One of the most conventional kind of ANNs used for regression is the Multilayer Perceptron (MLP), which is constructed by coupling the intelligent approaches usable for computation and biological principles; in Figure 1, a simple architecture of this type of ANN is illustrated (Ramezanizadeh et al., 2019a; Ramezanizadeh et al., 2019b).

In this network, there are several nodes in some layers. In its simplest form, with three layers, there is a hidden layer besides the input and output layers, while in networks with a higher degree of complexity, the number of hidden layers can be higher. In each network node, a weight vector is employed to link it to the upcoming layer. The summation of the nodes is the input of the next layer. By assumption of the input vector as X, n _j is the jth node’s input that is in the following layer which is determined by using Eq. (1) as follows (Zendehboudi and Li, 2017): n j = ∑ i = 1 n ω j i x i + θ j j = 1,2 , … . , K (1)

In Eq. (1), θ j _, ω j i , and K are the threshold, weight, and number of nodes, respectively. Thereafter, a transfer function is used to determine the inputs of the next layer as provided in Eq. (2). y j = f n j = f ( ∑ i = 1 n ω j i x i + θ j )   j = 1,2 , … , K (2)

There are different transfer functions applicable in the abovementioned equation. Multiplying the connecting weight and the hidden layer output will determine the output of the node. It should be noted that there is no specific rule to define the size of the hidden layer and its number. The number of this layer depends on the complexity of the problems, noise of data, etc. (Du and Swamy, 2006). In the process of training the network for regression, values of bias and weights are regulated. The backpropagation algorithm is among the most applied training approaches.

A standard Radial Basis Function (RBF) neural network is able to project linear data with low dimensions into non-linear data with high dimensions. This network is applied to estimate non-linear functions and set up mappings between the input and output parameters. Standard RBF forward-propagation is as follows (Tang and Liu, 2021): F x = ∑ i = 1 n w i G x , O i (3)

In Eq. 3, n is the training samples number, i.e., the hidden nodes number, and w i is the coefficients of weights. G x , O i is presented as follows (Tang and Liu, 2021): G x , O i = exp ⁡ ⁡ − 1 2 σ 2 x − O i 2 (4)

In Eq. 4, O i refers to the hidden nodes’ values and σ is the mean expansion constant. On the basis of the Galerkin method, F(x) can be replaced with F ∼ x as follows (Tang and Liu, 2021): F ∼ x = ∑ i = 1 m w i G x , t i = ∑ i = 1 m w i . exp ⁡ ⁡ − 1 2 σ 2 x − t i 2 (5)

In this equation, t i refers to the clustering centers’ means values. By minimizing the cost function, ε F ∼ = Y − F ∼ 2 , that means the minimization of forecasting error, a set of coefficients of weights could be calculated as follows (Tang and Liu, 2021): min ε F ∼ = min ⁡ ⁡ ∑ i = 1 n y i − ∑ j = 1 m 1 w j G x i , t j 2 + λ D F ∼ 2 = min ∑ i = 1 n y i − ∑ j = 1 m 1 w j . exp ⁡ ⁡ ( − 1 2 σ 2 x i − t j 2 2 + λ D F ∼ 2 ⁡ (6) where λ is the coefficient of regularization, y i refers to the ith training sample real fitness value, and D is the neural network output stability factor. The first term of Eq. 6 can be rewritten as Y − G W where Y = y 1 , y 2 , … , y n T G = G x 1 , t 1 G x 1 , t 2 G x 2 , t 1 G x 2 , t 2 … G x 1 , t m 1 … G ( x 2 , t m 1 ⋮ ⋮ G x n , t 1 G x n , t 2 ⋱ ⋮ … G x 1 , t m 1 W = w 1 , w 2 , … , w m 1 T (7)

In addition, D F ∼ 2 can be rewritten as follows (Tang and Liu, 2021): D F ∼ 2 = D F ∼ , D F ∼ = ∑ i = 1 m 1 w i G x , t i , D ∼ D ∑ i = 1 m 1 w i G x , t i = ∑ i = 1 m 1 w i G x , t i , ∑ i = 1 m 1 w i δ t i = ∑ i = 1 m 1 ∑ j = 1 m 1 w j w i G t i , t j = W T G 0 W (8)

In Eq. 8, D ∼ refers to the adjoint operator of D as the linear differential operator and δ t i denotes the Dirac Delta function at point x = ti. By combining Eq. 8, Eq. 6 would be equivalent to Eq. 9 for the matrix of weight as follows (Tang and Liu, 2021): G T G + λ G 0 W = G T Y (9)

