case featuring both a discontinuity and a phase shift between the two fidelity levels. This approach enables an accurate and cost-efficient description of the high-fidelity function. The results were compared to a conventional strategy based on the uncertainty associated with GP-based multi-fidelity method. The inability of GP methods to capture the discontinuity leads to significantly higher errors compared to the proposed strategy. Additionally, ANN-based multi-fidelity predictions with a naive, uniform distribution of high-fidelity points were also analyzed. Despite the inherent ability of neural networks to capture discontinuities, the absence of sampling points near the discontinuity prevented an accurate representation of the phase shift. The proposed strategy therefore appears appropriate for dealing with cases involving discontinuity and phase shift.

The strategy was also applied to aerodynamic data from RANS computations, using the Reynolds number as the fidelity parameter. This case was specifically designed to perform multi-fidelity analysis with data from numerical simulations while allowing precise evaluation of the high-fidelity curve by keeping low enough computational costs. As in the analytical case, the proposed strategy gives better results than the two usual ones. It has been shown that the point selection strategy is sensitive
to the size of the neural network. However, despite initial variations in point selection, the strategy consistently converges toward an accurate description of the target function. One of the perspectives of this work is therefore to find a way of choosing a relevant network size, which could potentially evolve during the procedure.

Finally, the strategy was tested in real conditions using RANS data for low fidelity and costly LES data for high fidelity. Promising results were obtained insofar as after only 3 iterations (i.e. 5 high-fidelity points evaluated), the stall zone is accurately captured. Then, although statistical RANS methods are known to be imprecise in the context of highly separated flows, they still provide valuable information on how to take full advantage of LES results while minimizing computational costs. This approach thus eliminates the need for prior knowledge of the quantity of interest when selecting points for LES evaluation, optimizing the overall process.

**Funding information.** The authors gratefully acknowledge support from NETHUNS project under grant ANR-21-CHIN-0001-01.

**Acknowledgements.** This work was granted access to the HPC resources of CINES/TGCC/IDRIS under the project 2A00611 made by GENCI. Part of this work has been initiated during the Extreme CFD Workshop & Hackathon (https://ecfd.coria-cfd.fr).

## A Comparison with a multi-fidelity prediction based on equally spaced high fidelity points

To illustrate how the proposed strategy allows to correct the RANS prediction by using only few LES predictions (see Section 4.3) in comparison with a more naive multi-fidelity strategy, a multi-fidelity strategy only based on equally spaced point selection is performed. To avoid recomputing costly LES simulations, we chose the following 5 points: $x_{HF} = 5°, 8°, 12.95°, 16°$ and $20°$, which are approximately equally spaced. Figure 12 shows the multi-fidelity prediction obtained with this naive strategy. This strategy fails to capture the correct phase shift, leading to significant errors at the evaluation points $x = 9.5°, 10.25°$, and $10.75°$.

Then, to achieve a level of accuracy comparable to that obtained in Figure 11, additional high-fidelity points would be required, which would increase the computational cost. Note that to maintain equally spaced points, it implies to recompute 4 new high-fidelity points ($x = 6.87°, 10.62°, 14.37°$, and $18.25°$), and not only one more, conversely to the proposed strategy. This would double the computational cost without guaranteeing a better result than the one obtained with the adaptive point selection strategy.

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<figure><img src="image_23.png" /><figcaption>**Fig. 12:** Lift coefficient in function of angle of attack. The red symbols correspond to low fidelity data $\mathbf{Y}_L$ (RANS). The black line corresponds to multi-fidelity prediction $\tilde{y}_{MF}$. The blue symbols correspond to high-fidelity $\mathbf{Y}_H$ (LES) data used for the multi-fidelity prediction. The orange symbols correspond to high-fidelity (LES) data for validation. The black crosses correspond to DNS data [33].</figcaption></figure>

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