## Abstract

Accurate determination of discharge capacity in radial gates as commonly designed check structures is of great importance in hydraulic engineering research. The discharge coefficient plays the most dominant role in calculating the flow discharge through the radial gates. The main goal of this study is to adopt Grey Wolf Optimization-based Kernel Extreme Learning Machine (KELM-GWO) to further improve the prediction accuracy of the discharge coefficient of radial gates. To compare the supreme performance of the proposed model, kernel-depend support vector machine (SVM) and Gaussian process regression (GPR) were employed. An extensive field database consisting of 546 data samples gathered from different types of radial gates was established for building prediction models. The modeling results indicated that the proposed KELM-GWO model (correlation coefficient [*R*] = 0.927, and root mean squared error [RMSE] = 0.018) and SVM model (correlation coefficient [*R*] = 0.940, and root mean squared error [RMSE] = 0.022) demonstrated better performance under free and submerged flow conditions, respectively. Moreover, it was found that the applied kernel-depend approaches can be suitable options to predict the discharge coefficient of radial gates under varied submergence conditions with a satisfactory level of accuracy.

## HIGHLIGHTS

Grey Wolf Optimization (GWO) and Kernel Extreme Learning Machine (KELM) were introduced for discharge coefficient prediction of radial gates.

Support Vector Machine (SVM) and Gaussian Process Regression (GPR) were employed for comparative purposes.

546 filed data points from different types of radial gates used to feed the utilized models.

Best input combinations for both free and submerged flow conditions were discussed.

## INTRODUCTION

Radial gates are known as the most common elements of water control structures in use today and can be found in most irrigation networks across the world. They are constructed by the use of a curved skin plate supported by a structural steel frame and play an essential role in facilitating water delivery, maintaining river environments, and controlling flooding. These types of gates are more cost-effective and simpler to utilize and maintain in comparison to sluice gates (Clemmens *et al.* 2003; Shahrokhnia & Javan 2006). The performance accuracy of radial gates as check structures can vary according to the type of flow condition (i.e., free or submerged flow). Under a free-flow condition, a free hydraulic jump is developed in the downstream part of the radial gate, and the flow is just affected by the upstream flow depth. On the other hand, with increasing tail-water depth, the jet emanating from the radial gate changes to partially submerged flow with an incomplete submerged jump and totally submerged flow with no jump. The discharge coefficient of gates constitutes the basis for the determination of the opening, which is of considerable importance for accurate simulation and controlling of canal flow. Furthermore, accurate determination of discharge is crucial for informing water-saving policies and efficient operational management. Numerous studies have attempted to enhance the discharge calibration of radial gates. Early work can be attributed to Metzler (1948), who developed a series of maps to determine the discharge coefficient of radial gates under submerged flow conditions. Toch (1955) and Buyalski (1983) applied the empirical relationship method in order to illustrate the variation of the discharge coefficients for both free and submerged flow conditions. Clemmens *et al.* (2003) acknowledged the inefficiency of empirical relationship methods in calibrating submerged radial gates and developed the Energy-Momentum (E-M) approach in order to calibrate the discharge through a single radial gate. Using Buyalski's experimental data, Wahl (2005) introduced relative gate opening to modify the correction factor. Shahrokhnia & Javan (2006) and Zahedani *et al.* (2012) derived equations for quantifying the submerged flow rate on the basis of the downstream flow depth using dimensional analysis and nonlinear regression. Abdelhaleem (2016) conducted research on a discharge estimation of three submerged parallel radial gates and found that the dimensional analysis method with the incomplete self-similarity concept excels any other calibration methods. The practical implementation of the aforementioned discharge calibration methods indicated inadequate performance when subjected to highly submerged flow. More recently, Guo *et al.* (2021a, 2021b, 2021c) proposed new criteria in order to subdivide the submerged flow into partially submerged and totally submerged flow. Using a new discharge calibration method, expressed as the identification method, they asserted that considering the classification method can satisfactorily enhance the discharge model accuracy for totally submerged flow.

Based on the aforementioned discussion, assessment of the hydraulic operation of radial gates can be implemented following different approaches: the energy balance approach, the momentum approach, and empirical equations. The literature demonstrates that employed approaches to the analysis of radial gates are unsuccessful in providing stable performance under various hydraulic conditions and may lead to unsatisfactory results. In such cases, machine learning (ML) methods can be a suitable tool to build hydraulic models of radial gates solutions under different hydraulic conditions. The high efficiency and speed of ML methods in modeling complex hydraulic problems have made them one of the most widely used methods in predicting the discharge coefficient of different types of weirs. Among them, the strong suitability of the Artificial Neural Network (ANN) algorithm for hydraulic problems has made it one of the leading models in the prediction of the discharge coefficient of weirs (Bilhan *et al.* 2011; Salmasi *et al.* 2013; Karami *et al.* 2018). However, the need to define the structure of ANN before training, as well as the possibility of getting trapped in local minima in the model training process, are among the reasons that encourage researchers to use an appropriate combination of neural and fuzzy systems in the form of Adaptive Neural Fuzzy Inference Systems (ANFIS) (Shamshirband *et al.* 2016; Haghiabi *et al.* 2018). On the other hand, the use of the kernel function as the core tool of ML methods increases their efficiency in terms of reducing overfitting. In this regard, the successful application of kernel-depend Support Vector Machine (SVM) (Azamathulla *et al.* 2016; Roushangar *et al.* 2021a, 2021b) and Gaussian Process Regression (GPR) (Akbari *et al.* 2019; Nourani *et al.* 2021) were reported in the literature. Kernel-depend methods are memory intensive, and trickier to tune due to the importance of picking the right kernel. In these models, we will be able to predict the proper behavior of the system, although we will not be able to characterize its intrinsic structure and behavior. A kernel method is an algorithm that depends on the data only through dot-products. When this is the case, the dot product can be replaced by a kernel function which computes a dot product in some possibly high-dimensional feature space (Smola 1996). This has two advantages: First, the ability to generate nonlinear decision boundaries using methods designed for linear classiﬁers. Second, the use of kernel functions allows the user to apply a classiﬁer to data that have no obvious ﬁxed-dimensional vector space representation.