Susbequently, the weight matrix could be directly solved by using Eq. 10 as follows (Tang and Liu, 2021): W = G T G + λ G 0 − 1 G T d (10)

Corresponding to the above theorem, coefficients of weight could be calculated. In standard RBF, the hidden nodes number is equal to the samples of the training number; consequently, a long convergence time and overfitting is typical when there is too large a number of training samples. When there are a small number of training data, mapping relation cannot be captured. To restrict the number of hidden nodes in standard RBF and enhance its performance, a combination scheme has been presented: GA-KM-RBF. A clustering algorithm of K-Means++ (KM) is employed in order to calculate the hidden nodes’ number. Subsequently, the Genetic Algorithm is utilized for optimization of network hyperparameters that further decreases the empirical coefficient dependence (Tang and Liu, 2021).

Adaptive neuro-fuzzy inference system (ANFIS) is another intelligent technique applicable for regression and proposing predictive models. The architecture of this method in its simplest form, in the case of a single output and two inputs, is illustrated in Figure 2. The first layer of this structure is used to convert input variables to fuzzy sets and project them based on fuzzy membership in span of 0 and 1. The inputs’ signals are created in the following layer and the weights are checked. Afterwards, normalized firing strength is determined in the 3rd layer. Subsequently, in the 4th layer, the obtained values are converted into defuzzy sets. In the last layer of this architecture, inputs from the previous part are summed up and the output is determined.

SVM is another intelligent method utilizable for regression and training (Sreedhara et al., 2019); however, this approach is not usable for solving overfitting problems. In order to resolve this issue, LS-SVM is applied, which will be introduced here. In general, Eq. (11) is applied as the non-linear function of this method (Suykens and Vandewalle, 2000; van Gestel et al., 2004; Ahmadi and Mahmoudi, 2016): f x = w t φ x + b (11)

In Eq. (11), f denotes the association between the input and output and w and b are weight vector and value of bias, respectively. φ is employed in order to convert inputs into the characteristics’ vector (Ahmadi and Mahmoudi, 2016; Ramezanizadeh et al., 2019a).

A fitting error function is applied to be minimized in order to have the highest accuracy. This function is obtained by using Eq. (12) as follows (Ahmadi and Mahmoudi, 2016; Ramezanizadeh et al., 2019a): Min   J w , e = 1 2 w T w + γ ∑ k = 1 m e k 2 (12)

In this regard, a limitation equation is defined as follows (Ahmadi and Mahmoudi, 2016; Ramezanizadeh et al., 2019a): y k = w T φ x k + b + e k , k = 1,2 , ⋯ , m (13)

where e _k and γ and are loose variables of kth x and margin parameter, respectively (Ahmadi and Mahmoudi, 2016; Ramezanizadeh et al., 2019a).

Lagrange multipliers are used in order to calculate the optimization process solution. The multiplier is defined as follows (Ahmadi and Mahmoudi, 2016; Ramezanizadeh et al., 2019a): L w , b , e , α = J w , e − ∑ k = 1 m α i w T ∅ x k + b + e k − Y k (14)

By implementing partial derivatives of Eq. (14) based on the variables, as provided in Eq. (15), it is possible to determine the optimal state. w = ∑ k = 1 m α i ∅ x i ∑ k = 1 m α i = 0 α i = γ e i w T ∅ x i + b + e i − Y i = 0 (15)

The linear form of the abovementioned equation is as follows (Ahmadi and Mahmoudi, 2016; Ramezanizadeh et al., 2019a): 0 − Y T Y Z Z T + 1 / γ b α = 0 1 (16)

Where α, Y, and Z are assumed as: α = [α ₁,…,α ₁], {Y = Y ₁;. . .; Y _ym }, and Z = φ(_X1)^T Y _i,…, φ(_Xm)^T Y _m, respectively. Applying kernel function of K (X,X_k) = φ(X)^T φ(X_k), i = 1,2,…,m, the regression is as follows (Ahmadi and Mahmoudi, 2016; Ramezanizadeh et al., 2019a): f x = ∑ k = 1 n α k K x , x k + b (17)

There are several functions such as RBF that can be used as a kernel in regression errors, as in Eq. (17). In regression models, radial basis function is a regular kernel (Ahmadi and Mahmoudi, 2016; Ramezanizadeh et al., 2019a). In the process of optimization, to obtain the parameter of kernel function, mean squared error (MSE) is considered as the objective function to be minimized (Fazeli et al., 2013; Ahmadi and Ebadi, 2014; Ahmadi et al., 2014).

