Francesco Romanò

Jean-Christophe Loiseau

Antoine Dazin

M A Elhawary

Machine learning for optimal flow control in an axial compressor

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Introduction

Axial compressors are standard components in numerous industrial applications such as power generation or aircraft engines. One long-term goal is to improve their performances while maintaining safe and reliable operating conditions. Rotating stall [START_REF] Tan | Spike-type compressor stall inception, detection, and control[END_REF] is a key problem hindering the stability and the progress of modern axial compressors. When it occurs, flow instabilities are generated, leading to degraded performances. In the worst-case scenario, rotating stall can eventually cause machine failure. Aiming at delaying the onset of this phenomenon, a lot of efforts have thus been devoted to designing and implementing effective flow control strategies, including both passive [START_REF] Hathaway | Passive endwall treatments for enhancing stability[END_REF] and active [START_REF] Margalida | Active flow control in an axial compressor for stability improvement: on the effect of flow control on stall inception[END_REF] methods. In particular, active flow control methods have the added advantage of being switched on only when control is needed, thus leading to an overall reduced energy consumption. Previous studies [START_REF] Margalida | Active flow control in an axial compressor for stability improvement: on the effect of flow control on stall inception[END_REF][START_REF] Moubogha Moubogha | Stall margin improvement in an axial compressor by continuous and pulsed tip injection[END_REF] T.I. : Quantitative AI in Complex Fluids and Complex Flows: Challenges and Benchmarks. Guest editors: Luca Biferale, Michele Buzzicotti, Massimo Cencini a e-mail: mohamed.elhawary@ensam.eu b e-mail: francesco.romano@ensam.eu c e-mail: jean-christophe.loiseau@ensam.eu d e-mail: antoine.dazin@ensam.eu (corresponding author) have demonstrated the effectiveness of air jets in postponing the onset of rotating stall in axial compressors. Such a control strategy relies on the injection of highmomentum fluid near the tip leading edge of the rotor blades by a discrete set of air jets positioned circumferentially upstream of the compressor. Despite the effectiveness of air jets for active flow control, achieving efficient and reliable control strategies remains challenging. One reason for that is the lack of efficient and accurate predictive models for control design.

A standard approach is thus to use surrogate models. Built on data, these models can approximate efficiently the relationship between the input parameters and the output of a given process. Examples of surrogate models include the response surface methodology [START_REF] Jang | Optimization of a stator blade using response surface method in a single-stage transonic axial compressor[END_REF], Kriging (also known as Gaussian process regression) [START_REF] Simpson | Kriging models for global approximation in simulationbased multidisciplinary design optimization[END_REF], or artificial neural networks [START_REF] Mengistu | Aerodynamic optimization of turbomachinery blades using evolutionary methods and ann-based surrogate models[END_REF]. Such surrogate models can be used to tackle local optimization using gradient-based techniques, or global optimization with the Nelder-Mead's simplex method [START_REF] Nelder | A simplex method for function minimization[END_REF] or genetic algorithms [START_REF] Bäck | An overview of evolutionary algorithms for parameter optimization[END_REF]. In turbomachinery applications, surrogate model applications include aerodynamic design optimization [START_REF] Mengistu | Aerodynamic optimization of turbomachinery blades using evolutionary methods and ann-based surrogate models[END_REF], flow modeling [START_REF] Li | Machine learning in aerodynamic shape optimization[END_REF], performance predictions [START_REF] Liu | Gas turbine performance prediction via machine learning[END_REF] and flow control [START_REF] Yu | Kriging surrogate model applied in the mechanism study of tip leakage flow control in turbine cascade by multiple dbd plasma actuators[END_REF], among others [START_REF] Li | Review of design optimization methods for turbomachinery aerodynamics[END_REF].

For flow control, data-driven techniques have recently gained a lot of attention [START_REF] Brunton | Closed-loop turbulence control: progress and challenges[END_REF][START_REF] Ren | Active flow control using machine learning: a brief review[END_REF]. Two prominent avenues are evolutionary algorithms [START_REF] Bäck | An overview of evolutionary algorithms for parameter optimization[END_REF][START_REF] Debien | Closedloop separation control over a sharp edge ramp using genetic programming[END_REF][START_REF] Li | Drag reduction of a car model by linear genetic program-ming control[END_REF][START_REF] Wu | Jet mixing enhancement using machine learning control[END_REF] and deep reinforcement learning (DRL) [START_REF] Rabault | Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control[END_REF][START_REF] Degrave | Magnetic control of tokamak plasmas through deep reinforcement learning[END_REF][START_REF] Garnier | A review on Deep Reinforcement Learning for Fluid Mechanics[END_REF][START_REF] Paris | Robust flow control and optimal sensor placement using deep reinforcement learning[END_REF][START_REF] Tang | Robust active flow control over a range of Reynolds numbers using an artificial neural network trained through deep reinforcement learning[END_REF][START_REF] Elhawary | Deep Reinforcement Learning for Active Flow Control around a Circular Cylinder Using Unsteady-mode Plasma Actuators[END_REF]. Deep reinforcement learning relies on deep neural networks to approximate the optimal state-action combination from repeated queries of the experimental or numerical environment. Genetic algorithms on the other hand borrow ideas from natural selection and genetic principles to evolve a population of possible control strategies. Such evolutionary algorithms have been used experimentally by [START_REF] Benard | Turbulent separated shear flow control by surface plasma actuator: Experimental optimization by genetic algorithm approach[END_REF] to optimize the control parameters of DBD actuators to control the flow separation downstream of a backward-facing step. Likewise, genetic algorithms have been coupled to artificial neural networks to optimize the parameters of a suction/blowing strategy to control the separation over SD7003 airfoil by [START_REF] Kamari | Optimization of sd7003 airfoil performance using tbl and cbl at low reynolds numbers[END_REF]. Intending for machine learning methods that automatically learn from data and improve over experience without being explicitly programmed, from the authors' point of view, GAs do not belong to them because GAs do not automatically learn, but they rather sort out, combine and compare sub-optimal solutions. On the other hand, DRL falls well within the category of ML methods. For an extensive overview, interested readers are referred to [START_REF] Ren | Active flow control using machine learning: a brief review[END_REF][START_REF] Brunton | Machine learning for fluid mechanics[END_REF] or the book on Machine Learning Control [START_REF] Duriez | Machine Learning Control -Taming Nonlinear Dynamics And Turbulence[END_REF].

