Discharge coefficient prediction of canal radial gate using neurocomputing models: an investigation of free and submerged flow scenarios

In the current study, three machine learning (ML) models, i.e. Gaussian process regression (GPR), generalized regression neural network (GRNN), and multigene genetic programming (MGGP), were developed for predicting the discharge coefficient (Cd ) of a radial gate under two different flow conditions, i.e. free and submerged. The modeling development of the flow and geometry input variables for the Cd was determined based on statistical correlations. We also performed a sensitivity analysis of the input variables for the Cd . The modeling results indicated that the developed ML models attained acceptable predictable performance; however, the prediction accuracy of the models was better under the free flow condition. In quantitative terms, the minimum root mean square error (RMSE) value was 0.010 using the GPR model and 0.019 using the MGGP model for the submerged and free flow conditions, respectively. The sensitivity analysis evidenced that the ratio of the gate opening height to the depth of water in the upstream (W:Y o) was the influential variable for the Cd under the free flow condition, whereas the ratio of the depth of water in the upstream to the depth of water in downstream (Y o:YT ) was the influential variable for the Cd under the submerged flow condition.


Research background
Most irrigation projects are characterized by the presence of radial gates (Clemmens et al., 2003), and the project operators normally rely on accurate measurements of discharge through radial gates for effective delivery of water to the end users. The presence of these radial gates also eliminates the need to dedicated separate structures for flow measurement (Abdelhaleem, 2016). A common problem for flow measurement is the calibration of radial gates due to the varied types of gates, channel configurations, and structures, as well as the sensitivity of calibrations to factors such as the type of gate seal and width of the downstream channel (Guo et al., 2021a). Calibrations for a gate operating under a free flow condition are available in standard references, which are easy to use and have high accuracy; however, for a gate operating in a submerged flow condition, calibrations are often inaccurate with reported error rates of up to 50%. Some of the available calibration methods depend on an energy equation, while other calibration methods involve a momentum equation in order to differentiate free flow conditions from submerged flow conditions. Designers are also interested in improving the performance of hydraulic structures in water engineering projects, and therefore they are always looking for reliable optimization models to achieve this feat (Vatankhah, 2013). The efficiency of hydraulic structures increases by optimizing their geometry; however, there are inherent weaknesses, such as sediment deposition in front of a structure, that cannot be solved (Kardan et al., 2017).