In this approach, a Volterra series is used for defining the relationship between the inputs and output. This series is similar to Kolmogorov discerete polynomial functions. In GMDH, the total estimator models are replaced by the incrementing iterative algorithm. In this method, polynomial neurons are generated and combined with each other in order to create a complex system with proper performance. GMDH is composed of a number of neurons with binomial transfer function that arise from the linking between various pairs of variables via the 2nd order Kolmogorov relation according to Eq. (18) (Moosavi et al., 2019). y = c 0 + ∑ i = 1 m c i x i + ∑ i = 1 m ∑ j = 1 m c i j x i x j (18) where x = x 1 , x 2 , … , x m is the input vector and c = c 1 , c 2 , … , c T is the weight vector which is obtained by the least-squares approach. Similar to MLP, GMDH is a multilayer feed forward network and the outputs of the previous layer are utilized as the inputs of the current layer. In this method, the architecture of the model and number of neurons and layers is determined in the process of training. In general, a large number of basic models are created in GMDH and some of them are selected to be combined amd utilized for reconstruction of other functions. Selection and generation steps are repeated till the best model is obtained (Moosavi et al., 2019).

In addition to modeling and regression, intelligent methods are applicable for clustering and classification. SVDD is a one-class classification approach. The principle idea of this method is finding a hypersphere with a small radius and as many as possible points in this. Support vectors are the points on the hypersphere surface. Eq. (19) is applied for the definition of the structure error function as follows: F R , a = R 2 s . t .   x i − a 2 ≤ R 2 , ∀ i (19) where R is the radius and a is the center. The distance from x i point to centre could not be strictly less than R in order to allow the training set to have outliers induced by noise or other parameters. A relaxation factor is introduced to punish the outliers. Consequently, the minimization error can be written as follows: F R , a = R 2 + C ∑ i ξ i s . t .   x i − a 2 ≤ R 2 + ξ i , ξ i ≥ 0 ∀ i (20)

In this equation, C is applied for controlling the compromise between the false treated points number and hypersphere radius. To solve this problem, Lagrange multipliers are used as follows: L R , a , α i , γ i , ξ i = R 2 + C ∑ i ξ i − ∑ i α i R 2 + ξ i − x i 2 − 2 a . x i − ∑ i γ i ξ i (21) where α i ≥ 0 and γ i ≥ 0 refer to Lagrange multipliers. By applying partial derivative to L, we obtain the following equations: ∂ L ∂ R = 0 , → ∑ i α i = 1 (22) ∂ L ∂ a = 0 , → a = ∑ i α i x i (23) ∂ L ∂ ξ i = 0 → C − α i − ξ i = 0 (24)

Therefore, Eq. (25) can be written as follows: L = ∑ i α i x i . x i − ∑ i , j α i α j x i . x i 0 ≤ α i ≤ C (25)

When the description of the data set by the hypersphere in the original space is not sufficient, the data set could be mapped from the space with a low dimension to a high-dimension one. In order to substitute the interproduct, kernel function K ( x i . x j ) is applied as follows: L = ∑ i α i K x i . x i − ∑ i , j α i α j K x i . x j (26)

The definition of distinguish function is as follows: I A = I ( z − a 2 ≤ R 2 = 1 A   i s   t r u e 0 A   i s   f a l s e (27)

As previously stated, intelligent methods can be applied for different turbomachines in order to model their behavior, predict the output, and design control systems (Javadi Moghaddam et al., 2011; Zhang et al., 2021). Different aspects of axial compressors, as the main components of gas turbine engines, have been modeled by using these approaches (Yu et al., 2007). In a study by Benini and Toffolo (2002), cascade performance was modeled by using ANN under generalized conditions. In this regard, quasi 3D CFD simulations were applied to prepare a database for the training network. As the model knew the characteristics of the geometry, including the effects of IGV depending on the mass flow rate and speed of rotation, it could directly determine the operating point of a compressor. In another work (Fei et al., 2016), three different intelligent techniques, namely, Back Propagation Neural Network (BPNN), SVM, and Gaussian kernel function Back Propagation Neural Network (GBPNN), were employed to estimate the performance of an axial compressor. They found that increments in the number of training datasets leads to enhancement in the exactness of the generated models by all of the applied approaches. Furthermore, it was demonstrated that making use of GBPNN provides the highest accuracy, followed by the SVM and BPNN. In another work, Sohail et al. (2021) focused on the forecasting tool of transonic compressor instability. In this regard, they utilized a dataset obtained from CFD for supervised learning of an ANN. The input variables and outputs of their model are shown in Figure 3. Obtained results from the deep learning ANN revealed that it is a promising method to forecast the performance of the compressor and rotor blade parameters.

