## Abstract

This investigation focuses on flow energy, a crucial parameter in the design of water structures such as channels. The research endeavors to explore the relative energy loss (Δ*E*_{AB}/*E*_{A}) in a constricted flow path of varying widths, employing Support Vector Machine (SVM), Artificial Neural Network (ANN), Gene Expression Programming (GEP), Multiple Adaptive Regression Splines (MARS), M5 and Random Forest (RF) models. Experiments span a Froude number range from 2.85 to 8.85. The experimental findings indicate that the Δ*E*_{AB}/*E*_{A} exceeds that observed in a classical hydraulic jump with constriction section. Within the SVM model, the linear kernel emerges as the best predictor of Δ*E*_{AB}/*E*_{A}, outperforming polynomial, radial basis function (RBF), and sigmoid kernels. In addition, in the ANN model, the MLP network was more accurate compared to the RBF network. The results indicate that the relationship proposed by the MARS model can play a significant role resulting in high accuracy compared to the non-linear regression relationship in predicting the target parameter. Upon comprehensive evaluation, the ANN method emerges as the most promising among the candidates, yielding superior performance compared to the other models. The testing phase results for the ANN-MLP are noteworthy, with *R* = 0.997, average RE% = 0.63%, RMSE = 0.0069, BIAS = −0.0004, DR = 0.999, SI = 0.0098 and KGE = 0.995.

## HIGHLIGHTS

This research reinforces the important of investigating the effect of arc-shaped constrictions in the flow path (such as constrictions from bridge piers).

This investigation improves the design of hydraulic control structures.

The performance of the ANN, GEP, MARS, M5, RF, SVM, and regression models has been evaluated using quantitative and qualitative indices (KGE, R, RE%, RMSE, BIAS, DR, scatter index (SI)).

## INTRODUCTION

One of the predominant challenges encountered in the downstream segments of many hydraulic structures is the effective management of the surplus kinetic energy within the flow. This need often requires the implementation of energy dissipating structures to avert potential structural damage. These energy dissipating structures are strategically positioned to regulate and diminish velocity, thereby facilitating the dissipation of excess energy downstream. The dissipation process is intricately linked to flow turbulence and disturbances, with a noteworthy portion of energy dissipated through the deliberate constriction of a specific channel area. This study shows the answer to a gap in the study of channel cross-sectional reduction for the purpose of enhancing energy dissipation.

To address elevated levels of energy, hydraulic jumps are often used to reduce kinetic energy. Additionally, impediments strategically positioned within the flow path contribute to energy absorption. In recent years, various studies have been conducted on hydraulic jumps and energy dissipation. Karbasi & Azamathulla (2016) investigated the characteristics of hydraulic jumps on a rough bed using the Gene Expression Programming (GEP) model. The results indicated that the GEP model is capable of predicting the features of a hydraulic jump on a rough bed with an acceptable level of accuracy. A comparison of Artificial Neural Network (ANN) and Support Vector Machine (SVM) models revealed that the performance of these models is slightly superior to the GEP model. Habibzadeh *et al.* (2019) investigated flow characteristics of downstream hydraulic jumps with and without blocks. The range of Froude numbers varied from 3.48 to 6.85. The results of their study indicated that this flow regime of submerged jumps can effectively be used as an energy dissipator within a stilling basin with a length approximately equal to that required for free hydraulic jumps. Ghaderi *et al.* (2020) conducted numerical simulations of free and submerged hydraulic jumps over various roughness shapes in different configurations and under varying Froude numbers using FLOW-3D software. The results indicated that the influence of roughness is more pronounced in reducing the maximum relative velocity in submerged jumps. Additionally, the greatest energy losses occur with triangular roughness elements compared to other models. Nouri *et al.* (2020) investigated the accuracy of M5P, Random Forest (RF) and stochastic M5P models in predicting the energy loss in cascade spillways. Their results showed that M5P model is more accurate compared to other models. Rahmanshahi & Shafai Bejestan (2020) carried out an experimental study focusing on inclined ramps featuring both smooth and rough surfaces. The investigation took into account various slopes and material sizes. The findings of the study revealed that a higher relative roughness corresponds to more significant energy loss. Moreover, an increase in the ramp's slope was a contributing factor to increased energy loss. In an effort to provide predictive tools, the researchers introduced two mathematical models utilizing a GEP model to estimate energy loss in ramps with both smooth and rough surfaces. Nasrabadi *et al.* (2021), utilizing the novel DGMDH technique, focused on predicting the characteristics of submerged hydraulic jumps. The results demonstrated that the DGMDH model, in comparison to the GMDH model, exhibits high accuracy in predicting relative depth, jump length, and relative energy dissipation. Furthermore, they recommended the utilization of this model for estimating the parameters of hydraulic jumps. Sauida (2022) investigated the relative energy loss of a hydraulic jump downstream of multi-gates using ANN model. The results show that ANN model is more accurate than regression; ANN models can be used to predict energy loss in multi-gates. Heydari *et al.* (2022) modeled the lengths of hydraulic jumps on rough beds using the Self-Adaptive Extreme Learning Machine (SAELM) machine learning approach. The superior performance of the SAELM model was compared with Multilayer Perceptron Neural Network (MLPNN) and SVM methods. The examination of the model results demonstrated the high effectiveness of the SAELM model. Mobayen *et al.* (2023) investigated computational models Multiple Adaptive Regression Splines (MARS) and EPR in estimating energy loss in gabion spillways. The results indicated that the regression equation derived from the EPR model was more complex than the regression equation derived from the MARS model. Abbaszadeh *et al.* (2023) experimentally investigated the hydraulic jump parameters in the threshold condition applied in sluice gates. The results revealed that the application of the threshold leads to an increase in energy dissipation and a reduction in the secondary flow depth.