To the best of our knowledge, limited references exist in the literature highlighting the potential of such control design strategies to mitigate rotating stall in axial compressors. This is the problem we tackle in this work. Given an experimental dataset [START_REF] Moubogha Moubogha | Stall margin improvement in an axial compressor by continuous and pulsed tip injection[END_REF], we train shallow neural networks to capture the relationship between our control parameters and two performance indices, i.e., the power balance and the surge margin improvement. The control parameters include the rotor rotational velocity and three air jet-related parameters: the absolute injection angle, the number of injectors pairs and the absolute injection velocity. First, the control parameters are optimized to maximize each of the performance indices independently. Then, a bi-objective optimization problem is formulated to explore the trade-off between these competing objectives. In both cases, we use shallow neural networks to approximate the cost function and genetic algorithms to solve the optimization problem. The paper is organized as follows: Sect. 2 presents a brief overview of the CME2 axial compressor, along with some definitions and an overview of experimental dataset used in the rest of this work. Then, details about our surrogate modeling and optimization procedures are given in Sect. 3. Finally, the key results are discussed in Sect. 4, while Sect. 5 provides the conclusion and some perspectives to this work.

Experimental flow configuration and dataset

Experimental configuration and definitions

The experimental setup considered in this work is the CME2 workbench, a single-stage low-pressure axial compressor with 30 and 40 blades for the rotor and sta- tor, respectively. The main characteristics of this experimental platform can be found in [START_REF] Veglio | Pressure measurements in an axial compressor: from design operating conditions to rotating stall inception[END_REF]. The actuators consist in 20 equally spaced air jets pairs, each with two injectors. These injectors are mounted circumferentially at 10 mm upstream of the leading edge of the rotor blades. Each injector pair can be supplied by an external power source [START_REF] Margalida | Active flow control in an axial compressor for stability improvement: on the effect of flow control on stall inception[END_REF], or from the high-pressure region of the compressor [START_REF] Raub | Ejector tip injection system for active aerodynamic compressor stabilization part ii[END_REF]. The control process can use either 10 or 20 pairs of injectors. By using 10 such pairs, the injectors can generate approximately the same injection velocity as when all 20 pairs are used, albeit at half the total injection flow rate. These actuators leverage the Coandǎ effect by injecting momentum near the tip clearance of the rotor blades as illustrated in Fig. 1.

Previous studies [START_REF] Margalida | Active flow control in an axial compressor for stability improvement: on the effect of flow control on stall inception[END_REF][START_REF] Moubogha Moubogha | Stall margin improvement in an axial compressor by continuous and pulsed tip injection[END_REF] have demonstrated the effectiveness of air jets to postpone the onset of rotating stall in axial compressors. Following [START_REF] Moubogha Moubogha | Stall margin improvement in an axial compressor by continuous and pulsed tip injection[END_REF], we focus our attention on three control parameters: the number of injectors pairs N inj , the injection angle α inj and the injection velocity V inj . Three operating conditions, characterized by the rotational velocity Ω, are moreover considered: Ω = 3200 RPM, 4500 RPM and 6000 RPM. The range of these different parameters is summarized in Table 1.

In turbomachinery applications, the effectiveness of a control strategy is often evaluated based on two performance indices: the surge margin improvement and the power balance. The surge margin improvement, hereafter denoted as SM I tot , is defined as

SM I tot = SM c -SM b SM b × 100, (1)
where SM b and SM c are the surge margins of the baseline and controlled configurations, respectively. These are given by

SM b = p tt,b Q s,b × Q nom p nom -1 × 100,
and

SM c = p tt,c Q s,c × Q nom p nom -1 × 100.
where Q s,b and p tt,b are the flow rate and total-to-total pressure in the baseline configuration, while Q s,c and p tt,c are the flow rate and total-to-total pressure in the controlled setup. Finally, Q nom and P nom are the nominal flow rate and power of the compressor. For a rotational velocity Ω = 3200RP M , the nominal power and flow rate are P nom = 2300W and Q nom = 5.3 kg/s, respectively. This performance index characterizes the effectiveness of the control strategy in terms of the margins in the flow rate and the total pressure ratio between the nominal and the critical operating points.

The second performance index considered is the power balance, hereafter denoted as P B. It is defined as

P B = (Q s,b + Q inj ) × p tt,c ρ c - Q s,b × p tt,b ρ b × 100 P nom , (2)
where ρ denotes the density. The subscript s, c, b, inj and nom refers to the values of these different variables for the last stable point, the controlled cased, the baseline case, injection and nominal conditions, respectively. Compared to the surge margin improvement, the power balance performance index evaluates the cost of the control process by comparing the output power gain by the compressor and the input power paid to activate the actuators. Both of these performance indices are schematically presented in Fig. 2. In Fig. 2a, two surge lines are plotted in red for the baseline case and the controlled case. The baseline configuration can be characterized by the stall pressure p s,b and the stall mass flow rate Q s,b at the last stable point LSP b . Likewise, the controlled configuration is characterized by p s,c and Q s,c at the last stable point LSP c . It should be emphasized finally that, compared to [START_REF] Moubogha Moubogha | Stall margin improvement in an axial compressor by continuous and pulsed tip injection[END_REF], the pressure rise p tt across the compressor inlet for the controlled case is computed based on the mass average value in the jet and the mean flow.

Experimental dataset

In this study, we use data from 175 experiments to train and test the two neural network models. The experiments are conducted at LMFL laboratory by [START_REF] Moubogha Moubogha | Stall margin improvement in an axial compressor by continuous and pulsed tip injection[END_REF] in the framework of ACONIT project [START_REF] Dazin | The aconit project: an innovative design approach of active flow control for surge prevention in gas turbines[END_REF]. The specifications of the experimental parameters are summarized in Table 1. The dataset was split into 96% for training and 4% for testing, but five experiments are excluded due to missing values of the power balance P B. Additionally, five extra experiments were performed, outside the range of the training and testing dataset, to test the extrapolation capabilities of the two neural network models.