Yue et al. (2022) generated a model for spanwise loss of a compressor stator. In order to construct the model, a significant number of numerical simulations were implemented to provide a database. Afterwards, intensity of secondary flow was introduced as the independent variable in order to implement feature engineering. The proposed model contained a selector that was obtained by using Support Vector Machine (SVM) regression in addition to estimators that were obtained on the basis of K-nearest neighbor regression. The proposed model reflected proper exactness for mid-span position loss coefficient with R² value of 0.97; however, its precision was lower for all spans. Intelligent methods are usable for modeling more than one output. For instance, in a work (Wu et al., 2019), two methods, Gaussian Process Regression (GPR) and Support Vector Regression (SVR), were applied to model adiabatic efficiency and total pressure ratio. In the case of applying SVR, the predicted error of the empirical model was reduced by 62.2% and 48.4% for the total pressure ratio and adiabatic efficiency, respectively. These values, in the case of employing GPR, were 55.2% and 50.1% for the denoted parameters, respectively. Gholamrezaei and Ghorbanian (2010) made use of ANN and rig data to generate compressor map. Two models were proposed in their research applicable for the prediction of mass flow rate and pressure ratio. In the first model, compressor pressure ratio was formulated based on the function of corrected mass flow rate for constant speed lines, while in the second model, the corrected mass flow rate was formulated as a function of pressure ratio for constant speed lines. It was observed that the first model is more precise for cases of relatively low speed lines while the second model showed higher robustness for cases of higher speed lines.

The performance of intelligent methods is dependent on several parameters, namely, the applied algorithm and the function (Ghorbanian and Gholamrezaei, 2007). Their performance could be enhanced by applying some modifications. A work was done to consider different Neural Networks (NNs) including General Regression NN (GRNN), Radial Basis Function (RBF), Multilayer Perceptron (MLP), and modified approach based on GRNN, called Rotated GRNN (RGRNN) (Ghorbanian and Gholamrezaei, 2009), to model compressor pressure ratio and mass flow rate by using two models. It was observed that making use of RGRNN leads to the highest accuracy; however, it is just applicable for representing characteristic curves for cases where experimental data is provided. The superior performance of RGRNN compared with GRNN in performance prediction of axial compressors has been observed in some other studies (Ghorbanian and Gholamrezaei, 2007). Integration of various intelligent techniques can determine outputs with better accuracy. For instance, Gholamrezaei and Ghorbanian (2015) assessed the performance of ANN and Fuzzy Inference System (FIS)+ANN for map generation of a compressor. The success rates in cases of using the mentioned approaches were 88% and 94%, respectively. In some studies, different functions and architectures are applied to find the optimal model in terms of accuracy. For instance, Yazar et al. (2014) applied various types of FIS structures and utilized different numbers of membership function. They found that number of membership function can affect the model exactness. Integration of optimization algorithm with intelligent techniques, used for modeling and regression, to tune the hyper parameters is another approach for modification of precision. Tang and Liu (2021) coupled Genetic Algorithm (GA) and generalized RBF. In their work, deviation angle and total pressure loss coefficient were predicted in the first step by applying surrogate model. Afterwards, these variables linked the model with throughflow theory. Efficiency and pressure ratio were predicted in the next stage and compared with the actual values determined in the experiments. In comparison with the traditional models and spanwise mixing model, the proposed approach provided much better performance.

There is a useful range for operation of axial compressors, as shown in Figure 4. At high mass flow rates, chocking occurs and sonic velocity is reached in some units. One the other hand, at low mass flow rate, there is possibility for the occurrence of stall and surge that causes instabilities in the fluid flow and performance of axial compressors. Stall is a type of instability caused by the disturbance of the circumferential flow (Gravdahl and Egeland, 1999; Wang et al., 2022). In terms of aerodynamics, there are two types of stall: individual blade stall and rotating stall. The first one happens when all of the blades around the annulus of the compressor simultaneously stall without propagation of stall. The second type, rotating stall, happens when there are large zones of stall that cover various passages of the blade and there is propagation in the rotation direction and at relative rotor speed (Boyce, 2012b). Surge refers to the flow axisymmtrical oscillation in the compressors and is featured by a limit cycle in the characteristics of these machines (Gravdahl and Egeland, 1999). This phenomenon is the flow reversal and continuous steady flow complete break down via the compressor. This phenomenon, surge, is the reversal of flow and complete break down of continuous flow in the compressor. Surge induces mechanical problems due to the high fluctuations of flow that causes changes in the direction of thrust force in the rotor section (Boyce, 2012b; Zhao et al., 2023).