Instances of channel constriction or abrupt reduction in cross-sectional area may arise due to the installation of structures such as bridge piers, causing impediments to the flow (Chow 1959; Henderson 1966). The presence of bridge piers alters the channel geometry, causing a constriction in the cross-sectional area. This alteration disrupts the smooth flow of water, leading to changes in velocity and pressure distribution. The constriction can result in increased turbulence and flow resistance, affecting the overall hydraulic behavior of the channel.

Hager & Dupraz (1985) experimentally investigated the characteristics of flow in a sudden constriction. They reported a good correlation between their research results and theoretical relationships. Wu & Molinas (2001) examined subcritical flow facing a short constriction along the flow. The relationship proposed for calculating the flow discharge showed good agreement with previous research findings. Dey & Raikar (2005) focused on experimental investigation of scour in a long constriction. Their results indicated that reducing the constriction width leads to an increase in scour depth. In the investigation conducted by Jan & Chang (2009), which focused on hydraulic jumps within a rapidly varied contracted flow, the conclusions from the experimental findings emphasized the substantial influence of bed angle on the relative length of hydraulic jumps. Remarkably, that study suggested that this dependence is distinct from the constriction angle of the sidewalls. Furthermore, they presented theoretical relationships for the secondary jump depth, considering factors such as the constriction cross-section and the bed slope. Similarly, Das *et al.* (2014) investigated the experimental study of relative energy loss in chutes with various slopes and constrictions. They explored energy loss within rapidly contracting flows. Their study revealed a positive correlation between energy dissipation and the slope of the rapid flow. This underscores the significance of bed slope as a determining factor in the dissipation process, as observed in their empirical results. Daneshfaraz *et al.* (2022a) investigated hysteresis in triangular constrictions experimentally. They reported that by increasing the cross-sectional area of the triangular constriction, the relative depth decreases, and with changes to the discharge under the same conditions, the flow depth changes.

The literature review reveals a gap in the study of channel cross-sectional reduction for the purpose of enhancing energy loss. Consequently, given the significance of this matter, the current research aims to experimentally investigate the impact of arc-shaped constrictions in the flow path. Additionally, the necessity for conducting new research in the field of intelligent modeling and soft computing for assessing essential relative energy loss, which has not been addressed so far, is underscored. To address these objectives, the present study employs artificial intelligence models, including ANN, SVM, RF algorithm, MARS, M5 algorithm, and GEP. The focus of this research is primarily on examining the accuracy of the mentioned intelligent models in arc-shaped constrictions. Furthermore, relationships are presented in the regression-based model (non-linear polynomial regression) alongside the MARS model. Various statistical indicators such as R, RMSE, average RE, KGE, BIAS, DR, and SI are scrutinized to evaluate the results of the models. In this research, energy loss in arc-shaped constrictions is investigated within the Froude number range of 2.85–8.85.

## METHODS

### Experimental equipment

### Relationships and parameters

*et al.*2023).

In the provided equations, the variables are defined as follows: *y*_{A} and *y*_{B} denote the flow depth at sections A and B, *y*_{SA} represents the submerged flow depth at section A, *V*_{A} and *V*_{B} are the flow velocities at sections A and B, *g* denotes gravitational acceleration, *E*_{A} and *E*_{B} stand for the specific energy at sections A and B, and *ΔE*_{AB} signifies the specific energy difference between the two sections. These parameters collectively contribute to the calculation of energy dissipation between the specified sections under both free and submerged flow conditions.

Determining and selecting the input parameters are important steps in modeling processes that use intelligent methods. In this section, the dimensionless parameters affecting the energy loss in the constriction of the flow path are introduced and different combinations of the parameters are used for modeling (Table 1).