Prediction of the discharge coefficient is required as a basis for discharge measurements in irrigation channels and plays a fundamental role in evaluating the hydraulic efficiency of radial gates. To the best of the authors’ knowledge, ML techniques have not been developed in design of the discharge coefficient of radial gates with an extensive number of field datasets. Therefore, the authors are eager to evaluate the ability of powerful kernel-depend approaches in the prediction of the discharge coefficient of radial gates. As major advantages, kernel-depend models have two main capabilities: (i) these methods can model nonlinear decision boundaries, and there are many kernels to choose from. They are also robust against overfitting, especially in high-dimensional space, (ii) in these models, we will be able to predict the proper behavior of the system, although we will not be able to characterize its intrinsic structure and behavior. Considering the advantages of kernel-depend methods, for the first time, Kernel Extreme Learning Machine (KELM), in combination with Grey Wolf Optimization (GWO), is developed for modeling the discharge coefficient of radial gates under free and submerged flow conditions. The performances of the proposed hybrid KELM-GWO model were compared with GPR and SVM that developed to this end, as well.

The rest of this paper is organized as follows. First, an extensive field database and the location of studied radial gates are presented. The development of dimensionless parameters under different hydraulic conditions is presented along with the corresponding histograms. After that, brief reviews of the theorem associated with the SVM, GPR, KELM, and GWO algorithms are presented, and consequently, the structure of the proposed KELM-GWO model and its optimization strategy is described. Next, the results are presented in terms of qualitative and quantitative performance. The conclusions are drawn in the final step of this research.

## METHODS

Check structure . | Seal type . | Number of gates . | Maximum upstream depth (m) . | Maximum downstream depth (m) . | Gate radius (m) . | Pinion height, α (m)
. | Gate width (each) (m) . | Number of data . |
---|---|---|---|---|---|---|---|---|

Velocity Barrier | Hard Rubber | 3 | 3.29 | 2.51 | 4.20 | 2.74 | 4.27 | 38 |

Fish Screen | Hard Rubber | 3 | 4.26 | 4.26 | 4.96 | 3.20 | 4.27 | 8 |

Coyote Canal | Hard Rubber | 2 | 4.26 | 4.26 | 6.48 | 4.11 | 5.49 | 4 |

West Canal | Hard Rubber and Music Note | 3 | 400.31 | 400.31 | 6.10 | 3.66 | 4.65 | 82 |

Putah South | Music Note | 2 | 3.13 | 3.13 | 3.44 | 2.13 | 3.05 | 4 |

Sand Creek | Music Note | 3 | 5.28 | 5.28 | 6.48 | 4.11 | 6.10 | 20 |

Dodge Avenue | Music Note | 3 | 5.28 | 5.28 | 6.48 | 4.11 | 6.10 | 22 |

Kaweah River | Music Note | 5 | 5.19 | 5.19 | 7.93 | 5.12 | 3.35 | 35 |

Fifth Avenue | Music Note | 3 | 5.19 | 5.19 | 6.48 | 4.11 | 5.49 | 9 |

Tule River | Music Note | 4 | 5.25 | 5.25 | 7.93 | 5.12 | 3.66 | 6 |

California Aqueduct | Hard Rubber | 3 | 8.11 | 8.11 | 11.44 | 7.62 | 7.62 | 201 |

Check structure . | Seal type . | Number of gates . | Maximum upstream depth (m) . | Maximum downstream depth (m) . | Gate radius (m) . | Pinion height, α (m)
. | Gate width (each) (m) . | Number of data . |
---|---|---|---|---|---|---|---|---|

Velocity Barrier | Hard Rubber | 3 | 3.29 | 2.51 | 4.20 | 2.74 | 4.27 | 38 |

Fish Screen | Hard Rubber | 3 | 4.26 | 4.26 | 4.96 | 3.20 | 4.27 | 8 |

Coyote Canal | Hard Rubber | 2 | 4.26 | 4.26 | 6.48 | 4.11 | 5.49 | 4 |

West Canal | Hard Rubber and Music Note | 3 | 400.31 | 400.31 | 6.10 | 3.66 | 4.65 | 82 |

Putah South | Music Note | 2 | 3.13 | 3.13 | 3.44 | 2.13 | 3.05 | 4 |

Sand Creek | Music Note | 3 | 5.28 | 5.28 | 6.48 | 4.11 | 6.10 | 20 |

Dodge Avenue | Music Note | 3 | 5.28 | 5.28 | 6.48 | 4.11 | 6.10 | 22 |

Kaweah River | Music Note | 5 | 5.19 | 5.19 | 7.93 | 5.12 | 3.35 | 35 |

Fifth Avenue | Music Note | 3 | 5.19 | 5.19 | 6.48 | 4.11 | 5.49 | 9 |

Tule River | Music Note | 4 | 5.25 | 5.25 | 7.93 | 5.12 | 3.66 | 6 |

California Aqueduct | Hard Rubber | 3 | 8.11 | 8.11 | 11.44 | 7.62 | 7.62 | 201 |

Check structure . | Seal type . | Number of gates . | Maximum upstream depth (m) . | Gate radius (m) . | Pinion height, α (m)
. | Gate width (each) (m) . | Number of data . |
---|---|---|---|---|---|---|---|

East Canal Headworks | Music Note | 2 | 6.32 | 8.01 | 4.57 | 4.88 | 37 |

Dez Irrigation Network, Main West Canal | Sharp | 2 | 3 | 7 | 3.62 | 15 | 80 |

Check structure . | Seal type . | Number of gates . | Maximum upstream depth (m) . | Gate radius (m) . | Pinion height, α (m)
. | Gate width (each) (m) . | Number of data . |
---|---|---|---|---|---|---|---|

East Canal Headworks | Music Note | 2 | 6.32 | 8.01 | 4.57 | 4.88 | 37 |

Dez Irrigation Network, Main West Canal | Sharp | 2 | 3 | 7 | 3.62 | 15 | 80 |

^{3}/s. It should be noted that the field data were gathered from a broad number of parallel gates with various gate openings, seal types, sill levels relative to the downstream channel bed, and flow conditions. The radial gate openings can be different from each other in cases such as the low discharge. In this case, some gates may be open while others are closed (Figure 2). Furthermore, in the case of simultaneous applications, different openings of the radial gates can lead to different flow characteristics in each gate as free, submerged, or transition flows. The gates can also be designed at higher or lower levels than the upstream or downstream channels or can be influenced by the curvature of the approaching flow lines in a channel intake. These concerns can challenge the accuracy of the proposed kernel-depend approaches to predict the discharge of radial gates.