In addition to performance prediction and the characteristics of fluid flow, intelligent methods are useful for other purposes in different energy-related systems (Hipple et al., 2020). For instance, Changzheng and Yong (Li and Lei, 2012) applied Support Vector Data Description (SVDD), developed on the basis of the SVM theory, to detect compressor surge. In their work, total pressure at the outlet was used as the characteristic signal and ten spectral lines were utilized in order to create vectors of the feature. For the applied technique, 400 vectors were determined to provide a training dataset. The outcomes of the training were applied to distinguish surge of the compressor. It was shown that the applied technique can provide a warning signal 50 ms prior to the surge point. In another work, Amanifard et al. (2008) proposed a model based on evolved Group Method of Data Handling (GMDH) by using experimental data to model multiple short-length-scale stall cells. Flow rate coefficient, pressure coefficient, and rotor rotational speed were the inputs of the generated model. Exactness of the model in prediction was acceptable with R² and Mean Absolute Percentage Error (MAPE) of around 0.86268 and 0.00065, respectively. In addition to modeling surge, it is possible to apply ANN to provide backstepping active control for axial compressor surge. In a study by Sheng et al. (2020), a closed coupled valve was used as the actuator in order to generate a compression system second order Moore-Greitzer surge model. The controller was designed on the basis of a wavelet neural network. They found that by using the proposed method, the compressor is allowed to stably work with high efficiency and pressure ratio beyond the boundary of surge. In another work, Methling et al. (2004) used wall static pressure signals to train a network to indicate when instability was being neared. They found that the provided monitoring system based on the ANN has the ability to cover the whole working range of the compressor to provide adequate training datasets.

Aerodynamic design of modern compressors is challenging because of the higher gradient of adverse pressure and increment in the interaction blade rows that induce internal turbulent flow. In this regard, it is beneficial to develop new approaches and algorithms for design and optimization of axial flow compressors (Ning et al., 2016). There is significant potential for applying intelligent methods for these intentions. In this condition, outputs of the models, e.g., characteristics of the compressor such as pressure ratio and mass flow rate, could be used for design or optimization in a faster process since these methods can generate databases for the mentioned intentions based on their prediction ability. As an example, Ghalandari et al. (2019) utilized a hybrid machine learning model composed of ANN, genetic algorithm, and design of experiment to optimize an aeromechanical aspect of compressor first row. In their work, ANN was trained by employing the results of 3D CFD simulation that was generated by design of experiment. The procedure of the optimization was initiated with values of approximated function by the ANN and the optimization algorithm (GA) was applied to reach the optimized point of the function; it was validated by the results of the simulation in order to update the model based on ANN. By using this process, the behavior of the blade and aerodynamic performance was increased by around 5.7%. Ju and Zhang (2011) applied a method consisting of Design of Experiment (DOE), GA, and ANN to optimize cascades of axial compressors. In their work, ANN was trained by using a back propagation algorithm and GA where the training sets were gathered by means of DOE approach and analyzed by CFD. The model based on ANN and GA was created to be used as a rapid flow solver to forecast aerodynamic performance. Multi-objective GA was employed in order to search a series of Pareto-optimum solutions. It was found that, depending on the incidence angle, the pressure loss coefficient can significantly decrease by employing the proposed approach.

Uelschen and Lawerenz (2000) applied ANN and GA for design of compressor airfoils. In their study, ANN was employed to map coherence between aerodynamics and geometry. In comparison with flow calculations, which are very time-consuming processes, much lower computational effort is required for the assessment of the trained network; consequently, the optimization and design is accelerated. Giassi et al. (2003) applied ANN for optimization of aerodynamic aspects of axial compressors and inverse design problem resolution. Their approach was based on the integration of ANN and a classical optimizer. Flow solver, on the basis of Navier-stokes, was applied for precise determination of the objective function. The trained network in the first stage, by using the primary datasets, was applied to reach a new design point. It was determined by the solver to update datasets used for training for more iterative steps. The proposed method showed significantly lower computational effort in comparison with the classical optimization approaches. In another work by Pakatchian et al. (2019), optimization of blade shape was implemented by making use of ANN. In their study, ANN was employed to develop datasets for 2D sections. Subsequently, trained ANNs were employed for optimization of 3D shapes along with the stacking line parametrization. This algorithm was applied for the first four stages. Utilization of the optimization process led to 1.5% increment in the isentropic efficiency of first four stages of a 16-stage axial compressor.

In Table 1, studies related to the employment of machine learning techniques in axial flow compressors are summarized.