Parameters . | Min. . | Max. . | Average . | Model no. . | Parameters . |
---|---|---|---|---|---|

ΔE_{AB}/E_{A} | 0.524 | 0.890 | 0.670 | 1 | Fr_{A}, B/W |

Fr_{A} | 2.854 | 8.858 | 5.795 | 2 | Fr_{A}, y_{B}/y_{A} |

B/W | 0.333 | 0.500 | 0.415 | 3 | B/W, y_{B}/y_{A} |

y_{B}/y_{A} | 1.299 | 2.750 | 2.061 | 4 | Fr_{A}, B/W, y_{B}/y_{A} |

Parameters . | Min. . | Max. . | Average . | Model no. . | Parameters . |
---|---|---|---|---|---|

ΔE_{AB}/E_{A} | 0.524 | 0.890 | 0.670 | 1 | Fr_{A}, B/W |

Fr_{A} | 2.854 | 8.858 | 5.795 | 2 | Fr_{A}, y_{B}/y_{A} |

B/W | 0.333 | 0.500 | 0.415 | 3 | B/W, y_{B}/y_{A} |

y_{B}/y_{A} | 1.299 | 2.750 | 2.061 | 4 | Fr_{A}, B/W, y_{B}/y_{A} |

*Q*represents discharge,

*W*represents the channel width,

*B*represents the constriction width dimensions,

*L*represents the constriction length,

*ρ*represents the specific gravity of fluid and

*μ*represents the dynamic viscosity. According to the π-Buckingham theorem and selecting

*y*

_{A},

*ρ*, and

*g*, as repeated parameters, dimensionless parameters are obtained.where

*Fr*

_{A}denotes the Froude number and

*Re*

_{A}denotes the Reynolds number. In the present research, the flow is turbulent, so the effect of

*Re*

_{A}is ignored (Nasrabadi

*et al.*2021; Norouzi

*et al.*2023). In addition, some parameters of Equation (3) have certain values and are not part of the research objectives, so they were ignored, too (Rahmanshahi & Shafai Bejestan 2020). White's theorem provides a useful insight that dimensionless parameters can be obtained through various mathematical operations such as division, multiplication, addition, or subtraction of other dimensionless parameters (White 2016; Daneshfaraz

*et al.*2022b). In the present study, the most significant dimensionless parameters affecting energy dissipation are expressed as follows:

### Support Vector Machine

*ε*) (Norouzi

*et al.*2021).

*W*is the coefficient vector,

*T*is the transpose of

*W*,

*b*is a constant term included in the regression function, and

*ø*is the kernel function. The goal is to discover the function

*f*(

*x*) by training the SVM model with a set of examples (training set). The SVM regression function can be written as:

*a*corresponds to the average Lagrange coefficients. The computation of

_{i}*ø*(

*X*) can be complex. In the SVM regression model, a kernel function is employed, and its intricacy is contingent on the scale of the training data and the dimensions of the feature vector. Four commonly utilized kernel types in practice are the Linear kernel, Polynomial kernel, Sigmoid kernel, and radial basis function (RBF) kernel. These kernels play a pivotal role in shaping the SVM regression model and cater to diverse modeling scenarios (Hassanzadeh & Abbaszadeh 2023).

In the above equations, *K* (*X _{i}*,

*X*) represents the covariance or kernel function, calculated at points

_{j}*X*and

_{i}*X*. The functions

_{j}*a*,

*C*,

*d*, and

*σ*denote kernel functions. The term

*d*represents the degree of the polynomial,

*σ*is the variance and hyper parameter, and

*C*is a positive integer that acts as a penalty factor when model training errors occur. Here, the values of Capacity, Epsilon and Gamma are 10, 0.1 and 10.

### Artificial Neural Network

*et al.*2012). Here, the values of Min hidden units and Max hidden units are 3 and 21, respectively. The values of Networks to train and Networks to retain were introduced to the Statistica 12 software as 20 and 5, respectively. In Figure 3, the architecture of the artificial neural network model is presented. In the present research, the ANN model with three input neurons, one hidden layer (with 21 neurons), and one output neuron has been employed (Norouzi

*et al.*2020; Ayaz

*et al.*2024).

### Random Forest

*et al.*2020). RFs can be used for both classification and regression problems. A random forest, maps input data to outputs in the training or model fitting phase. During training, the model is fed data that is relevant to the problem domain that the model needs to learn to make predictions (Jahed Armaghani

*et al.*2020). The model learns the relationships among the data and the values the user wants to predict. In the RF, number of trees = 500, minimum no. of cases = 5, maximum no. of levels = 10, minimum no. in child node = 5, and max. no. of nodes = 100. The tree graph of the present model is shown in Figure 4.

### Multiple Adaptive Regression Splines

In Equation (14), *Y* is the estimated value of the response variable, *X* is the vector of explanatory variables, *B _{k}* is the basis function, and

*C*are coefficients determined by minimizing the sum of squared residuals. Each basis function may take the form of a linear spline function or the product of two or more of them, indicating interaction effects. The Multiple Adaptive Regression Splines model divides the space of explanatory variables into distinct regions with specific nodes, which result in the maximum reduction in the sum of squared errors. The fitting of the MARS model occurs in two stages. In the forward stage, a large number of basis functions with different nodes are successively added to the model, producing a complex and overfit model. In the backward stage or pruning stage, basis functions with less importance and impact on the estimation are removed. For the MARS model, the following settings were defined: Maximum number of Basis Functions: 21, Degree of interaction: 1, and Penalty and Threshold values: 2 and 0.005, respectively.