*y*

_{1}is the upstream depth (m),

*y*

_{2}is the water depth at the vena contracta (m),

*y*

_{3}is the downstream flow depth (m),

*w*is the average gate opening (m),

*r*is the radial gate radius (m),

*θ*is the gate leaf angle from horizontal, and

*α*is the gate trunnion height above the invert (m). The discharge coefficient of radial gates under free and submerged flow conditions was obtained through and , respectively, where,

*q*is the discharge per unit width (m

^{2}/s),

*C*

_{d}is the discharge coefficient, and

*g*is the gravitational constant (m/s

^{2}).

### Input parameters

*et al.*2012; Bijankhan

*et al.*2013):

In which *Q* is the flow discharge, *y*_{1} is the upstream flow depth, *y*_{3} is the downstream flow depth, *L* is the channel width, *w* is the gate opening, is the water viscosity and *ρ* is the water density. Dimensional analysis may reduce the dimensions of the input matrix, so would create a low-dimensions space where the number of the studied parameters is low. Moreover, further application of experimental results can be achieved through the dimensionless groupings of variables (Granata & de Marinis 2017).

*w/r*,

*y*

_{1}

*/r*,

*y*

_{3}

*/r*,

*a/r*, and

*θ*on the discharge coefficient of radial gates. The gate leaf angle from horizontal can be expressed as a function of the radial gate radius (

*r*), the gate opening (

*w*), and the gate trunnion height above the invert (

*α*). Neglecting the influence of the Reynolds number and rearranging the dimensionless groups’ yields:

Selected input combinations for the prediction of discharge coefficient under both free and submerged conditions were extracted through a large number of trial-and-error processes and listed in Table 3. Statistical information of input and output parameters in terms of the maximum (max), minimum (min), mean and standard deviation (S.td.) values is represented in Table 4.

Free flow . | Submerged flow . | ||
---|---|---|---|

Model . | Input parameters . | Model . | Input parameters . |

F(I) | S(I) | ||

F(II) | S(II) | ||

F(III) | S(III) | ||

F(IV) | S(IV) |

Free flow . | Submerged flow . | ||
---|---|---|---|

Model . | Input parameters . | Model . | Input parameters . |

F(I) | S(I) | ||

F(II) | S(II) | ||

F(III) | S(III) | ||

F(IV) | S(IV) |

. | . | Parameter . | Min . | Max . | Mean . | St.d. . |
---|---|---|---|---|---|---|

Free flow | Input parameters | 0.255 | 0.559 | 0.418 | 0.063 | |

0.065 | 0.788 | 0.442 | 0.152 | |||

1.54 | 64.55 | 6.06 | 8.50 | |||

0.011 | 0.315 | 0.115 | 0.068 | |||

Output parameter | 0.432 | 0.684 | 0.597 | 0.047 | ||

Submerged flow | Input parameters | −0.949 | 28.5 | 2.37 | 4.58 | |

0.568 | 1.76 | 0.776 | 0.299 | |||

0.299 | 0.845 | 0.584 | 0.123 | |||

1.09 | 37.03 | 5.02 | 4.59 | |||

0.010 | 0.642 | 0.187 | 0.119 | |||

Output parameter | 0.005 | 6.14 | 0.616 | 0.376 |

. | . | Parameter . | Min . | Max . | Mean . | St.d. . |
---|---|---|---|---|---|---|

Free flow | Input parameters | 0.255 | 0.559 | 0.418 | 0.063 | |

0.065 | 0.788 | 0.442 | 0.152 | |||

1.54 | 64.55 | 6.06 | 8.50 | |||

0.011 | 0.315 | 0.115 | 0.068 | |||

Output parameter | 0.432 | 0.684 | 0.597 | 0.047 | ||

Submerged flow | Input parameters | −0.949 | 28.5 | 2.37 | 4.58 | |

0.568 | 1.76 | 0.776 | 0.299 | |||

0.299 | 0.845 | 0.584 | 0.123 | |||

1.09 | 37.03 | 5.02 | 4.59 | |||

0.010 | 0.642 | 0.187 | 0.119 | |||

Output parameter | 0.005 | 6.14 | 0.616 | 0.376 |

*y*

_{1}

*/r*is satisfactorily large (i.e., from 0.1 to 0.8) with an approximately uniform frequency distribution within it (skewness = 0.91). On the contrary, the frequency distribution for other dimensionless namely, the ratio of the upstream flow depth to the gate opening

*y*

_{1}

*/w*is asymmetric and positively skewed (skewness = 5.21). The tail of the distribution tends to the largest value of 64.6, but most of the samples (around 84%) were assisted with

*y*

_{1}

*/w*around 1.5–7.8. Finally, Figure 4 shows that under submerged hydraulic conditions, the ratio of the downstream flow depth to the gate opening (

*y*

_{3}/w) had an asymmetric distribution of the frequency, with the highest fraction of the relative frequency (70%) in the range of (skewness = 2.88). The distribution pattern of the submergence ratio parameter (

*y*

_{1}–

*y*

_{3}/w) indicated that the majority of samples (around 70%) were conducted with

*y*

_{1}

*-y*

_{3}

*/w*around (−0.9 to 2), which confirms the rule of submerged flow conditions. Moreover, although the distribution of the

*w/r*and

*y*

_{3}

*/r*parameters is fully fragmented, the pattern of the distributions is not symmetrical.

### Support vector machine

SVM is known as one of the most potent kernel-depend models in the predicting field rooted in structural risk minimization (Vapnik 1999). This capability grants SVM a better generalizability to overcome the shortcomings of traditional ML approaches such as the ANN algorithm. Using the concept of kernel functions, SVM maps the complicated nonlinear problem input factors into high-dimension space to transform complicated nonlinear problems into linear problems. Therefore, the kernel function plays a significant role in the performance of SVM. The central concept of SVM is based on establishing an optimal hyperplane as the decision surface that maximizes the separation edge between the positive case and the counter-example. Introducing the kernel function neatly solves the inner product operation in the high-dimensional space and is one of the main advantages of SVM. Therefore, the kernel function plays a significant role in the performance of SVM. In the present study, the prediction performance of employed SVM methods is discussed with different kernel functions including linear, polynomial, sigmoid and Radial Basis Function (RBF). The optimum values of kernel parameters were obtained through a trial-and-error process. Furthermore, optimization of SVM hyper-parameters has been carried out by a systematic grid search of the parameters using cross-validation on the training dimensionless measures.