Empirical formulation
One common calibration method is the empirical relationship method which uses an empirical relationship to explain variations in the discharge coefficient for radial gates under free and submerged flow conditions ( Figure 1). A radial gate exhibits a complex rating curve, and only a few empirical equations are currently available in the literature. An extensive experimental investigation was conducted by Buyalski (1983), on the flow through a radial gate under free and submerged flow conditions and several maps were developed for calculating the discharge coefficient. In another study, Tel (2000) recognized the contraction coefficient δ as being dependent on the gate leaf angle θ and proposed an empirical equation for a sharp-edged radial gate. In China, hydraulic textbooks express the discharge coefficient for submerged flow (C sub ) as a product of the submergence coefficient σ and the discharge coefficient for free flow (C free ) with the same upstream flow depth (y1) and gate opening (w) hydraulics teaching and research group of Tsinghua University (Guo et al., 2021b). According to Wang (1955), the submergence coefficient σ is an empirical function of the submergence ratio (X r ); however, the empirical equations proposed by Wang (1955) were experimentally demonstrated by (Mu et al., 2009) to have unacceptable accuracies when X r > 0.8. The calibration of a submerged radial gate using empirical relationship methods has not always been successful, for example, a study by Clemmens et al. (2003) reported an error rate of up to 50%. The hydraulic performances of gates have been studied around the world, for instance, the early studies on gates conducted by Gibson (1919) and Baines (1950). Baines (1950) relied on a dimensionless form of effective parameters in both free and submerged flow conditions and presented the discharge coefficients for sluice gates as a graph. The study provided the basis for most C d equations. Later, new relationships for determining C d in both free and submerged flow conditions were proposed by Rajaratnam and Subramanya (1967) and Sepúlveda et al. (2009).
A study by Silva and Rijo (2017) focused on C d determination using experimental and laboratory data and showed that the energy-momentum-based method (EM) performed better than the orifice flow rate equationbased methods as well as those methods based on the π theorem of the dimensional analysis. Velocity profile and flow characteristics near a slide gate were experimentally studied by Esmailzadeh et al. (2015), who presented V x , V y , and V z which were the x, y, and z components of flow velocity, respectively, in the vicinity of the slide gate.
According to the literature abstracted from the Scopus database, the commonest types of gates are the radial and slice gates ( Figure 2). Figure 2 exhibits the occurrence of the major keywords for the adopted studies in this domain. Since the force required to open gates is significant, hydraulic engineers normally rely on radial gates rather than the other types of gates (Belaud et al., 2009).  The special shape of radial gates and the existence of a cylindrical shell ensures that the pressure of the input water passes through the axis of the gate without creating a torque around it. The calculation of the radial gate C d is important for solving some nonlinear equations and complex graphs that are associated with its magnitude. Furthermore, most of the gates' C d equations rely on some assumptions that sometimes lead to unrealistic results. Soft computing has emerged as a promising tool in hydraulic engineering (Sharafati, Haghbin et al., 2020) and has been reportedly successful in modeling the C d of hydraulic structures (Granata et al., 2019;Karami et al., 2018;Roushangar et al., 2017). The limitations of empirical formulas for C d calculation include the huge number of effective parameters and their interactions, high uncertainty, large number of assumptions, and solution complexity, etc., and therefore soft computing has been considered to be a better alternative for finding solutions to complex hydraulic engineering problems. Soft computing can be used to solve the governing equations that are subject to boundary conditions; hence, soft computing models can effectively determine the C d and the related parameters. Soft computing models for predicting the C d require a reasonable level of laboratory experiments to build the matrix of predictive models.

Machine learning models adopted in the literature
Research in engineering and sciences has experienced significant progress in applications of soft computing and ML (Moosavi et al., 2019;Yaseen et al., 2019), due to their capacity for solving complex problems associated with nonlinearity and non-stationarity. Over the past decade, ML models have been successfully applied in hydrology (Salih et al., 2019), climate (Malik et al., 2020), fluid mechanic (Ansari et al., 2020), surface water quality (Sharafati, Asadollah et al., 2020), geo-science (Mehajan & Verma, 2020), hydraulic engineering (Sharafati et al., 2018) and several others.
Within the focus of the current research, several soft computing models have been introduced to predict the C d of several types of discharge gates (e.g. weir gate, piano key weir, sluice gate, inclined slide gate, and triangular side orifice) such as an adaptive neuro-fuzzy inference system (Parsaie et al., 2017(Parsaie et al., , 2019, Gaussian process regression (Akbari et al., 2019), random forest Salmasi et al., 2021), group method of data handling (Parsaie et al., 2018), deep learning , genetic programming (Salmasi & Abraham, 2020), locally weighted learning regression (Jamei et al., 2021), gene expression programming (Ebtehaj, Bonakdari, Zaji et al., 2015), artificial neural network (Ebtehaj, Bonakdari, Khoshbin et al., 2015), hybrid inclusive multiple model , extreme learning machine (Zarei et al., 2020), multivariate adoptive regression spline (Yousif et al., 2019), support vector machine (Hu et al., 2021), and several others. Although there have been numerous soft computing models implemented for C d prediction, there is still a need for studies on new robust ML models.

Research gap and motivation
According to the literature, there are several studies on empirical formulas for determining the discharge coefficient; however, they are limited with certain datasets and channel properties, and they are not applicable for all types of channels or types of flow. In addition, the applied machine learning models, in the literature, exhibited some limitations such as the necessity of internal parameter tuning, network adjustment, and several other drawbacks that require developer interaction. Hence, in this study, we aimed at overcoming some of these limitations and developing a new more appropriate and reliable technology for discharge coefficient prediction.