_{k}### M5 algorithm

*sd*denotes the standard deviation,

*T*includes the samples reaching the considered node, and

*T*represents the samples obtained from the division of the considered node based on the selected feature. The M5 algorithm examines all possible scenarios for branching based on a specific feature and ultimately selects a scenario that can reduce the error function more than others. After completing the tree-building algorithm, a multiple-variable linear regression model is fitted to the existing samples in each internal node. In the current research, the M5Rule option of WEKA software was used to model the M5.

_{i}### Gene expression programming

This approach is considered part of evolutionary algorithms, all of which are grounded in the principles of Darwinian evolution. These algorithms define an objective function in the form of criteria and then employ a learned function to measure and compare various solution methods (Najafzadeh 2019). In a step-by-step process of refining data structures, they ultimately present a suitable solution method. Gene expression programming is a recent method among these evolutionary algorithms, and due to its sufficient accuracy, it is considered the most conventional and widely used approach. The primary domain of gene expression programming is the same as genetic algorithms, with the distinction that this method uses branches instead of bit strings. Each branch consists of a set of terminals (problem variables) and a set of functions (primary operators) (Mohammed & Sharifi 2020). Table 2 shows the values of the parameters defined for the GEP model in the GenXpro Tools 4 software. The parameters and their rates to estimate the desired parameter if the population is considered up to 10,000 are listed in Table 2.

Parameters . | Value . |
---|---|

Head size | 7 |

Chromosomes | 30 |

Number of genes | 3 |

Mutation rate | 0.044 |

Inversion rate | 0.1 |

One point recombination rate | 0.3 |

Two point recombination rate | 0.3 |

Gene recombination rate | 0.1 |

IS transposition rate | 0.1 |

RIS transposition rate | 0.1 |

Gene transposition rate | 0.1 |

Fitness function error rate | RMSE |

Linking function | + |

Generation number | 10,000 |

Parameters . | Value . |
---|---|

Head size | 7 |

Chromosomes | 30 |

Number of genes | 3 |

Mutation rate | 0.044 |

Inversion rate | 0.1 |

One point recombination rate | 0.3 |

Two point recombination rate | 0.3 |

Gene recombination rate | 0.1 |

IS transposition rate | 0.1 |

RIS transposition rate | 0.1 |

Gene transposition rate | 0.1 |

Fitness function error rate | RMSE |

Linking function | + |

Generation number | 10,000 |

### Statistical indicators

*et al.*2020; Daneshfaraz

*et al.*2022b; Agarwal

*et al.*2023; Najafzadeh

*et al.*2023; Nourani

*et al.*2023; Najafzadeh & Mahmoudi-Rad 2024):Here, RE indicates the relative error.Here, RMSE indicates the root mean square error;

*n*indicates the total data.where SI indicates the scatter index.Here, KGE indicates the Kling Gupta Efficiency;

*R*indicates the correlation coefficient;

*β*indicates the average calculated data relative to the average observed data;

*γ*indicates the standard deviation of the calculated data relative to the standard deviation of the observed data.

If the resulting calculations yield 0.7 < KGE < 1, then the performance is characterized as ‘very good’. If 0.6 < KGE < 0.7, the results are ‘good’. For 0.4 < KGE < 0.5 or 0.5 < KGE < 0.6; respective descriptors ‘acceptable’ and ‘satisfactory’ are used. If, however, KGE < 0.4, the results are ‘unsatisfactory’ (Gupta *et al.* 2009). If DR = 1, the soft computing technique shows the most efficient performance, DR > 1 shows over predictions, and DR > 0, under prediction (Najafzadeh *et al.* 2022). If the BIAS index equal 0 shows the most efficient performance, when BIAS > 0 indicates over prediction and BIAS < 0 indicates under prediction.

## RESULTS AND DISCUSSION

No. . | ΔE_{AB}/E_{A}
. | Fr_{A}
. | B/W
. | y_{B}/y_{A}
. | no. . | ΔE_{AB}/E_{A}
. | Fr_{A}
. | B/W
. | y_{B}/y_{A}
. |
---|---|---|---|---|---|---|---|---|---|