### Gaussian process regression

*x*and

*y*refer to the input and output vectors, respectively;

*f*stands as the function values of GPR and

*ε*is noise. In GPR models, patterns are captured in the data by means of the covariance function. Various types of covariance (kernel) functions can be implemented in the core tool of GPR model as expressed in Equations (11)–(16):

The hyper-parameters can be calculated via Equation (19) utilizing a gradient descent algorithm.

### Kernel extreme learning machine

*β*stands as the connection weight between the hidden layer and the

_{j}*i*th output,

*ω*stands as the connection weight between the hidden layer and the inputs,

_{j}*b*stands as the bias of the

_{j}*j*th hidden neurons,

*L*Stands as the number of the hidden nodes, and

*g*is the activation function. The network architecture of ELM model is depicted in Figure 6.

*(*.*)*Therefore, *β* can be obtained byusing the formula , where stands as the pseudo-inverse of *H*.

*h(x)*is unknown, the kernel-depend ELM model can be built as follows (Huang

*et al.*2011):where

*i,j*= 1,2, …,

*M*.

### Grey wolf optimization

*et al.*(2014) proposed GWO as a novel metaheuristic algorithm. GWO is inspired by two social modes. The first is hierarchical. The pack of grey wolves is divided into four levels as depicted in Figure 7 based on their social rank. The highest level is alpha (

*α*) and plays the leadership role. The second is hunting. The GWO is one of the commonly used metaheuristic algorithms that has been implemented in the areas of water engineering, parameter optimization and image classification (Guo

*et al.*2021a, 2021b, 2021c; Roushangar

*et al.*2021a, 2021b).

Here, the GWO algorithm is employed to find the optimum values of KELM hyper-parameters (kernel parameters and regularization coefficient). The optimized parameters indicate the prey and the dynamic optimization process of the algorithm represents the grey wolves hunting.

*n*denotes the number of iterations, and denote the position vector of the prey and grey wolf, respectively; and denote the coefficient vectors as follows:where component of is linearly reduced from 2 to 0 during the iteration, and , are random vectors in [0,1].

GWO algorithm ends with the completion of the maximum number of iterations. The return value of *α* is the optimum value of kernel parameters and regularization coefficient.

### Modeling framework

^{®}platform. Optimal adjustment of constant parameters including the regularization coefficient and especially the kernel parameters can substantially influence the prediction performance of the proposed KELM-GWO model. The best values of the above-mentioned effective hyper-parameters were attained through the GWO algorithm. Consequently, the GWO was coupled with the KELM model to improve the prediction capability of the discharge coefficient of the radial gates. The flowchart of the proposed hybrid model is shown in Figure 9. The data sets for both free and submerged flow conditions were partitioned into two sub-sets. The first separation as 75% of the total data were associated with the training set while the remaining 25% of data were used for testing goals.

### Performance evaluation criteria

*R*), the Scatter Index (SI), BIAS, and the Root Mean Square Error (RMSE) were utilized as statistical indexes for evaluating the models (Najafzadeh

*et al.*2018; Saberi-Movahed

*et al.*2020).

## RESULTS AND DISCUSSIONS

Owing to the importance of choosing the proper kernel function in modeling accuracy, in the first step, different structures of employed kernel-depend approaches were analyzed through various kernel functions. Table 5 lists the obtained results of employed kernel-depend approaches with different kernels for models F(IV) and S(IV). Moreover, optimal parameters of employed models are listed in Tables 6–8, respectively.

Model . | Method . | Kernel type . | Performance criteria . | |||
---|---|---|---|---|---|---|

Testing . | ||||||

R
. | SI . | BIAS . | RMSE . | |||

F(IV) | KELM-GWO | Linear | 0.609 | 0.049 | 0.000473 | 0.038 |

Polynomial | .0927 | .0023 | .−0005946 | .0018 | ||

RBF | 0.490 | 0.067 | −0.010038 | 0.052 | ||

Wavelet | 0.725 | 0.040 | −0.001569 | 0.031 | ||

GPR | Exponential | .0911 | .0023 | .−0002162 | .0018 | |

Squared Exponential | 0.899 | 0.025 | −0.003053 | 0.019 | ||

Matern 3/2 | 0.900 | 0.025 | −0.002817 | 0.019 | ||

Matern 5/2 | 0.899 | 0.025 | −0.003014 | 0.019 | ||

Rational Quadratic | 0.899 | 0.025 | −0.003053 | 0.019 | ||

SVM | Linear | 0.785 | 0.037 | 0.005165 | 0.029 | |

Polynomial | 0.620 | 0.045 | −0.002548 | 0.035 | ||

RBF | .0896 | .0027 | .−0001713 | .0021 | ||

Sigmoid | 0.403 | 1.01 | −0.131587 | 0.785 | ||

S(IV) | KELM-GWO | Linear | 0.085 | 0.328 | 0.019855 | 0.254 |

Polynomial | 0.365 | 0.341 | 0.035280 | 0.266 | ||

RBF | .0839 | .0185 | .−0027257 | .0144 | ||

Wavelet | 0.839 | 0.185 | −0.027258 | 0.144 | ||

GPR | Exponential | .0944 | .0102 | .0009989 | .0079 | |

Squared Exponential | 0.877 | 0.156 | 0.000228 | 0.121 | ||

Matern 3/2 | 0.931 | 0.117 | 0.005591 | 0.090 | ||

Matern 5/2 | 0.917 | 0.130 | 0.003291 | 0.100 | ||

Rational Quadratic | 0.877 | 0.156 | 0.000228 | 0.121 | ||

SVM | Linear | 0.158 | 0.310 | −0.026804 | 0.239 | |

Polynomial | 0.559 | 1.33 | 0.257247 | 1.026 | ||

RBF | .0940 | .0106 | .−0000572 | .0022 | ||

Sigmoid | 0.064 | 0.315 | −0.002886 | 0.243 |

Model . | Method . | Kernel type . | Performance criteria . | |||
---|---|---|---|---|---|---|