Surrogate model and optimization problem

We present in Sect. 3.1 a brief overview of the architecture and training procedure of the shallow neural networks used as surrogate models to approximate the input-output relationship between the control parameters and the two performance indices defined in the previous section. Then, Sect. 3.2 describes the formulation of the single-and multi-objective optimization problems and their resolution using genetic algorithms.

Shallow neural networks

Given the experimental dataset, we approximate the relationship between the control parameters (i.e., N inj , α inj , V inj and Ω) and the two performance indices with a surrogate model built using a shallow feedforward neural network. Each of these two relationships is approximated using a different network, albeit with the same architecture. A schematic representation is shown in Fig. 3. The architecture is as follows:

• Each network takes as input a vector x ∈ R 4 defined as

x = [α inj V inj N inj Ω] .
• The first hidden layer has 64 artificial neurons performing the following operations

z (1) i = σ ⎛ ⎝ b (1) i + 4 j=1 W (1) ij x j ⎞ ⎠ for i = 1, • • • , 64,
where z

(1) i denotes the i th output of this layer, and b (1) ∈ R 64 (the bias vector) and W (1) ∈ R 64×4 are learnable parameters. The activation function σ : R → R is the ReLU defined as σ(x) = max(0, x).

• The second hidden layer, with 32 neurons, acts in a similar way on z (1) , i.e., z

(2) i = σ ⎛ ⎝ b (2) i + 64 j=1 W (2) ij z (1) j ⎞ ⎠ .) for i = 1, • • • , 32.
Once again, the activation function σ is set to the ReLU function. • Finally, the output layer takes as input z (2) and maps it to the desired performance index. This map- Denoting by y ∈ R the ground truth value of either performance index, the learnable parameters of each layer are obtained by minimizing the following mean square error loss function L W (1) , b (1) , W (2) , b (2) ,

W (o) , b (o) = 1 M M i=1 (y i -ŷi ) 2 , ( 3
)
where M = 168 is the total number of data points in the training set. Both of the networks are implemented using Keras with the TensorFlow back end, while the ADAM optimizer [START_REF] Kingma | A Method for Stochastic Optimization[END_REF] is used for the minimization of the cost function. The different parameters of the networks and optimizer are summarized in Table 2.

Optimization problem and genetic algorithm

The aim of this study is to find the set of control parameters x ∈ R 3 maximizing either of the performance indices defined in Sect. 2.1, or a combination thereof. Letting J 1 : R 3 → R be the surrogate model approximating the surge margin improvement SM I tot and J 2 : R 3 → R be the corresponding model for the power balance P B, we can thus formulate the following multi-objective unconstrained optimization problem maximize

(1 -λ) J 1 (x) + λJ 2 (x), (4)
where λ ∈ [0, 1] is a trade-off parameter. Note that setting λ = 0 or λ = 1 reduces Eq. 4 to a singleobjective optimization problem. The maximizer x(λ) of Eq. 4 thus forms a parameterized family of optimal solutions forming a Pareto front. It cannot be guaranteed in general that the entries of the solution x will be in a given range when solving an unconstrained optimization problem. As such, we also consider the following constrained variation

maximize (1 -λ) J 1 (x) + λJ 2 (x) subject to -45 • ≤ α inj ≤ 30 • 46m/s ≤ V inj ≤ 202m/s N inj = 10 (or 20).
(

) 5
The benefit of using the constrained formulation of the optimization problem is that practitioners can force the solution (if it exists) to be in a given range of safe operating conditions. On the other hand, using an unconstrained formulation allows for the possibility of finding a global optimal solution outside of the range of the experimental data. It should be noted that our control system only supports N inj values of 10 and 20 due to design constraints. Therefore, N inj is set to either 10 or 20 in both constrained and unconstrained optimization processes. Additionally, compressors are typically designed to operate at a constant rotational velocity Ω; thus, the optimal control parameters are explored at Ω = 3200, 4500 and 6000 RPM. The range of all experimental data is summarized in table creftab: set of configurations.

A genetic algorithm (GA) is employed to solve both problems Eqs. 4 and 5. The main search operators in GAs are crossover and mutation. These two operators can function as an exploration or exploitation operators [START_REF] Lim | Crossover and mutation operators of genetic algorithms[END_REF]. In our genetic algorithm (1), random (uniform) mutation operation with a single-point crossover operator [START_REF] Lim | Crossover and mutation operators of genetic algorithms[END_REF]

Model selection and hyperparameter optimization

In this section, we discuss the methodology of tuning the neural network hyperparameters and the model selection criteria. Given a dataset in the domain D, we split them into training and testing sets D train and D test , respectively. In the process of splitting the original dataset D, we use K-fold cross-validation method [START_REF] Yates | Cross validation for model selection: a review with examples from ecology[END_REF], which divides D into K equally sized samples. The K folds take the following form: D train = {D test for i = {1, . . . , K} where K in the number of folds. Let A = {A (1) , . . . , A (N ) } denote a set of learning algorithms, and let the associated hyperparameters x of each algorithm A (j) having a domain Λ (j) . Finally, let L (A x (j) , D (i) train , D (i) test ) be the loss function of a certain learning algorithm A (j) trained on D (i) train and tested on D (i) test utilizing the hyperparameter x. This allows the model selection and the hyperparameter optimization equation to be written as

minimize 1 K K i=1 L (A x (j) , D (i) train , D (i) test ). ( 6
)
The goal of Eq. 6 is to find the optimal neural network algorithm and hyperparameters, A x * ( †) , that minimize the loss function, where x * denotes the optimal hyperparameter and † refers to the optimal algorithm. Various methods are discussed in the literature [START_REF] Akiba | Optuna: A next-generation hyperparameter optimization framework[END_REF][START_REF] Kotthoff | Auto-weka: Automatic model selection and hyperparameter optimization in weka. Automated machine learning: methods, systems[END_REF][START_REF] Feurer | Hyperparameter optimization. Automated machine learning: Methods, systems, challenges[END_REF] for tackling the optimization problem Eq. ( 6). In our study, we use a genetic algorithm for random search of the neural networks hyperparameters and 25 folds, i.e., K = 25, for our model selection. Furthermore, to simplify the hyperparameter optimization process, we use a neural network architecture with two hidden layers (encompassing 64 and 32 neurons, respectively). ReLU is employed as activation function and Adam as optimizer. Then we employ the GA for optimizing two parameters, namely the learning rate and the weight initialization.