1 | 0.559 | 3.430 | 0.330 | 1.642 | 57 | 0.830 | 8.686 | 0.330 | 2.540 |

2 | 0.571 | 3.514 | 0.330 | 1.681 | 58 | 0.841 | 8.858 | 0.330 | 2.580 |

3 | 0.575 | 3.651 | 0.330 | 1.722 | 59 | 0.524 | 2.889 | 0.500 | 1.298 |

4 | 0.585 | 3.714 | 0.330 | 1.755 | 60 | 0.531 | 3.054 | 0.500 | 1.380 |

5 | 0.595 | 3.852 | 0.330 | 1.781 | 61 | 0.541 | 3.145 | 0.500 | 1.440 |

6 | 0.601 | 3.942 | 0.330 | 1.825 | 62 | 0.551 | 3.265 | 0.500 | 1.460 |

7 | 0.608 | 3.958 | 0.330 | 1.855 | 63 | 0.562 | 3.314 | 0.500 | 1.470 |

8 | 0.614 | 3.968 | 0.330 | 1.903 | 64 | 0.578 | 3.385 | 0.500 | 1.475 |

9 | 0.618 | 4.452 | 0.330 | 1.911 | 65 | 0.582 | 3.407 | 0.500 | 1.482 |

10 | 0.624 | 4.158 | 0.330 | 1.924 | 66 | 0.591 | 3.565 | 0.500 | 1.500 |

11 | 0.625 | 4.142 | 0.330 | 1.932 | 67 | 0.601 | 3.612 | 0.500 | 1.550 |

12 | 0.631 | 4.280 | 0.330 | 1.941 | 68 | 0.605 | 3.785 | 0.500 | 1.600 |

13 | 0.635 | 4.369 | 0.330 | 1.950 | 69 | 0.612 | 3.854 | 0.500 | 1.620 |

14 | 0.642 | 4.457 | 0.330 | 1.967 | 70 | 0.628 | 3.985 | 0.500 | 1.650 |

15 | 0.641 | 4.551 | 0.330 | 1.971 | 71 | 0.625 | 4.025 | 0.500 | 1.655 |

16 | 0.645 | 4.624 | 0.330 | 1.983 | 72 | 0.629 | 4.112 | 0.500 | 1.672 |

17 | 0.652 | 4.765 | 0.330 | 1.991 | 73 | 0.634 | 4.265 | 0.500 | 1.690 |

18 | 0.658 | 4.737 | 0.330 | 2.044 | 74 | 0.639 | 4.245 | 0.500 | 1.717 |

19 | 0.645 | 4.865 | 0.330 | 2.051 | 75 | 0.645 | 4.614 | 0.500 | 1.720 |

20 | 0.651 | 4.952 | 0.330 | 2.067 | 76 | 0.650 | 4.453 | 0.500 | 1.750 |

21 | 0.655 | 5.015 | 0.330 | 2.071 | 77 | 0.655 | 4.548 | 0.500 | 1.770 |

22 | 0.658 | 5.114 | 0.330 | 2.082 | 78 | 0.658 | 4.632 | 0.500 | 1.780 |

23 | 0.658 | 5.254 | 0.330 | 2.085 | 79 | 0.662 | 4.785 | 0.500 | 1.790 |

24 | 0.665 | 5.345 | 0.330 | 2.089 | 80 | 0.668 | 4.856 | 0.500 | 1.814 |

25 | 0.671 | 5.425 | 0.330 | 2.100 | 81 | 0.665 | 4.844 | 0.500 | 1.821 |

26 | 0.672 | 5.458 | 0.330 | 2.114 | 82 | 0.701 | 4.915 | 0.500 | 1.840 |

27 | 0.672 | 5.476 | 0.330 | 2.157 | 83 | 0.675 | 5.021 | 0.500 | 1.850 |

28 | 0.665 | 5.585 | 0.330 | 2.165 | 84 | 0.682 | 5.145 | 0.500 | 1.920 |

29 | 0.671 | 5.625 | 0.330 | 2.171 | 85 | 0.685 | 5.225 | 0.500 | 1.928 |

30 | 0.675 | 5.745 | 0.330 | 2.182 | 86 | 0.691 | 5.336 | 0.500 | 1.940 |

31 | 0.683 | 5.836 | 0.330 | 2.197 | 87 | 0.695 | 5.445 | 0.500 | 1.980 |

32 | 0.685 | 5.914 | 0.330 | 2.200 | 88 | 0.701 | 5.585 | 0.500 | 1.985 |

33 | 0.689 | 5.981 | 0.330 | 2.218 | 89 | 0.705 | 5.685 | 0.500 | 1.998 |

34 | 0.692 | 6.415 | 0.330 | 2.214 | 90 | 0.711 | 5.758 | 0.500 | 2.000 |

35 | 0.698 | 6.245 | 0.330 | 2.225 | 91 | 0.714 | 5.811 | 0.500 | 2.069 |

36 | 0.704 | 6.365 | 0.330 | 2.234 | 92 | 0.720 | 6.150 | 0.500 | 2.080 |

37 | 0.708 | 6.565 | 0.330 | 2.244 | 93 | 0.734 | 6.180 | 0.500 | 2.140 |

38 | 0.714 | 6.776 | 0.330 | 2.252 | 94 | 0.740 | 6.254 | 0.500 | 2.145 |

39 | 0.718 | 6.834 | 0.330 | 2.268 | 95 | 0.750 | 6.361 | 0.500 | 2.170 |

40 | 0.725 | 6.956 | 0.330 | 2.278 | 96 | 0.765 | 6.895 | 0.500 | 2.216 |

41 | 0.722 | 7.010 | 0.330 | 2.249 | 97 | 0.780 | 7.219 | 0.500 | 2.256 |

42 | 0.728 | 7.156 | 0.330 | 2.278 | 98 | 0.801 | 7.385 | 0.500 | 2.280 |

43 | 0.731 | 7.230 | 0.330 | 2.287 | 99 | 0.811 | 7.454 | 0.500 | 2.285 |

44 | 0.740 | 7.365 | 0.330 | 2.287 | 100 | 0.820 | 7.558 | 0.500 | 2.310 |

45 | 0.745 | 7.454 | 0.330 | 2.294 | 101 | 0.825 | 7.632 | 0.500 | 2.350 |

46 | 0.755 | 7.587 | 0.330 | 2.301 | 102 | 0.830 | 7.758 | 0.500 | 2.380 |

47 | 0.765 | 7.654 | 0.330 | 2.306 | 103 | 0.838 | 7.865 | 0.500 | 2.420 |

48 | 0.758 | 7.712 | 0.330 | 2.314 | 104 | 0.845 | 7.948 | 0.500 | 2.450 |

49 | 0.765 | 7.836 | 0.330 | 2.325 | 105 | 0.850 | 8.025 | 0.500 | 2.480 |

50 | 0.770 | 7.958 | 0.330 | 2.325 | 106 | 0.855 | 8.114 | 0.500 | 2.500 |

51 | 0.771 | 8.025 | 0.330 | 2.345 | 107 | 0.865 | 8.225 | 0.500 | 2.550 |

52 | 0.791 | 8.145 | 0.330 | 2.355 | 108 | 0.870 | 8.365 | 0.500 | 2.580 |

53 | 0.801 | 8.226 | 0.330 | 2.355 | 109 | 0.875 | 8.445 | 0.500 | 2.620 |

54 | 0.815 | 8.354 | 0.330 | 2.458 | 110 | 0.880 | 8.526 | 0.500 | 2.650 |

55 | 0.822 | 8.425 | 0.330 | 2.488 | 111 | 0.885 | 8.652 | 0.500 | 2.680 |

56 | 0.825 | 8.523 | 0.330 | 2.500 | 112 | 0.890 | 8.858 | 0.500 | 2.750 |

No. . | ΔE_{AB}/E_{A}
. | Fr_{A}
. | B/W
. | y_{B}/y_{A}
. | no. . | ΔE_{AB}/E_{A}
. | Fr_{A}
. | B/W
. | y_{B}/y_{A}
. |
---|---|---|---|---|---|---|---|---|---|