Testing . | ||||||

R
. | SI . | BIAS . | RMSE . | |||

F(IV) | KELM-GWO | Linear | 0.609 | 0.049 | 0.000473 | 0.038 |

Polynomial | .0927 | .0023 | .−0005946 | .0018 | ||

RBF | 0.490 | 0.067 | −0.010038 | 0.052 | ||

Wavelet | 0.725 | 0.040 | −0.001569 | 0.031 | ||

GPR | Exponential | .0911 | .0023 | .−0002162 | .0018 | |

Squared Exponential | 0.899 | 0.025 | −0.003053 | 0.019 | ||

Matern 3/2 | 0.900 | 0.025 | −0.002817 | 0.019 | ||

Matern 5/2 | 0.899 | 0.025 | −0.003014 | 0.019 | ||

Rational Quadratic | 0.899 | 0.025 | −0.003053 | 0.019 | ||

SVM | Linear | 0.785 | 0.037 | 0.005165 | 0.029 | |

Polynomial | 0.620 | 0.045 | −0.002548 | 0.035 | ||

RBF | .0896 | .0027 | .−0001713 | .0021 | ||

Sigmoid | 0.403 | 1.01 | −0.131587 | 0.785 | ||

S(IV) | KELM-GWO | Linear | 0.085 | 0.328 | 0.019855 | 0.254 |

Polynomial | 0.365 | 0.341 | 0.035280 | 0.266 | ||

RBF | .0839 | .0185 | .−0027257 | .0144 | ||

Wavelet | 0.839 | 0.185 | −0.027258 | 0.144 | ||

GPR | Exponential | .0944 | .0102 | .0009989 | .0079 | |

Squared Exponential | 0.877 | 0.156 | 0.000228 | 0.121 | ||

Matern 3/2 | 0.931 | 0.117 | 0.005591 | 0.090 | ||

Matern 5/2 | 0.917 | 0.130 | 0.003291 | 0.100 | ||

Rational Quadratic | 0.877 | 0.156 | 0.000228 | 0.121 | ||

SVM | Linear | 0.158 | 0.310 | −0.026804 | 0.239 | |

Polynomial | 0.559 | 1.33 | 0.257247 | 1.026 | ||

RBF | .0940 | .0106 | .−0000572 | .0022 | ||

Sigmoid | 0.064 | 0.315 | −0.002886 | 0.243 |

Italic values indicate superior results.

Method . | Model . | Kernel type . | Parameter . | ||||||
---|---|---|---|---|---|---|---|---|---|

Ρ
. | c
. | d
. | γ
. | ω
. | α
. | β
. | |||

KELM-GWO | F(IV) | Linear | 9.2969 | – | – | – | – | – | – |

Polynomial | 1 | 4.6017 | 5.1627 | – | – | – | – | ||

RBF | 201 | – | 10 | – | – | – | |||

Wavelet | 160.8304 | – | – | – | 7.8560 | 5.3051 | 8.6632 | ||

S(IV) | Linear | 1.6319 | – | – | – | – | – | – | |

Polynomial | – | – | – | – | |||||

RBF | 201 | – | – | 0.1037 | – | – | – | ||

Wavelet | 201 | – | – | – | 0.1036 | 10 | 0.2307 |

Method . | Model . | Kernel type . | Parameter . | ||||||
---|---|---|---|---|---|---|---|---|---|

Ρ
. | c
. | d
. | γ
. | ω
. | α
. | β
. | |||

KELM-GWO | F(IV) | Linear | 9.2969 | – | – | – | – | – | – |

Polynomial | 1 | 4.6017 | 5.1627 | – | – | – | – | ||

RBF | 201 | – | 10 | – | – | – | |||

Wavelet | 160.8304 | – | – | – | 7.8560 | 5.3051 | 8.6632 | ||

S(IV) | Linear | 1.6319 | – | – | – | – | – | – | |

Polynomial | – | – | – | – | |||||

RBF | 201 | – | – | 0.1037 | – | – | – | ||

Wavelet | 201 | – | – | – | 0.1036 | 10 | 0.2307 |

Method . | Model . | Kernel type . | Parameter . | ||
---|---|---|---|---|---|

σ
. _{L} | σ
. _{F} | Log likelihood . | |||

GPR | F(IV) | Exponential | .106417 | 0.0700 | 168.7399 |

Squared Exponential | 1.9743 | 0.0793 | .1709388 | ||

Matern 3/2 | 4.3681 | 0.0799 | 170.8195 | ||

Matern 5/2 | 3.1036 | 0.0792 | 171.2564 | ||

Rational Quadratic | 1.9743 | 0.0793 | 171.2564 | ||

S(IV) | Exponential | 0.2721 | 0.4724 | 57.1633 | |

Squared Exponential | 0.1200 | 0.5359 | 90.8263 | ||

Matern 3/2 | 0.1618 | 0.5011 | 75.6284 | ||

Matern 5/2 | 0.1439 | 0.5154 | 82.3447 | ||

Rational Quadratic | 0.1200 | 0.5359 | 90.8263 |

Method . | Model . | Kernel type . | Parameter . | ||
---|---|---|---|---|---|

σ
. _{L} | σ
. _{F} | Log likelihood . | |||

GPR | F(IV) | Exponential | .106417 | 0.0700 | 168.7399 |

Squared Exponential | 1.9743 | 0.0793 | .1709388 | ||

Matern 3/2 | 4.3681 | 0.0799 | 170.8195 | ||

Matern 5/2 | 3.1036 | 0.0792 | 171.2564 | ||

Rational Quadratic | 1.9743 | 0.0793 | 171.2564 | ||

S(IV) | Exponential | 0.2721 | 0.4724 | 57.1633 | |

Squared Exponential | 0.1200 | 0.5359 | 90.8263 | ||

Matern 3/2 | 0.1618 | 0.5011 | 75.6284 | ||

Matern 5/2 | 0.1439 | 0.5154 | 82.3447 | ||

Rational Quadratic | 0.1200 | 0.5359 | 90.8263 |

Italic values indicate superior results.

Method . | Model . | Kernel type . | Parameter . | ||||
---|---|---|---|---|---|---|---|

c
. | d
. | γ
. | C
. | ε
. | |||

SVM | F(IV) | Linear | – | – | – | 1 | 0.001 |

Polynomial | 3 | 3 | 50 | 1 | 0.001 | ||

RBF | – | – | 200 | 1 | 0.001 | ||

Sigmoid | 2 | – | 200 | 14 | 0.4 | ||

S(IV) | Linear | – | – | – | 12 | 0.001 | |

Polynomial | 1 | 5 | 150 | 0 | 0.201 | ||

RBF | – | – | 90 | 3 | 0.001 | ||

Sigmoid | 2 | – | – | 1 | 0.001 |

Method . | Model . | Kernel type . | Parameter . | ||||
---|---|---|---|---|---|---|---|