Results

This section summarizes the key results of our study. First, Sect. 4.1 describes the in-sample (training) and out-of-sample (testing and extrapolation) performances of the surrogate models. Then, the results of the optimization task pertaining to finding the optimal control parameters to delay rotating stall are being presented in Sect. 4.2.

Surrogate models

Given the training set, neural network surrogate models approximating the power balance P B and surge margin improvement SM I tot have been trained until the loss function has stabilized. More precisely, the training procedures are performed for 550 epochs until a mean square error less than 10 -3 is achieved. Figure 4a compares the prediction of the surrogate models with the ground truth data for the power balance P B, while Fig. 4b provides the same comparison for the surge margin improvement SM I tot . The quality of this approximation can be quantified using the coefficient of determination R 2 defined as

R 2 = 1 - M i=1 (y i -ŷi ) 2 M i=1 (y i -ȳ) 2 ,
where ȳ denotes the sample mean of the performance index considered. In both cases, the surrogate models achieve a coefficient of determination R 2 ≥ 0.98. The two surrogate models moreover exhibit similar performances when the test set is being considered, as shown in Fig. 5. Furthermore, Fig. 6 compares the predictions of the surrogate models with the actual data for the extrapolation dataset. The surrogate models achieve R 2 = 0.89 and 0.82 for P B and SM I tot , respectively.

To better understand the capabilities and limitations of our model in extrapolating, we study the impact of splitting the dataset D between training D train and testing D test ; hence, we aim to investigate different combinations of D (i) train and D (i) test . We found that the split of the input database between training and testing affects the robustness of the extrapolation prediction. To identify the cause of this sensitivity, we first obtained the optimal neural network architecture by employing a genetic algorithm for hyperparameter optimization as discussed in Sect. 3.3. Then, we trained the optimal neural network architecture on 2000 random data splits of the original database (i.e., different training/testing sets). Out of the analysis of our 2000 random data splits, we identified a narrow set of data that consistently resulted in poor predictions for both extrapolation and testing conditions. In order to confirm that these ill-conditioned testing points unconditionally undermines both testing and extrapolation predictions, we trained another 2000 different neural network architectures by forcing such data to belong to the testing dataset. As a result, none of these 2000 trials yielded a model that gives good predictions in terms of testing and extrapolation sets. In this process, we also utilized the GA for hyperparameter optimization, discussed in Sect. 3.3, but without cross-validation.

To further understand whether the ill-conditioned testing data could be correlated with the sensitivity of the underlying physics, we compute the gradients of the neural network outputs, i.e., P B and SM I tot , with respect to the relevant inputs for the flow control system, i.e., the velocity ratio V R = V inj /U , where U = πDΩ Fig. 6 Comparison between the experimental results (ground truth) of the extrapolation set and the shallow network's predictions for a the power balance P B and b the surge margin improvement SM I tot tive injection angle β inj minus the compressor stagger angle γ st , which is fixed at 45 • . We therefore computed ∂ VR SM I tot , ∂ βatt SM I tot , ∂ VR P B and ∂ βatt P B keeping constant Ω and N inj . This protocol is therefore supposed to allow comparisons for different air jet control parameters (V inj and α inj ) with the compressor parameters (U and γ st ), making it easier to compare the compressor's performance under different operating conditions (i.e., Ω = 3200RP M, 4500RP M and6000RP M ).

Figure 7 shows ∂ VR P B, and ∂ βatt P B, and similarly in Fig. 8 for the gradient components of SM I tot at Ω = 3200 RPM and N inj = 20. Figures 7 and8 show that the ill-conditioned testing data (marked by red squares) are partially located in regions where there is a high sensitivity of the output gradients (sharp change of gradient components on the right end of the figures). We further stress, however, that most of the red squares are located well within the bulk region of the domain, where the outputs are not sensitive to the injection parameters V R and β att . Hence, we speculate that the physical features of the are not responsible of the observed ill-conditioning of the debated testing inputs. A possible reason for the ill-conditioning of such testing data could be the limited dataset. In addition, the results suggest that data points far away from the interpolation/extrapolation boundary may severely affect the training of the models up to ill-condition the coherence of the model training. In fact, to have a reliable model, we need to include such ill-conditioned testing points in the training dataset.

We conducted two additional trials to test the significance of the ill-conditioned testing points. The first trial involved training 2000 neural network models using the same process discussed in Sect. 3.3, while intentionally including the ill-conditioned points in the training dataset. The goal was to decrease the K-fold number (10 folds instead of 25 folds), i.e., increase the testing dataset from 4% to 10%. Although the models converged rapidly for the same K folds (K= 25), we observed no significant benefits for lower K-fold tests (K= 10). In the second trial, we trained 2000 neural network models using the same process discussed in Sect. 3.3, after removing the ill-conditioned points from the dataset completely. The results of these tests showed that excluding the ill-conditioned points under-mined the predictions for both testing and extrapolations.

In conclusion, including the ill-conditioned points in the testing dataset or excluding them completely from the dataset unconditionally undermines the predictions of the testing and the extrapolation data. On the other hand, including them in the training dataset benefits the model training convergence, but it cannot be used to increase the testing percentage of the datasets. These results might indicate the deficiency of the limited number of dataset, but they also give an insight about the possible amount of data required to build accurate predictive models for similar problems.

Optimization results

Given the surrogate models presented in the previous section, we now turn our attention to the solutions of both optimization problems from Sect. 3.2, beginning with their single-objective variations in Sect. 4.2.1, while the results of the bi-objective formulation are presented in Sect. 4.2.2.