1 | 0.559 | 3.430 | 0.330 | 1.642 | 57 | 0.830 | 8.686 | 0.330 | 2.540 |

2 | 0.571 | 3.514 | 0.330 | 1.681 | 58 | 0.841 | 8.858 | 0.330 | 2.580 |

3 | 0.575 | 3.651 | 0.330 | 1.722 | 59 | 0.524 | 2.889 | 0.500 | 1.298 |

4 | 0.585 | 3.714 | 0.330 | 1.755 | 60 | 0.531 | 3.054 | 0.500 | 1.380 |

5 | 0.595 | 3.852 | 0.330 | 1.781 | 61 | 0.541 | 3.145 | 0.500 | 1.440 |

6 | 0.601 | 3.942 | 0.330 | 1.825 | 62 | 0.551 | 3.265 | 0.500 | 1.460 |

7 | 0.608 | 3.958 | 0.330 | 1.855 | 63 | 0.562 | 3.314 | 0.500 | 1.470 |

8 | 0.614 | 3.968 | 0.330 | 1.903 | 64 | 0.578 | 3.385 | 0.500 | 1.475 |

9 | 0.618 | 4.452 | 0.330 | 1.911 | 65 | 0.582 | 3.407 | 0.500 | 1.482 |

10 | 0.624 | 4.158 | 0.330 | 1.924 | 66 | 0.591 | 3.565 | 0.500 | 1.500 |

11 | 0.625 | 4.142 | 0.330 | 1.932 | 67 | 0.601 | 3.612 | 0.500 | 1.550 |

12 | 0.631 | 4.280 | 0.330 | 1.941 | 68 | 0.605 | 3.785 | 0.500 | 1.600 |

13 | 0.635 | 4.369 | 0.330 | 1.950 | 69 | 0.612 | 3.854 | 0.500 | 1.620 |

14 | 0.642 | 4.457 | 0.330 | 1.967 | 70 | 0.628 | 3.985 | 0.500 | 1.650 |

15 | 0.641 | 4.551 | 0.330 | 1.971 | 71 | 0.625 | 4.025 | 0.500 | 1.655 |

16 | 0.645 | 4.624 | 0.330 | 1.983 | 72 | 0.629 | 4.112 | 0.500 | 1.672 |

17 | 0.652 | 4.765 | 0.330 | 1.991 | 73 | 0.634 | 4.265 | 0.500 | 1.690 |

18 | 0.658 | 4.737 | 0.330 | 2.044 | 74 | 0.639 | 4.245 | 0.500 | 1.717 |

19 | 0.645 | 4.865 | 0.330 | 2.051 | 75 | 0.645 | 4.614 | 0.500 | 1.720 |

20 | 0.651 | 4.952 | 0.330 | 2.067 | 76 | 0.650 | 4.453 | 0.500 | 1.750 |

21 | 0.655 | 5.015 | 0.330 | 2.071 | 77 | 0.655 | 4.548 | 0.500 | 1.770 |

22 | 0.658 | 5.114 | 0.330 | 2.082 | 78 | 0.658 | 4.632 | 0.500 | 1.780 |

23 | 0.658 | 5.254 | 0.330 | 2.085 | 79 | 0.662 | 4.785 | 0.500 | 1.790 |

24 | 0.665 | 5.345 | 0.330 | 2.089 | 80 | 0.668 | 4.856 | 0.500 | 1.814 |

25 | 0.671 | 5.425 | 0.330 | 2.100 | 81 | 0.665 | 4.844 | 0.500 | 1.821 |

26 | 0.672 | 5.458 | 0.330 | 2.114 | 82 | 0.701 | 4.915 | 0.500 | 1.840 |

27 | 0.672 | 5.476 | 0.330 | 2.157 | 83 | 0.675 | 5.021 | 0.500 | 1.850 |

28 | 0.665 | 5.585 | 0.330 | 2.165 | 84 | 0.682 | 5.145 | 0.500 | 1.920 |

29 | 0.671 | 5.625 | 0.330 | 2.171 | 85 | 0.685 | 5.225 | 0.500 | 1.928 |

30 | 0.675 | 5.745 | 0.330 | 2.182 | 86 | 0.691 | 5.336 | 0.500 | 1.940 |

31 | 0.683 | 5.836 | 0.330 | 2.197 | 87 | 0.695 | 5.445 | 0.500 | 1.980 |

32 | 0.685 | 5.914 | 0.330 | 2.200 | 88 | 0.701 | 5.585 | 0.500 | 1.985 |

33 | 0.689 | 5.981 | 0.330 | 2.218 | 89 | 0.705 | 5.685 | 0.500 | 1.998 |