c
. | d
. | γ
. | C
. | ε
. | |||

SVM | F(IV) | Linear | – | – | – | 1 | 0.001 |

Polynomial | 3 | 3 | 50 | 1 | 0.001 | ||

RBF | – | – | 200 | 1 | 0.001 | ||

Sigmoid | 2 | – | 200 | 14 | 0.4 | ||

S(IV) | Linear | – | – | – | 12 | 0.001 | |

Polynomial | 1 | 5 | 150 | 0 | 0.201 | ||

RBF | – | – | 90 | 3 | 0.001 | ||

Sigmoid | 2 | – | – | 1 | 0.001 |

The results given in Table 5 demonstrated that under free-flow conditions, the proposed KELM-GWO model with polynomial kernel function surpasses all other approaches with values of *R* equal to 0.927, SI = 0.023, BIAS = −0.005946, and RMSE = 0.018 for the testing phase. In order to implement the proposed KELM-GWO, the population (pack size) was applied as 300, and iteration was set to 50. The regulation coefficient parameter of *ρ* and polynomial kernel parameters, including the free parameter of *c* and polynomial degree (*d*) were optimized through the GWO algorithm as *ρ* = 1, *C* = 4.6017 and *d* = 5.1627, respectively. To tune the GRP hyper-parameters, a Standard Gradient Descent (SGD) optimizer was utilized to maximize the log marginal likelihood. In consideration of the SI values, small variations can be observed throughout the employed kernel functions. GPR performance ranges between 0.025 and 0.023 (for Exponential kernel). Using the SGD optimizer, the ideal values of Exponential kernel parameters were obtained as the length scale parameter (*σ*_{l}) equal to 10.6417 and the signal standard deviation (*σ*_{f}) equal to 0.07. For the implementation of the embedded SVM method, the best values of kernel parameters were achieved through a trial-and-error procedure. Additionally, the optimization process of the cost factor (*C*) and the loss function (*ε*) was conducted by a grid search in the *C* and *ε* parameter space using the cross-validation technique. The result proved that SVM-RBF had the best prediction accuracy (*R* = 0.896, SI = 0.027, BIAS = −0.001713, and RMSE = 0.021) as suggested by researchers (Komasi *et al.* 2018; Najafzadeh & Oliveto 2020; Amininia & Saghebian 2021; Roushangar & Shahnazi 2021). It was followed by a linear kernel function (*R* = 0.785, SI = 0.037, BIAS = 0.005165, and RMSE = 0.029). The results showed that using the sigmoid kernel function as a core tool of SVM led to under performance. Taking into account the obtained results, kernel functions of polynomial, Exponential, and RBF were selected as core tools of KELM-GWO, GPR, and SVM methods, respectively, to predict the discharge coefficient of radial gates under free-flow conditions. Furthermore, model S(IV) was selected in order to better understand the performance of the employed kernel-depend approaches under submerged flow conditions. Based on the results of Table 5, the SVM model with the kernel function of RBF can predict the discharge coefficient of submerged radial gates better than other employed kernel-depend methods with *R* = 0.940, SI = 0.106, BIAS = −0.000572, and RMSE = 0.022. It was followed by GPR-Exponential (*R* = 0.944, SI = 0.102, BIAS = 0.009989, and RMSE = 0.079). As can be seen, in contrast to the free-flow condition, the proposed KELM-GWO model with polynomial kernel had poor performance, and RBF kernel function performs better structure of KELM-GWO. However, Table 5 suggests that the proposed KELM-GWO model has a lower efficiency in comparison with other kernel-depend methods for the prediction of the discharge coefficient of submerged radial gates. As a result, for submerged flow conditions, KELM-GWO-RBF, GPR-Exponential, and SVM-RBF were used for the rest of the input combinations. According to the results presented in Table 5, it was found that two chief factors including the kernel type and good setting of parameters play a key role in the improving accuracy level of kernel-depend models. However, there is no method to determine how to choose an appropriate kernel and its parameters for a given dataset to achieve high generalization performance. On the other hand, coupling kernel-based methods with different optimization methods create more complex models and requires longer computation time.

*γ*), which may cause under-fitting and overfitting problems (Roushangar & Shahnazi 2020a, 2020b). Figure 10 illustrates the statistical parameter of RMSE via gamma values of the SVM models (models F(IV) and S(IV)). From the figure, it can be observed that the accuracy of the SVM model shows various behavior with changes in gamma values for different developed models. Fluctuations with the introduction of model S(IV) as an input to SVM are more evident. The best-fitting gamma values for the F(IV) are obtained when

*γ*≥ 160. The SVM model with small values of

*γ*has tended to overfit, or memorize data. Thus, for solving the objective problem (fed with model F(IV)) with employed datasets, the SVM model with gamma values less than 160 leads to overfitting. According to the output, the ideal value of the RBF kernel parameter was obtained

*γ*= 200 for model F(IV). The optimal hyper-parameters of SVM for model F(IV) were achieved as (

*C*= 1,

*ε*= 0.001). On the other hand, optimum

*γ*was 90 for model S(IV) and hyper-parameters of SVM were optimized as

*C*= 3, and

*ε*= 0.001 for this model. As can be seen, there is no definite relationship between gamma value and assessing parameters and may exhibit different performances for different inputs.

Table 9 provides the statistical indicators of the employed kernel-depend models during the testing phase. It can be seen that under free-flow conditions, using the ratio of the downstream flow depth to the gate opening (*y*_{3}/*w*) as the only input parameter provided the desired prediction accuracy while using only the ratio of the downstream flow depth to the radial gate radius (*y*_{3}/*r*) as input did not yield the appropriate prediction accuracy. This can show the importance of the gate opening on the prediction process of the discharge coefficient under free-flow conditions. According to SI values, when comparing F(II) and F(III), adding the ratio of the gate opening to the radial gate radius (*w*/*r*) improves the average prediction accuracy of employed KELM-GWO and GPR approaches by 4 and 7%, respectively. It can be seen that the parameter *y*_{1}*/w* is the most influential parameter for the prediction of the discharge coefficient of radial gates under free-flow conditions. Taking into consideration the statistical indices, KELM-GWO-Polynomial and GPR-Exponential models generated the most accurate results with model F(IV), where *y*_{1}*/w*, *w/r*, and *α-w/r* were utilized as the input parameters. As mentioned in the previous section, under free-flow conditions, the comparison results of the employed kernel-depend methods revealed that the hybridized KELM-GWO-Polynomial with model F(IV) as input combination outperformed the SVM and GPR standalone models with *R* = 0.927, SI = 0.023, BIAS = −0.005946, and RMSE = 0.018. It should be noticed that the prediction of discharge coefficient under field conditions for multiple gates with different openings is complicated. It may become more intractable if one gate is free-flowing and another in the transition zone.