Single-objective optimization

Maximizing the power balance P B compared to the baseline configuration, optimizing the power balance performance index amounts to maximize the output power generated by the compressor for a given set of operating conditions (e.g., fix mass flow rate Q). Tables 3 and4 summarize the results of the corresponding constrained and unconstrained maximization problems, respectively. Values highlighted in red correspond to optimized parameters which are at or beyond the extrema specified in the constrained case. These extrema correspond to those of the training set. It can be observed that the injection velocity V inj increases almost linearly with the rotational velocity Ω, while the injection angle α inj appears to decrease with increasing Ω. Note moreover that, for Ω = 3200 RPM and 4500 RPM, the injection angle is at or below the lower bound specified in the constrained problem. As discussed in Sect. 4.2.2, the Pareto-optimal points occur within a specific range of the velocity ratio V R and the relative angle of attack β att . The formu- Note that for 3200RPM and 4500 RPM, the optimizer sets the injection angle at its maximum absolute value (see values in bold) lation of V R and β att is discussed in Sect. 4.1. As Ω increases, U increases, and thus, V inj tends to increase in order to maintain the optimal V R range. Similarly, as both V inj and Ω increase, the injection angle α inj tends to decrease to maintain the optimal β att range. Maximizing the surge margin improvement SM I tot while we optimized for the power balance and let the surge margin improvement unchecked in the previous section, we now do the opposite: Optimize for the surge margin improvement and leave the power balanced unhampered. Optimizing for this performance index primarily aims at extending the range of conditions at which the compressor can safely operate. Tables 5 and6 summarize the results of the corresponding constrained and unconstrained maximization problem, respectively. Note that for 6000 RPM, the optimizer sets the both the injection angle and the injection velocity to their maximum allowed value (see values in bold) Note that for 6000 RPM, the optimizer sets the injection velocity beyond its maximum allowed value (see values in red)

As before, values highlighted in red correspond to optimized parameters at or beyond the bounds specified in the constrained formulation. While the evolution of the injection velocity with respect to the rotational velocity of the rotor appears to follow a similar trend as what is observed when maximizing the power balance (i.e., increasing V inj with increasing Ω), the optimized values are comparatively larger (as much as 40% larger for Ω = 3200 RPM). More importantly, the evolution of the injection angle α inj is different. While α inj decreases with Ω when maximizing the power balance, it now appears to stay constant (if not increase) as Ω increases when the surge margin performance index is considered. As discussed in Sect. 4.2.2, the Pareto-optimal points are achieved within a specific range of V R and β att . This can help explaining the evolution trends of V inj . Additionally, it can be observed that at a rotational velocity of 6000 RPM, the optimal value of V inj for surge margin improvement is higher than that for power balance. This difference in V inj could explain the reverse trend in the evolution of α inj . In other words, as V inj is higher in the case of SM I tot , for the same rotational velocity, α inj tends to increase in order to maintain the same relative injection angle β inj and thus achieve the same optimal value of β att .

Bi-objective optimization

As discussed in the previous section, maximizing the power balance P B and the surge margin improvement SM I tot leads to conflicting evolution of the optimized parameters, most notably for the injection angle. Aiming at achieving the best of these two objectives, we now turn our attention to the bi-objective optimization Eq. 5. In particular, we look at how the solution of this optimization problem changes as the trade-off parameter λ evolves. The bi-objective optimization processes are performed based on constrained and unconstrained conditions. This process begins by analyzing the case where control parameters are constrained. Figure 9 shows the non-dominated optimal points (i.e Pareto front) at different rotational velocities (3200 RPM, 4500 RPM and 6000 RPM). Each Pareto front is labeled by three optimal points: A i , C i and B i . Points A i correspond to the maximization of surge margin improvement, points C i correspond to the maximization of power balance, and points B i represent the Pareto-optimal solutions. The specifications of points A i , C i and B i are summarized in Table 7. It can be observed that as we pass from points A i to C i at each rotational velocity, the value of V R increases linearly. The value of β att follows a similar trend as V R , except at Ω = 6000 RPM. The physical interpretation of these observations is the subject of ongoing study.

Comparisons between the non-dominated optimal points for the constrained case are also shown in Fig. 10. A notable feature is the similarity in the range of V R and β att where all Pareto-optimal points (B i ) are achieved. In particular, as shown in Fig. 10 and Table 7, the Pareto-optimal points (B i ) can be obtained at V R between 1.2 and 1.4 and β att between 7 • and 11 • . These similarities in the range of V R and β att at different operating conditions may be due to the self-similar flow topology of the air jets, as discussed in [START_REF] Mokkadem | Small widths wallattached coandȃ jets for flow control[END_REF]. This issue is also subjected to the ongoing research study.

For the case where control parameters are unconstrained, Fig. 11 shows similar trends compared to the constrained optimization Fig. 10, except for some intersections between rotational velocities of 3200RPM and 4500RPM. Additionally, small changes can be observed in the range of V R and β att where the Pareto-optimal points (B i ) are achieved. This range, in the unconstrained case, is between 1.1 and 1.6 and between 1 • and 11 • for V R and β att , respectively. The similarities in the range of V R and β att at all operating conditions suggest the possibility of developing non-dimensional forms of the control parameters. These dimensionless quantities aim to generalize the active flow control strategy and make it applicable to other compressors.

Conclusion

A machine learning optimization algorithm is used to identify the optimal air jet control parameters for active flow control in an axial compressor CME2. Two neural networks are first created to model the relationship between our control parameters and two performance metrics, namely power balance and surge mar- The two neural networks are trained for 550 epochs until a mean square error less than 10 -3 is reached. All neural networks achieve a coefficient of determination of R 2 ≥ 0.98 for the training and testing dataset. When extrapolating to new data, the networks had R 2 = 0.89 and 0.82 for the power balance and surge margin improvement performance metrics, respectively. A sensitivity analysis has been carried out by investigating the gradients of the neural network outputs with respect to the flow control parameters. We demonstrated that having limited dataset can allow to build a reliable neural network model. Special attention must be paid to the database splitting, as limited databases could require sensitive data to be part of the training rather than the testing dataset. Hence, if a study aims to produce a neural network model for applications with a parameter space comparable to ours and their model relies on a limited database, a sensitivity analysis is here advised. It is worth noting that, to the best of our knowledge, the dataset we utilized is one of the most comprehensive databases available in the literature for active flow control in axial compressors, given the large dimensionality of the control parameter space (i.e., injection velocity, injection angle, number of injectors and rotational velocity). This means that limited databases are a challenge to face in active flow control applied to axial compressors, which further proves the widespread importance of sensitivity analyses.