34 | 0.692 | 6.415 | 0.330 | 2.214 | 90 | 0.711 | 5.758 | 0.500 | 2.000 |

35 | 0.698 | 6.245 | 0.330 | 2.225 | 91 | 0.714 | 5.811 | 0.500 | 2.069 |

36 | 0.704 | 6.365 | 0.330 | 2.234 | 92 | 0.720 | 6.150 | 0.500 | 2.080 |

37 | 0.708 | 6.565 | 0.330 | 2.244 | 93 | 0.734 | 6.180 | 0.500 | 2.140 |

38 | 0.714 | 6.776 | 0.330 | 2.252 | 94 | 0.740 | 6.254 | 0.500 | 2.145 |

39 | 0.718 | 6.834 | 0.330 | 2.268 | 95 | 0.750 | 6.361 | 0.500 | 2.170 |

40 | 0.725 | 6.956 | 0.330 | 2.278 | 96 | 0.765 | 6.895 | 0.500 | 2.216 |

41 | 0.722 | 7.010 | 0.330 | 2.249 | 97 | 0.780 | 7.219 | 0.500 | 2.256 |

42 | 0.728 | 7.156 | 0.330 | 2.278 | 98 | 0.801 | 7.385 | 0.500 | 2.280 |

43 | 0.731 | 7.230 | 0.330 | 2.287 | 99 | 0.811 | 7.454 | 0.500 | 2.285 |

44 | 0.740 | 7.365 | 0.330 | 2.287 | 100 | 0.820 | 7.558 | 0.500 | 2.310 |

45 | 0.745 | 7.454 | 0.330 | 2.294 | 101 | 0.825 | 7.632 | 0.500 | 2.350 |

46 | 0.755 | 7.587 | 0.330 | 2.301 | 102 | 0.830 | 7.758 | 0.500 | 2.380 |

47 | 0.765 | 7.654 | 0.330 | 2.306 | 103 | 0.838 | 7.865 | 0.500 | 2.420 |

48 | 0.758 | 7.712 | 0.330 | 2.314 | 104 | 0.845 | 7.948 | 0.500 | 2.450 |

49 | 0.765 | 7.836 | 0.330 | 2.325 | 105 | 0.850 | 8.025 | 0.500 | 2.480 |

50 | 0.770 | 7.958 | 0.330 | 2.325 | 106 | 0.855 | 8.114 | 0.500 | 2.500 |

51 | 0.771 | 8.025 | 0.330 | 2.345 | 107 | 0.865 | 8.225 | 0.500 | 2.550 |

52 | 0.791 | 8.145 | 0.330 | 2.355 | 108 | 0.870 | 8.365 | 0.500 | 2.580 |

53 | 0.801 | 8.226 | 0.330 | 2.355 | 109 | 0.875 | 8.445 | 0.500 | 2.620 |

54 | 0.815 | 8.354 | 0.330 | 2.458 | 110 | 0.880 | 8.526 | 0.500 | 2.650 |

55 | 0.822 | 8.425 | 0.330 | 2.488 | 111 | 0.885 | 8.652 | 0.500 | 2.680 |

56 | 0.825 | 8.523 | 0.330 | 2.500 | 112 | 0.890 | 8.858 | 0.500 | 2.750 |

*et al.*2023). The selection of the best kernel was made by ensuring that statistical indicators such as R, RMSE, average RE, KGE, BIAS, DR, and SI had satisfactory performance compared to experimental results. The laboratory and predicted values of relative energy loss for various kernels are shown in Figure 6(a) and 6(b). In addition, Figure 6(c) shows the results of the training and testing phases for different datasets. As observed, the linear kernel exhibits high accuracy compared to other kernels and predicts relative energy loss with high precision. As indicated in Figure 6(d) and 6(e), the statistical outcomes for the Linear kernel during the training phase are