Flow condition . | Method . | Model . | Performance criteria . | |||
---|---|---|---|---|---|---|

Testing . | ||||||

R
. | SI . | BIAS . | RMSE . | |||

Free flow | KELM-GWO | F(I) | 0.305 | 0.057 | −0.005410 | 0.044 |

F(II) | 0.866 | 0.028 | 0.002324 | 0.022 | ||

F(III) | 0.907 | 0.026 | 0.008926 | 0.021 | ||

F(IV) | .0927 | .0023 | .−0005946 | 0.018 | ||

GPR | F(I) | 0.549 | 0.050 | 0.000421 | 0.038 | |

F(II) | 0.896 | 0.025 | −0.003127 | 0.019 | ||

F(III) | 0.908 | 0.024 | −0.002365 | 0.018 | ||

F(IV) | .0911 | .0023 | .−0002162 | 0.018 | ||

SVM | F(I) | 0.442 | 0.052 | −0.004215 | 0.040 | |

F(II) | 0.870 | 0.028 | 0.002492 | 0.022 | ||

F(III) | 0.880 | 0.028 | −0.004524 | 0.021 | ||

F(IV) | .0896 | .0027 | .−0001713 | 0.021 | ||

Submerged flow | KELM-GWO | S(I) | 0.657 | 0.276 | 0.080632 | 0.214 |

S(II) | 0.680 | 0.254 | 0.031145 | 0.197 | ||

S(III) | 0.767 | 0.255 | 0.035795 | 0.197 | ||

S(IV) | .0839 | .0185 | .−0027117 | 0.144 | ||

GPR | S(I) | 0.727 | 0.213 | −0.000803 | 0.165 | |

S(II) | 0.828 | 0.176 | 0.012242 | 0.136 | ||

S(III) | 0.880 | 0.148 | 0.006356 | 0.115 | ||

S(IV) | .0944 | .0102 | .0009989 | 0.079 | ||

SVM | S(I) | 0.864 | 0.179 | −0.017431 | 0.139 | |

S(II) | 0.822 | 0.183 | 0.010703 | 0.141 | ||

S(III) | 0.897 | 0.162 | −0.015473 | 0.125 | ||

S(IV) | .0940 | .0106 | .−0000572 | 0.022 |

Flow condition . | Method . | Model . | Performance criteria . | |||
---|---|---|---|---|---|---|

Testing . | ||||||

R
. | SI . | BIAS . | RMSE . | |||

Free flow | KELM-GWO | F(I) | 0.305 | 0.057 | −0.005410 | 0.044 |

F(II) | 0.866 | 0.028 | 0.002324 | 0.022 | ||

F(III) | 0.907 | 0.026 | 0.008926 | 0.021 | ||

F(IV) | .0927 | .0023 | .−0005946 | 0.018 | ||

GPR | F(I) | 0.549 | 0.050 | 0.000421 | 0.038 | |

F(II) | 0.896 | 0.025 | −0.003127 | 0.019 | ||

F(III) | 0.908 | 0.024 | −0.002365 | 0.018 | ||

F(IV) | .0911 | .0023 | .−0002162 | 0.018 | ||

SVM | F(I) | 0.442 | 0.052 | −0.004215 | 0.040 | |

F(II) | 0.870 | 0.028 | 0.002492 | 0.022 | ||

F(III) | 0.880 | 0.028 | −0.004524 | 0.021 | ||

F(IV) | .0896 | .0027 | .−0001713 | 0.021 | ||

Submerged flow | KELM-GWO | S(I) | 0.657 | 0.276 | 0.080632 | 0.214 |

S(II) | 0.680 | 0.254 | 0.031145 | 0.197 | ||

S(III) | 0.767 | 0.255 | 0.035795 | 0.197 | ||

S(IV) | .0839 | .0185 | .−0027117 | 0.144 | ||

GPR | S(I) | 0.727 | 0.213 | −0.000803 | 0.165 | |

S(II) | 0.828 | 0.176 | 0.012242 | 0.136 | ||

S(III) | 0.880 | 0.148 | 0.006356 | 0.115 | ||

S(IV) | .0944 | .0102 | .0009989 | 0.079 | ||

SVM | S(I) | 0.864 | 0.179 | −0.017431 | 0.139 | |

S(II) | 0.822 | 0.183 | 0.010703 | 0.141 | ||

S(III) | 0.897 | 0.162 | −0.015473 | 0.125 | ||

S(IV) | .0940 | .0106 | .−0000572 | 0.022 |

Italic values indicate superior results.

*y*

_{1}

*-y*

_{3}

*/w*,

*y*

_{3}

*/r*,

*y*

_{1}

*/r*, and

*w/r*performed better than the other developed input combinations. Applying this input model in the structure of the SVM-RBF model yielded

*R*test of 0.940, SI test of 0.106, BIAS test of −0.000572, and RMSE test of 0.022. A noteworthy point is the disappointing prediction performance of the hybridized KELM-GWO-RBF model in predicting the discharge coefficient of submerged radial gates. The resulting

*R*, SI, BIAS, and RMSE of the KELM-GWO-RBF method for all the developed models were less than those for the GPR-Exponential and SVM-RBF approaches. Comparing the obtained results of S(III) and S(IV) demonstrated that considering the parameter

*w/r*led to a significant increase in prediction accuracy of GPR-Exponential (64%) and SVM-RBF (53%) in terms of SI performance metric. In contrast, the KELM-GWO-RBF method showed the lowest sensitivity to the parameter

*w/r*in the modeling of the discharge coefficient of submerged radial gates. It confirms that different kernel-depend techniques utilize various degrees of input parameter characteristics for proper modeling of the relationship between input and output parameters. The results presented in Table 9 also showed that the submergence ratio (

*y*

_{1}–

*y*

_{3}/w) as a common parameter in all developed input models has an essential role in the modeling process and is the most influential parameter in predicting the discharge coefficient of submerged radial gates. The results are consistent with the findings of Ansar & Ferro (2001), and Shahrokhnia & Javan (2006) that the discharge of a submerged radial gate can be considered as a function of the differential flow depth. The scatter plots of the predicted discharge coefficient (

*C*

_{d}) vs. observed values for the best models are depicted in Figure 11.