Research objectives
The objectives of this study are: (i) to develop three different neurocomputing models, i.e. Gaussian process regression (GPR), generalized regression neural network (GRNN), and multigene genetic programming (MGGP) for predicting C d radial gate; (ii) to evaluate two different flow conditions, i.e. free and submerged, based on the predictability performance of the adopted machine learning models; and (iii) to identify the influential variables for C d determination through a sensitivity analysis. Figure 1 shows the sketch plan of a radial gate and the related variables that influence the C d , as reported in the literature, for both investigated flow conditions, i.e. free and submerged. The variables are grate radius (R), gate opening height (W), depth of water in the upstream (Y o ), depth of water in downstream (Y T ), trunnion pin height (h), and width (B). For the model development, in this study, the experimental dataset was collected from the open access literature (Buyalski, 1983). The C d magnitude is influenced by geometry and flow properties for both water flow conditions. The C d free flow function is defined as:

Dataset description
The C d submerged flow function is defined as: On the basis of the energy equation and Bernoulli hydraulic engineering concept, the C d is calculated as follows: where Q is the magnitude of the discharge following under the gate. The current study was adopted on the calculation of the C d indicated in Equation (3) (Buyalski, 1983).     For the predictive models for accurately estimating the discharge coefficients of radial gates in both free and submerged flow conditions, 458 and 2136 experimental datasets were separately implemented, respectively. The statistical descriptive analysis of the flow, geometry, and discharge coefficient for both free and submerged flow conditions are reported in Tables 1 and 2.
The distribution function of the normalized values of all variables used in both free and submerged flow conditions are depicted in Figure 3. A comparison of the distribution of the normalized data are for both scenarios shows that the datasets for free flow conditions are close to a normal distribution, and the range of skewness values (i.e. [−0.6414, 2.282]) fully confirms this claim. Figure 4 shows the correlogram of each scenario (free and submerged conditions) which describes the linear correlation between input and output based on the Pearson correlation coefficient. The correlation analysis assists to identify the most significant input parameters for estimating the discharge coefficient in both scenarios. The results show that the parameter, Y T Y 0 , with the highest magnitude of Pearson correlation coefficient (P c = −0.9202), is the most effective feature for modeling the discharge coefficient in the submerged scenario and W Y 0 (P c = −0.894) is the most influential feature for modeling the discharge coefficient in the free flow scenario. However, the sensitivity analysis can confirm the results of the correlation analysis.

Gaussian process regression (GPR)
GPR is a robust nonparametric kernel-based probabilistic model whose processes (Gaussian processes (GP)) (West et al., 2021) are in the form of a set of random variables in a multivariant Gaussian distribution (Schulz et al., 2018). By considering x as the input domain, while y is the output domain, the n pairs (x i , y i ) sphere of influence is extracted. The domains of the sphere are equally distributed and independent. The average function (μ = Y → Re) is hypothesized to define the Gaussian process of x variables. Then, the covariance function of k : X * X → Re is implemented. The merit of the GPR model is that it recognizes the random variability of f(x) for each supplied predictor of (x), in which it presents the (f ) function that is randomly featured (Jamei, Ahmadianfar, Olumegbon et al., 2020). In this study, we assume the observation error is independent and has equivalent zero mean value distribution (μ(x) = 0), variance σ 2 , and f (x) of the GP on x (denote with k): where I represents the identity matrix, and Owing to the normality of y x ∼ N(o, K + σ 2 I), the test label's conditional distribution with the condition of the testing and training data pair whereK(X, X ) represents the matrix n × n * of the evaluated covariance in each testing and training data pair which are similar for the other values of (K(X, X), K(X, X * ) and K(X * , X * ); X is the training vector label; Y is the training data label y i ; and X * is the test data. The specified covariance function for the creation of a semifinite positive covariance matrix of K ij = k(x i , x j ). The specification of the kernel k and the noise degreeσ 2 qualifies Equations (5) and (6) for a deduction. For a proper training of the GPR model, the appropriate covariance function K(X, X ) and its associated parameters must be selected as the covariance function determines the actual function of the GPR model. The geometric structure of the training samples is embedded in this function, meaning that the realization of precise predictions requires estimating the mean and covariance functions from the employed data (called hyperparameters) (Pasolli et al., 2010).