After this, single-and bi-objective optimization using these neural network models are performed under both constrained and unconstrained control parameter conditions. The results of the bi-objective optimization showed that the Pareto-optimal points were achieved at specific ranges of V R and β att . When control parameters were constrained, the optimal range for V R was between 1.2 and 1.4, and for β att was between 7 and 11 degrees. When control parameters were unconstrained, the optimal range for V R was between 1.1 and 1.6, and for β att was between 1 and 11 degrees. This similarity in the range of the optimal control parameters suggests the possibility of developing control laws that could be applied to other compressor geometries. Additionally, this work opens up opportunities for further research to explore the underlying physics behind the observations made in this study. For example, further study could be done to understand the reasons behind the reversed trend of the relative angle of attack of the nondominated optimal points at 6000 RPM.

Fig. 1

 1 Fig. 1 Schematic view for air jet control system

Fig. 2 Fig. 3

 23 Fig. 2 Schematic definitions of a the surge margin improvement SM I tot and b the Power Balance P B. Both panels depict an uncontrolled (i.e., baseline) and controlled surge line. In panel a, LSP b denotes the last stable point in

  train } for the training and D test = {D

Fig. 4 5

 45 Fig. 4 Comparison between the experimental results (ground truth) of the training set and the shallow network's predictions for a the power balance P B and b the surge margin improvement SM I tot

Fig. 7 Fig. 8

 78 Fig. 7 Derivatives of PB with respect to βatt a and VR b at Ω = 3200 RPM and Ninj = 20. The ill-points are highlighted in red squares

Fig. 9

 9 Fig.[START_REF] Bäck | An overview of evolutionary algorithms for parameter optimization[END_REF] Results of the bi-objective optimization problem in the (P B-SM I tot) plane for different rotational velocities Ω. Points in red highlight the Pareto front of this biobjective optimization problem, i.e., the set of optimal values obtained for varying trade-off λ. Points in black correspond to the values of the two objectives obtained for ran-

27 4. 1

 1 Points labeled Ai and Ci correspond to the maximization of the surge margin improvement or power balance, respectively. Points labeled B i correspond to the best trade-off between the two conflicting objectives. The optimal values highlighted in red are the limits of the training dataset range

Fig. 10 Fig. 11

 1011 Fig. 10 Comparison between the non-dominated optimal points of the Pareto fronts for constrained optimization for a the power balance P B and b the surge margin improvement SM I tot. The points show non-dominated optimal points at Ω = 3200 RPM, 4500 RPM and 6000 RPM. Each

Table 1

 1 Range of experimental parameters considered

	Rotor's rotational	3200 RPM, 4500RPM and
	velocity Ω	6000RPM
	Number of injectors	10 and 20
	pairs N inj
	Absolute injection	-45
	angle αinj

• , -

Table 2

 2 Specification

	Optimizer	ADAM
	Learning rate	0.005
	Epochs	550

of shallow neural network's architecture and optimizer used for the learning process Number of neurons per layer 4 → 64 → 32 → 1 Activation function ReLU(x) = max(0, x)

  is applied.

	Algorithm 1 : Genetic algorithm
	1: Population initialization;
	2: Specify λ in (1 -λ) J 1(x) + λJ2(x);
	3: Set constant value of Ω;
	4: Set optimization boundaries for N inj, αinj and Vinj;
	5: while Stopping criteria is not met do
	6:	Objective function calculations;
	7:	Parents selection;
	8:	Crossover;
	9:	Mutation;
	10:	New-generation selection;
	11: end while

Table 4

 4 Results

	Rotational	Injector	Injection	Injection
	velocity	pairs	angle ( • )	velocity
	(RPM)			(m/s)
	3200	20	-65	103
	4500	20	-59	137
	6000	20	-18	161
	Note that for 3200RPM and 4500 RPM, the optimizer sets
	the injection angle (see values in bold)

of the unconstrained optimization procedure when aiming at maximizing the power balance performance index P B for the different operating conditions

Table 5

 5 Results of the constrained optimization procedure when aiming at maximizing the surge marring improvement SM I tot for the different operating conditions

	Rotational	Injector	Injection	Injection
	velocity	pairs	angle ( • )	velocity
	(RPM)			(m/s)
	3200	20	-33	144
	4500	20	-33	165
	6000	20	-45	202

Table 6

 6 Results of the unconstrained optimization procedure when aiming at maximizing the surge marring improvement SM I tot for the different operating conditions

	Rotational	Injector	Injection	Injection
	velocity	pairs	angle ( • )	velocity
	(RPM)			(m/s)
	3200	20	-32	144
	4500	20	-31	166
	6000	20	-45	206

Table 7

 7 Specifications of non-dominated optimal points resulting from the bi-objective optimization problem

	Point	Ω (RPM)	Ninj	αinj ( • )	Vinj (m/s)	VR	βatt ( • )	SM Itot	P B
	A1	3200	20	-33	144	1.81	0.2	59	0.2
	B 1	3200	20	-45	105	1.31	10.4	52	1.9
	C 1	3200	20	-45	92	1.14	11.9	45	2.0
	A 2	4500	20	-31	165	1.45	0.4	50	2.3
	B 2	4500	20	-45	151	1.33	10.1	46	2.7
	C 2	4500	20	-45	129	1.13	12	42	2.9
	A 3	6000	20	-45	202	1.34	10.1	33	3.9
	B 3	6000	20	-41	185	1.22	8.7	32	3.9
	C 3	6000	20	-19	159	1.04	-0.5

Author contribution statement

All the authors were involved in preparing the manuscript. All the authors have read and approved the final manuscript.

Data availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Financial interests This paper is supported by Clean Sky European Union's Horizon 2020 research and innovation program under grant agreement No 886352, project ACONIT.