*et al.*(2012) were used as depicted in Table 10. The results illustrated in Figure 13 and Table 11 reflect that the employed kernel-depend methods give superior performance than the selected dimensionless formulas.

Flow condition . | Name . | Formula . |
---|---|---|

Submerged | Shahrokhnia & Javan (2006) | |

Shahrokhnia & Javan (2005) | ||

Zahedani et al. (2012) | ||

Free | Shahrokhnia & Javan (2003) | |

Zahedani et al. (2012) |

Flow condition . | Name . | Formula . |
---|---|---|

Submerged | Shahrokhnia & Javan (2006) | |

Shahrokhnia & Javan (2005) | ||

Zahedani et al. (2012) | ||

Free | Shahrokhnia & Javan (2003) | |

Zahedani et al. (2012) |

Flow condition . | Name . | Performance criteria . | |||
---|---|---|---|---|---|

R
. | SI . | BIAS . | RMSE . | ||

Submerged | Shahrokhnia & Javan (2006) | −0.102 | 0.618 | 0.086736 | 0.485 |

Shahrokhnia & Javan (2005) | −0.001 | 0.890 | 0.587679 | 0.699 | |

Zahedani et al. (2012) | −0.027 | 0.575 | −0.248229 | 0.452 | |

Free | Shahrokhnia & Javan (2003) | 0.582 | 2.68 | 2.06514 | 2.081 |

Zahedani et al. (2012) | 0.491 | 0.209 | −0.059289 | 0.161 |

Flow condition . | Name . | Performance criteria . | |||
---|---|---|---|---|---|

R
. | SI . | BIAS . | RMSE . | ||

Submerged | Shahrokhnia & Javan (2006) | −0.102 | 0.618 | 0.086736 | 0.485 |

Shahrokhnia & Javan (2005) | −0.001 | 0.890 | 0.587679 | 0.699 | |

Zahedani et al. (2012) | −0.027 | 0.575 | −0.248229 | 0.452 | |

Free | Shahrokhnia & Javan (2003) | 0.582 | 2.68 | 2.06514 | 2.081 |

Zahedani et al. (2012) | 0.491 | 0.209 | −0.059289 | 0.161 |

Existing equations and related constant parameters are developed in special laboratories with specific flow conditions, therefore, these equations show acceptable results in particular conditions, but their applicability to field data with various hydraulic conditions is questionable. Another major flaw in the equations obtained from laboratory studies is that they are all based on upstream and downstream channels that are the same width and have the same floor elevation as the gate. This rarely occurs in practice. The main advantage of the kernel-depend approaches against the traditional methods is that kernel-depend models find the best solution for the problem through a high-dimensional domain by creating several candidate computer programs and choosing the best one with the highest fitness. On the other hand, proposed traditional methods are not flexible enough to present uniform results under different conditions and their applicability is limited to the specific cases of their development.

*y*

_{1}

*–y*

_{3}

*/w*was considered as a representative of subdivision criteria for submerged flow through a radial gate. Various intervals of submergence ratio were evaluated through the trial-and-error processes and the generalization capability of proposed kernel-depend techniques with the introduced best input combinations were investigated for each interval. Results of the testing parts are plotted in Figure 14, which indicates an obvious ascending trend in the performance of GPR-Exponential and SVM-RBF techniques. These findings show that the prediction performance of SVM-RBF and GPR-Exponential techniques with the appointed best input combinations tends to be more robust with decreasing levels of submergence. According to the obtained results, the proposed hybrid KELM-GWO method with RBF kernel function gave the most accurate results (

*R*= 0.980, SI = 0.071, BIAS = 0.004973, and RMSE = 0.053) for extremely highly submerged flow with

*S*

_{r}< 0.1 (where

*S*

_{r}=

*y*

_{1}−

*y*

_{3}/

*w*). The plot indicates that the KELM-GWO-RBF model exhibits outstanding performance for the high submerged flow where

*S*

_{r}< 0.1 and

*S*

_{r}< 1. As the submergence decreased, the performance of this method also fluctuated and reached its lowest level (

*R*= 0.861 SI = 0.249, BIAS = −0.011391, and RMSE = 0.194) when

*S*

_{r}< 3. The results showed a more stable performance of GPR and SVM techniques under varied submergence conditions. However, employed SVM-RBF and GPR-Exponential techniques demonstrate disappointing performance under extremely highly submerged flow. It is due to the fact that the coefficient of discharge values under submerged flow conditions show large variations with submergence and this influences the prediction capability of employed kernel-depend techniques.

## CONCLUSIONS

The present paper is an attempt to implement a new kernel-depend ELM GWO algorithm (KELM-GWO), GPR, and support vector machine (SVM) for modeling the discharge coefficient of radial gates. To build prediction models, an extensive field database was established, consisting of 546 data samples gathered from different types of radial gates. The optimization capability of GWO that is naturally inspired by the swarm evolutionary feature is utilized to tune the KELM model for the implemented application. The obtained results showed that the proposed KELM-GWO gives better prediction accuracy than the employed GPR and SVM approaches for the prediction of the discharge coefficient under free-flow conditions. From the current research, the following conclusions were drawn:

- •
Among the developed input combinations, the model having input parameters of the

*y*_{1}*/w*,*w/r*, and*α-w/r*provides the best performance. The results also showed that under submerged hydraulic conditions, the GPR model has better predicting capability with input parameters of*y*_{1}*–y*_{3}*/w*,*y*_{3}*/r*,*y*_{1}*/r*and*w/r*. - •
Results comparison of modeling under free-flow conditions reveals the superiority of polynomial, exponential, and RBF kernel functions as the core tool of KELM-GWO, GPR, and SVM, respectively.

- •
Under submerged flow conditions, KELM-GWO model with RBF kernel function, GPR with exponential, and SVM with RBF presented better prediction accuracy.

- •
According to the obtained results, the proposed hybrid KELM-GWO method with RBF kernel function gave the most accurate results for extremely highly submerged flow. Furthermore, the prediction performance of SVM and GPR techniques with the selected best input combinations tends to be more robust with decreasing levels of submergence.

- •
Finally, the tested data sets of all cases were predicted by different traditional models based on dimensional analysis. Results comparison showed that employed kernel-depend methods have superior capability for prediction of discharge coefficient of radial gates under free and submerged flow conditions.

However, it is noteworthy to mention that the proposed kernel-depend techniques are data-driven, so further studies should be conducted using data ranges beyond this study and field data to confirm the high capabilities of the proposed prediction tools to model the discharge coefficient of submerged radial gates.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

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