Generalized regression neural network (GRNN)
The GRNN model is a neural network approach to classification problems and its working principle depends mainly on nonparametric regression, while other NN techniques rely mainly on parametric regression (Sanikhani et al., 2019). The GRNN model is mainly useful in prediction and regression tasks. Its advantages in these application areas include the following: (i) it performs the best transitions between a specific determined value with another, (ii) it is suitable for prediction problems that places less premium on linearity, (iii) it uses an instantaneous learning algorithm which rules out the need for tedious codes in its backpropagation, (iv) it demonstrates structural flexibility as compared with other NN techniques, and (v) it can always handle all forms of noise in the input dataset. The GRNN model, as depicted in Figure 5(a), has four layers: the input layer, hidden layer, summation layer, and output layer (Jamei, Ahmadianfar, Olumegbon, et al., 2020;Specht, 1991). The pattern unit (p1, p2, . . . , pn) contains the radial basis function (RBF) neurons that require no weight connection to be directly mapped to the hidden space. For each input unit, the number of neurons is equal to the number of input samples. For the summation layer, it has two units which are the numerator (Sn) and the denominator neuron vectors, while Sd is the arithmetic sum of the RBF neurons. The estimated value (Y) is obtained by dividing Sn by Sd. The prediction results are considered to be theoretically constant if the mapping relationship has been determined. Nonlinear regression analysis forms the algebraic basis of the GRNN model. Assume that f (x, y) is the joint probability density function of vector random variable X and scalar random variable y, the calculation of the estimated value Y can be done thus: The proposed density functionf (X, y) by Parzen using a nonparametric estimator is expressed as (Parzen, 1962): where X i is the sample value of the vector random variable X, y i is the sample value of the scalar random variable y, p is the dimension of X, n is the number of samples observed, and s is the width coefficient (also called the spread parameter). The substitution of Equation (7) into Equation (8) provides the estimated value as follows:

Multigene genetic programming (MGGP)
GP is an optimization method that relies on the Darwin's theory of evolution (Gandomi et al., 2010); its concept is similar to that of a genetic algorithm (GA), since both techniques depend on three main operators which are selection, crossover, and mutation (Danandeh Mehr & Nourani, 2017). However, both techniques differ in how they present their solutions; for the GA, the solutions are presented as strings of fixed lengths, while the solutions of GP are expressed by tree structures of different sizes (Garg et al., 2014). The first step of the GP method is the random generation of a set of models which are created via a combination of the components called functions (F) and terminals (T). For a set of F, the basic arithmetic operators(−, +, ×,Ã., etc.) are involved, but for a set of T, it consists of the input variables and numerical constants of a problem. The selection of crossover and mutation operators is the basis for choosing models using the GP algorithm. Hence, new populations are created for new generations based on these operators. The iterative process is sustained in the GP method until a termination criterion is met. In this study, we consider the optimum number of generations to be the stopping criterion. The MGGP model was developed as an extension of the GP method (Gandomi & Alavi, 2011), which unlike the traditional GP, relies on the evaluation of an individual tree. The formation of the MGGP model is based on the weighted linear combination of the output of several trees in the GP method (Gandomi & Alavi, 2011;). The MGGP model considers each tree as a 'gene' or a GP tree. The development of the MGGP model follows the steps listed below: Step 1: Setting up of the initial parameters, such as the population size, the number of generations, the functions and terminals, the maximum number of genes, and the rate of genetic operators.
Step 2: Random initiation of the initial gene population.
Step 3: Model formation based on a set of genes.
Step 4: Evaluation of the model efficiency based on specified objective function, such as the RMSE.
Step 5: New population generation using the genetic operators.
Step 6: Check if the termination condition is met, otherwise, revert to Step 5. Figure 5(b) exhibits the formal MGGP model structure.