Conflict of interest

The authors declare no conflict of interest


Almost universally, music uses scales consisting of a small number of notes. Could this increase the fitness of melodies for oral transmission? By reproducing the process online, a new study reveals how cognition, sound and culture may interact to shape music.

In most idioms that have arisen across the world [START_REF] Dowling | Music Cognition[END_REF][START_REF] Savage | Statistical universals reveal the structures and functions of human music[END_REF][START_REF] Brown | Universals in the world's musics[END_REF][START_REF] Mehr | Universality and diversity in human song[END_REF] , and quite possibly since prehistoric times [START_REF] Conard | New flutes document the earliest musical tradition in southwestern Germany[END_REF] , music uses tones moving up and down in pitch along scales that have a small number of permissible notes. This is in spite of pitch perception being remarkably accurate, with justnoticeable differences down to about 0.2% in trained listeners [START_REF] Micheyl | Influence of musical and psychoacoustical training on pitch discrimination[END_REF] . So, if adjacent note discriminability was the sole constraint, about 350 notes per octave would be available. Instead, most known cultures use between 5 and 12 notes per octave. Why? Imagine that cave painters had settled upon a small set of shapes to draw, and then everybody else followed suit to this day?

Although the comparison has many caveats, it highlights how puzzling the enduring existence of musical scales actually is. From the perspective of music cognition, such a peculiar arrangement has long been thought as deriving from universal constraints on pitch perception and memory 1,7- that discrete musical scales may arise through a delicate interplay between motor, perceptual, and cognitive biases at the individual level, which in turn interact with the shared biases of cultural transmission.

To study cultural transmission within a time period compatible with research grants, Anglada-Tort et al. [START_REF] Anglada-Tort | Large-scale iterated singing experiments reveal oral transmission mechanisms underlying music evolution[END_REF] used a clever technique called 'iterative learning'. In the last ten years or so, iterative learning has been used to study how oral transmission shapes language [START_REF] Kirby | Cumulative cultural evolution in the laboratory: An experimental approach to the origins of structure in human language[END_REF] , musical rhythm [START_REF] Jacoby | Integer Ratio Priors on Musical Rhythm Revealed Cross-culturally by Iterated Reproduction[END_REF] , or even melodies [START_REF] Lumaca | Cultural transmission and evolution of melodic structures in multi-generational signaling games[END_REF][START_REF] Verhoef | Melodic universals emerge or are sustained through cultural evolution[END_REF][START_REF] Popescu | Core principles of melodic organisation emerge from transmission chains with random melodies[END_REF] . What the new study [START_REF] Anglada-Tort | Large-scale iterated singing experiments reveal oral transmission mechanisms underlying music evolution[END_REF] offers, thanks to a truly impressive online tour de force, is a systematic investigation of the transmission process on an unprecedented, well, scale. The nature of the experiment allowed for the causal manipulation of important transmission parameters, such as the length of the seed melodies or the native culture of the participants, in ways that were simply not attainable for previous corpus analyses or in-lab studies.

Let us first sketch the basic assumptions underlying the iterative learning paradigm. For melodies, the experimental task is seemingly straightforward: you are being played a short tune and politely asked to sing it back, as accurately as possible. Unbeknownst to you, however, the tune is derived from a previous vocal production of another participant, all within a long 'transmission chain'. The small errors made when singing back are what matter: throughout a transmission chain, random errors should even out, but shared biases should accumulate (Figure 1). Arguably, the process should mimic the way that tunes have been carried over generations through oral transmission, resulting in the musical systems we ponder about now.

In the first experiment of Anglada-Tort et al. [START_REF] Anglada-Tort | Large-scale iterated singing experiments reveal oral transmission mechanisms underlying music evolution[END_REF] , three-tone seed melodies were used. This amounts to two successive frequency ratios or intervals between notes. Except for those with the mixed blessing of absolute pitch, intervals are what matter for melodies [START_REF] Dowling | Music Cognition[END_REF] . Initially, the seed melodies were randomly sampled from a uniform distribution of intervals. Thus, they were very different from our familiar melodies made out of scales, as the very point of a scale is to unevenly distribute intervals. Seed melodies were then passed through the transmission process, with chains of 10 iterations. The results were clear-cut: starting from a shapeless pitch soup, melodies morphed into orderly scales consisting of a limited number of intervals. This was reflected by the distribution of intervals at the end of the chain, and also by complexity measures, which decreased precipitously over iterations. As a result, the copying errors from one iteration to the next also decreased: melodies mapped to scales were much fitter for transmission.

Several variations on the theme extended the main findings. When the length of the seed melodies was increased, the complexity measures dropped even more quickly, presumably to keep copying error low as the seed melodies were more complex to start with. Adding distractor melodies between listening and singing, thus loading memory, also induced quicker drops in complexity. Having participants reproduce melodies with a slider, and not with their voice, led to more copying errors overall but still a reduction in complexity over time. When the main experiment was replicated with Indian participants instead of US participants, similar patterns were observed, although the intervals that emerged were different across the two populations.

In all experiments described so far, melodies had to survive the diversity of individual biases across listeners, presumably converging to shared biases. The authors also investigated what happened when the same participant performed all iterations within a chain. In this case, complexity decreased at a shallower rate, and the resulting distribution of intervals was more varied. So, self-transmitted melodies were more complex than socially transmitted ones.

Interestingly, copying errors were quite small in this case, suggesting that the melodies quickly adapted to the idiosyncratic biases of each listener.

In summary, the Anglada-Tort et al. [START_REF] Anglada-Tort | Large-scale iterated singing experiments reveal oral transmission mechanisms underlying music evolution[END_REF]

study brings decisive experimental evidence

confirming what music theorists had conjectured all along: because of how human cognition works, information transmitted through a small number of categories (say, seven plus or minus two) is much more robust than information coded over 380 discriminable values [START_REF] Dowling | Music Cognition[END_REF] . In the context of cultural transmission, this amounts to having a powerful self-correcting code across generations.