Model development
In order to validate the provided AI models, seven efficient goodness-of-fit tests were considered which are as follows (Khozani et al., 2019;Yaseen, 2021): (10) (ii) Root mean square error (RMSE): (iii) Mean absolute percentage error (MAPE): (iv) Relative absolute error (RAE): (a) Standard deviation (SD): (b) Nash-Sutcliffe coefficient (NS) (Jamei, Ahmadianfar, Chu et al., 2020;Nash & Sutcliffe, 1970): (vii) Willmott's agreement index (I A ) (Willmott et al., 2012): where C do,i and C dp,i are the ith experimental and predicted discharge coefficients, respectively; and C do,i and C dp are mean values of experimental and predicted discharge coefficients at the radial gates, respectively.
In this study, three robust machine learning approaches were provided including GPR, MGGP, and GRNN models, for which all setting and tuning parameters of the aforementioned models are listed in Table 3.
The road map of assessing the discharge coefficient into the nose and wake zone using the provided models is illustrated in Figure 6. To develop the GPR model, five kernel functions including ARD exponential, rational quadratic, squared exponential, ardmatern32, ard-matern52 were examined to achieve the best performance for predicting the discharge coefficient under both free and submerged conditions (see Table 3). The results of the optimal kernel function performance evaluations, for each scenario, are shown in Figure 7. The statistical criteria in radar plots (Figure 7) show that, in the free flow scenario, the ardmatern52 kernel (R = 0.9514 and RMSE = 0.020) and, in the submerged scenario, the ardmatern32 kernel (R = 0.9975 and RMSE = 0.01087) have superior performance for predicting C d values, respectively. The parameter settings and the mathematical operators of the MGGP are reported in Table 3. In addition, the MGGP model for both scenarios provided a predictive relationship based on the chosen operators that are listed in Table 4. As before, the input neurons number in the GRNN approach is equal to the input variables and the main setting parameter is the spread value, whose range for achieving optimum results is between 0.02 and 0.05 variables.