Further aspects of the new results bring exciting new questions. First, the intervals to which participants eventually converged were different from the most common ones found in corpus analyses. World-wide, small intervals of zero, two, and three semitones, called the unison, second, and minor third, are by far the most prevalent [START_REF] Mehr | Universality and diversity in human song[END_REF] . Focusing on Western music, intervals around two semitones are the most common, with some variation depending on the historical period [START_REF] Zivic | Perceptual basis of evolving Western musical styles[END_REF] . However, using iterative learning, the most common intervals after transmission were comparatively larger, from five to seven semitones, encompassing musical fourths and fifths. Why such a difference? Both approaches could have revealed the same underlying structures, as corpus analysis seems like an ecological way to survey the musical forms that have survived transmission across generations.

Anglada-Tort et al. [START_REF] Anglada-Tort | Large-scale iterated singing experiments reveal oral transmission mechanisms underlying music evolution[END_REF] tested two plausible models to explain the interval distributions they observed after iterative learning. Maybe US participants favored intervals that are considered consonant in Western music, of which the fifth is a prime example? Even though this idea helped accounting for some of the differences between US and Indian listeners, it did not capture the full pattern of results. Maybe vocal production constraints played a part, as singing large intervals require more vocal effort? This was consistent with the general reduction in interval size observed throughout transmission chains, but again it failed to capture the full pattern of results.

An unexpected finding points towards yet another factor. Many of the end-products of transmission chains were archetypal, following what the authors termed an 'arched contour': a simple up-down melody mostly expressing the same intervals in ascending and descending forms. What made participants so depressingly predictable, when the random seed melodies could have opened up a wild range of possible tunes to settle on? One possibility is that they tried to establish a tonality or a mode while reproducing melodies. Tonalities and modes are hierarchies amongst the different notes of a scale: for music theory, even if all notes of a scale are permissible, not all are of equal structural importance. If participants had wanted to establish a tonality, an arched contour and the use of, for example, a fifth would have served them well.

But remember that this is not what was asked of them! They only had to sing back the melody as accurately as possible. Seeing the emergence of proto-tonalities or modes in a simple transmission experiment confirms that the concepts are firmly ingrained in the mind of even nonmusical listeners of various cultures [START_REF] Tillmann | Implicit learning of tonality: a selforganizing approach[END_REF] , and suggests that it further increases fitness for transmission by adding yet another error-correcting layer to the musical code.

To conclude, the large-scale investigation of iterative learning for melodies by Anglada-Tort et al. [START_REF] Anglada-Tort | Large-scale iterated singing experiments reveal oral transmission mechanisms underlying music evolution[END_REF] has revealed a sort of a 'snap to grid' process, transforming random pitch distributions into orderly scales even over short oral transmissions chains. The process is likely complex, involving a mix of perceptual, cognitive, motor, and cultural factors, with bootstrapping relationships among them that remain to be untangled. Ultimately, though, the resulting 'internalized scales' [START_REF] Shepard | Auditory illusions demonstrating that tones are assimilated to an internalized musical scale[END_REF] enable the daily wonder accomplished by each and every one of us when turning mere acoustic vibrations into music. A random seed melody is initially drawn from an almost-uniform distribution of possible pitch intervals. The first participant in the transmission chain listens to the seed, then attempts to sing it back as accurately as possible. However, the participant has a set perceptual, cognitive, and motor biaseslumped together as the distributions in white in the illustrationthese these biases introduce systematic errors in the way the melodies are heard, represented, and sung back.

As the transmission chain progresses, shared biases get reinforced. As a result, the final melody has a statistical distribution of pitch intervals that is not flat anymore: the distribution peaks are akin to degrees in a musical scale, reducing the complexity of the melody.

Figure 1 .

 1 Figure 1. Illustration of iterated learning for melodies.

. In a new study reported in this issue of Current Biology, Anglada-Tort et al.

further show

Declaration of Interests:

The authors declare no competing interests.


domain: sdv spi

text:
Pauline Assemat

Anne Gomez-Brouchet

Anthony Mancini

Romain Alis

Michel Quintard

Pascal Swider

DIGITAL FUNCTIONAL IMAGING AT MICROSCALE: OSTEOSARCOMA MICROENVIRONMENT AND TREATMENT RESISTANCE

Introduction

Osteosarcoma (OST) is a malignant primary bone tumor that mainly affects children and young adults. It is characterized by the anarchic production of bone by tumor cells. The high genomic complexity of OST has not allowed the identification of recurrent therapeutic molecular targets or specific treatment resistance biomarkers. This genomic complexity leads to an intra tumoral and inter individual heterogeneity [1] which can be explored at different scales: metabolomic, molecular, tissue or macroscopic scales. OST components are moreover subject structural and fluid mechanical effects adding supplementary complexity in the understanding tumor microenvironment. Using porous media theory, we developed a new digital functional imaging technique to explore mechanical and transport properties of tissue from histological and immunohistological human data. We found correlations between elastic properties, convective properties and cell distribution into the microenvironment depending upon treatment response and we highlighted the impact on neo-formed bone microarchitecture.

Materials and Methods

We propose a consistent upscaling method to characterize the mechanical properties of tumoral microenvironment which is a strongly heterogeneous porous medium. The core is a sequential grid-block approach [2] after applying dedicated image segmentations. The methodology is adapted to large size binarized images (50000px*50000px) extracted from histological and immunohistological slices obtained in clinical routines. Interstitial fluid flow and elastic models are explored. Piecewise constant equivalent parameters such as tissular permeabilities and stiffness coefficients are determined with reliability.

Results

On a cohort of 4 patients, it was found that chemotherapy was more efficient in lacy-like neo-formed bone regions compared to trabecular-like neo-formed bone regions with areas of good response matching with a decreasing number of residual cells and a decreasing local tissue permeability and stiffness (fig 1).

Discussion and Conclusions

We have developed an innovative approach based on heterogeneous porous media theory and based on data from clinical setting to explore tumoral microenvironment properties. This new in silico functional imaging technique, is generic and could be adapted for different physics and other cancers.

Figure 1 :

 1 Figure 1: OST histological section, cell density map (bad response to treatment corresponds to high cell density) and stiffness map.