Modeling results and assessment
In this section, the capability of the three machine learning models (i.e. GPR, GRNN, and MGGP) for predicting the coefficient discharge of a radial gate are evaluated and discussed in detail. All models are assessed for predicting C d of a radial gate in two scenarios: (i) submerged and (ii) free flow. As mentioned in the data preprocessing section, 2136 and 479 data points for the  Table 5). In addition, in terms of R, RMSE, MAPE, RAE, NS, and IA, the GPR model, in the testing phase, proved to have more precise predictions than the MGGP and GRNN models. It should be noted that, the standard deviation (SD) obtained by the GRNN model is less (better) than the other models in both training (SD = 0.150205) and testing phases (SD = 0.149758).
In the case of the second scenario (free flow condition), the GRNN model in the training phase had an R value of approximately 0.99795 versus 0.94773 (MGGP) and 0.94544 (GPR). The RMSE, MAPE, and IA values are 0.01923, 2.1530, and 0.9723 (GRNN) versus .01932, 2.2324, and 0.972485 (MGGP), and 0.01973, 2.2676, and 0.97106 (GPR), respectively. In the testing phase, the MGGP model has better efficiency, and was more accurate than the GPR and GRNN models (see Table 6). Its    Tables 5 and 6 show that, for the first scenario, the GPR model is more accurate than the MGGP and GRNN models. In addition, in the second scenario, the MGGP model had a better performance than that of the GPR and GRNN models. Figure 8 illustrates the scatter plots of the predicted C d values for the three machine learning models (i.e. GPR, GRNN, and MGGP) as compared with the observed C d during the training and testing phases for both scenarios (submerged and free flow). As shown from this figure, the plotted data points calculated by the GPR model correlated more closely towards the 1:1 line than the other models for the first scenario. Besides, the MGGP model yielded the best second results and GRNN was ranked as the poorest model for the first scenario. For the second scenario, the MGGP model had a better correlation as compared with the GRP and GRNN models. Accordingly, these results reaffirm that the GPR and MGGP models have better estimative skills than the GRNN model considered in this study.  Figure 9 demonstrates the probability distribution function (PDF) of predicted and observed discharge coefficient values in both submerged and free flow scenarios. In the submerged scenario, the results of the GPR and MGGP models with similar performance are more consistent with observed values than those of the GRNN model. In addition, in the free flow scenario, all three models yield acceptable predictive performance. However, the MGGP model showed the best agreement with the observed values, and therefore was identified as the superior approach, followed by the GPR and GRNN models.
Moreover, the Taylor diagram of the C d prediction efficiency for the GPR, GRNN, and MGGP models in two scenarios are displayed in Figure 10. For the first scenario (Figure 10(a)), the predicted values obtained by the GPR model are more accurate and closer to the target point. In the case of second scenario (Figure 10(b)), the GPR model, and especially the MGGP model, are located closer to the target point, demonstrating more accurate efficiency than the GRNN model. Figure 11 depicts the relative deviation (RD) of all models versus the observed C d for training and testing dataset and for both scenarios. The smaller the RD range of the model, the higher efficiency of the model results.
According to Figure 11, in the first scenario, the RD for the GPR model is in the range of [−32, 32] and [−27, 21] for the training and testing phases and provides better predicted values as compared with the MGGP model (training [−75, 26], testing [−100, 75]) and the 58], testing: [−130, 25]). For the second scenario, the RD for the MGGP model is within the range of [−5.8, 8] for both the training and testing phases and provides less RD as compared with the MGGP ([−5.9, 9]) and the GRNN ([−6, 10]) models. Consequently, for the first and second scenarios the GPR and MGGP models are more accurate models for predicting C d .
Finally, the cumulative frequency against the absolute relative deviation (ARD) percentage is shown in Figure 12. For the first scenario, the figure clearly shows that the GRNN model can yield an ARD of approximately 2% for 80% predictive values, while the corresponding values for the GPR and MGGP models are 3% and 5.8%, respectively. In the case of second scenario, all models have almost the same performance. However, in the test phase, the MGGP model shows relatively better performance than the other two models.

Discussion
The aim of this study was to establish a better and reliable soft computing model for C d prediction, inspired by the necessity to have an accurate C d prediction as it can contribute to the design, control, and management of canals, weirs, and other hydraulic structures. The adopted predictive models (GPR, GRNN, and MGGP) were selected based on their remarkable progression in the research domain of hydraulic structure problems (Akbari et al., 2019;Dou et al., 2020;Ebtehaj, Bonakdari, Zaji, et al., 2015;Han et al., 2020;Salmasi et al., 2021;Yan & Mohammadian, 2019). However, in this study, the model for radial grate C d is new, as there are few studies in the literature for predicting it. The predictive models revealed a sophisticated prediction accuracy for two flow scenarios (free flow and submerged). However, there was a noticeable limitation in the prediction accuracy for the free flow scenario, which could best be explained by the fact that the free flow scenario had more complex phenomena and the type of the discharge stochasticity required more external variables to better comprehend the physical meaning. However, it is important to highlight that the results support the aim of providing a robust intelligence model. It is worthwhile mentioning that the experimental datasets were 458 and 2136 for free and submerged conditions, respectively. The data span indicated sufficient information to construct the predictive models. Figure 11. The relative deviation of all ML models in submerged scenario for training and testing phases.

Sensitivity analysis
The simulated C d is highly influenced by several physical variables, as the flow discharge is associated with several hydraulic variables. Hence, the sensitivity analysis to identify the parameters influencing the magnitude of the C d is an essential procedure for this type of modeling. For the sensitivity analysis, all the input variables were sequentially excluded and the effect of removing each of them in the GPR model was explored using statistical criteria for separately implemented models. This procedure was conducted for each scenario and the corresponding results were compared with the original situation (in the presence of all variables). Tables 7 and 8 report the statistical measures for the submerged and free flow scenarios over the testing modeling phase for the optimal predictive models. On the one hand, for the submerged flow Figure 12. The cumulative frequency of absolute relative deviation (%5) for each scenario and AI models. Table 7. The sensitivity analysis for submerged scenario in the testing stage using GPR model.  scenario, Y T /Y 0 is the influential variable in the C d computations. On the other hand, in the free flow scenario, W/Y 0 is a highly influential variable in the C d computations. In a more descriptive manner, the performance index is expressed as follows: PI is an evaluation indicator of the ath sensitivity analysis combination (see Figure 13). In order to address the effect of input parameters on the discharge coefficient for each scenario, an efficient contour plot, as shown in Figure 14, was produced on the basis of the most influential input variables versus the predicted Cd values. It should be mentioned that the outcomes of the best model in each scenario were considered, which included the GPR outcome for the submerged and MGGP outcome for the free type of flow. Figure 14(a) shows that, in the free flow scenario, the maximum discharge coefficient values were in ranges of W/Y 0 < 0.15 and h/Y 0 > 0.75, and the minimum values were in ranges of W/Y 0 > 0.7 and h/Y 0 < 1.75, respectively. In addition, according to Figure 14(b), for the submerged flow, the maximum and minimum C d values were in the ranges of 0.1 < W/Y 0 < 0.5 and 0.45 < Y t /Y 0 < 0.6 and 0.8 > W/Y 0 and Y t /Y 0 > 0.95, respectively.

Validation comparison with the literature review studies
It is worthwhile to validate the predictability performance of the models in this study, although a comparison with previously conducted ML models is not valid for all the cases, because the results of the predictive models varied from one case to another with respect to the range of the C d value, data span, and the maximum values of the C d . Nevertheless, the validation reported here provides a general perspective about the models' performances. A study was conducted on the C d prediction of a weir gate structure using genetic expression programming (GEP), multivariate adaptive regression spline (MARS), and group of method data handling (GMDH) models (Parsaie & Haghiabi, 2020), which attained correlation values of 0.92 (GEP), 0.94 (MARS), and 0.91 (GMDH). In another study (Ebtehaj, Bonakdari, Zaji, et al., 2015), the authors developed a GEP model for C d prediction of a rectangular slide weir and the proposed model attained a maximum correlation value of 0.93. It is clearly evidenced from the reported correlation values that the models in this study revealed better predictability performance as compared with those in the above-mentioned studies in the literature. Figure 13. The sensitivity analysis of each scenario in testing phase Figure 14. Variation of the most influential inputs versus predicted discharge coefficient (a) using GPR approach in the free condition and (b) using MGGP approach in the submerged condition.

Conclusions and future research direction
The current study investigated the reliability of new predictive models for discharge coefficient prediction of radial gate flow discharge for two different flow conditions, free and submerged. The motivation of the C d prediction was inspired from its significant contribution to diverse hydraulic engineering applications such as gate adjustment in irrigation canals and water discharge regulation control. Three ML models were developed for this purpose (i.e. GPR, GRNN, and MGGP). The experimental datasets were collected from the open access literature and used for the model development. The influential variables that are presented the flow and geometry used and determined based on the statistical correlation. The prediction accuracy was sufficient for both flow conditions. However, the predictive models reported better prediction accuracy for the free flow condition. The ranking of the accuracy of the prediction models was MGGP, GPR, and GRNN for the free flow condition and GPR, MGGP, and GRNN for the submerged flow. The sensitivity analysis was essential to determine the importance of the predictors, and thus W:Y o was identified as the influential variable for the C d prediction under the free flow condition, whereas Y o :Y T was the influential variable for the C d prediction under the submerged flow condition.