*R*= 0.992, RMSE = 0.0114, average RE% = 1.41, BIAS = −0.0005, DR = 0.999, SI = 0.0161 and KGE = 0.982. Correspondingly, in the testing phase, these values are 0.994, 0.0101, 1.33%, −0.0002, 0.999, 0.015, and 0.980, respectively. Figure 6(f) and 6(g) illustrates that, for the superior kernel, a substantial portion of data in both training and testing phases falls within the ±3% relative error band. This observation underscores high solution accuracy, with over 97% of the data residing within the ±3% error band during both training and testing phases.

Statistical indicator . | Kernels . | |||
---|---|---|---|---|

Linear . | Polynomial . | RBF . | Sigmoid . | |

R (test) | 0.994 | 0.966 | 0.992 | 0.436 |

KGE (test) | 0.980 | 0.887 | 0.985 | −1.514 |

RMSE (test) | 0.010 | 0.024 | 0.014 | 0.362 |

Average RE% (test) | 1.334 | 3.047 | 1.442 | 34.91 |

BIAS | −0.0002 | −0.0022 | −0.0012 | −0.168 |

DR | 0.999 | 0.999 | 0.998 | 0.787 |

SI | 0.015 | 0.035 | 0.017 | 0.525 |

Statistical indicator . | Kernels . | |||
---|---|---|---|---|

Linear . | Polynomial . | RBF . | Sigmoid . | |

R (test) | 0.994 | 0.966 | 0.992 | 0.436 |

KGE (test) | 0.980 | 0.887 | 0.985 | −1.514 |

RMSE (test) | 0.010 | 0.024 | 0.014 | 0.362 |

Average RE% (test) | 1.334 | 3.047 | 1.442 | 34.91 |

BIAS | −0.0002 | −0.0022 | −0.0012 | −0.168 |

DR | 0.999 | 0.999 | 0.998 | 0.787 |

SI | 0.015 | 0.035 | 0.017 | 0.525 |

The results of SVM, ANN, RF, MARS, GEP, and M5 models to predict the relative energy loss are presented in Table 5. According to Table 5, it was observed that model no. 4 with three input parameters provides favorable statistical results compared to other models and was selected as the superior model in the processing. Model no. 3 does not have sufficient accuracy to predict the relative energy loss. Also, the comparison of two models no. 1 and 2 shows that replacing the use of parameter *Fr*_{A} significantly improves the accuracy of modeling, which indicates the high impact of parameter *Fr*_{A} in predicting relative energy loss.

Model no. . | method . | Train . | Test . | ||||||
---|---|---|---|---|---|---|---|---|---|

R (–) . | Avg. RE (%) . | RMSE (–) . | KGE (–) . | R (–) . | Avg. RE (%) . | RMSE (–) . | KGE (–) . | ||

1 | SVM | 0.795 | 3.54 | 0.0982 | 0.794 | 0.814 | 3.52 | 0.0980 | 0.815 |

ANN | 0.802 | 2.94 | 0.0854 | 0.800 | 0.818 | 2.57 | 0.0842 | 0.824 | |

RF | 0.741 | 4.14 | 0.0992 | 0.755 | 0.752 | 4.05 | 0.0991 | 0.755 | |

GEP | 0.800 | 2.98 | 0.0868 | 0.810 | 0.810 | 2.58 | 0.0867 | 0.807 | |

M5 | 0.748 | 4.01 | 0.0972 | 0.757 | 0.757 | 4.00 | 0.0945 | 0.750 | |

MARS | 0.814 | 2.45 | 0.0792 | 0.814 | 0.825 | 2.24 | 0.0795 | 0.821 | |

2 | SVM | 0.868 | 2.55 | 0.0765 | 0.868 | 0.870 | 2.82 | 0.0700 | 0.872 |

ANN | 0.894 | 1.48 | 0.0545 | 0.884 | 0.892 | 1.54 | 0.0514 | 0.892 | |

RF | 0.825 | 2.92 | 0.0923 | 0.825 | 0.826 | 2.85 | 0.0852 | 0.814 | |

GEP | 0.884 | 1.46 | 0.0548 | 0.868 | 0.892 | 1.84 | 0.0714 | 0.884 | |

M5 | 0.840 | 2.85 | 0.0892 | 0.845 | 0.825 | 2.70 | 0.0801 | 0.825 | |

MARS | 0.892 | 1.35 | 0.0546 | 0.887 | 0.892 | 1.58 | 0.0500 | 0.898 | |

3 | SVM | 0.558 | 4.95 | 0.1455 | 0.560 | 0.545 | 5.25 | 0.1580 | 0.545 |

ANN | 0.585 | 5.14 | 0.1865 | 0.465 | 0.582 | 4.88 | 0.1784 | 0.485 | |

RF | 0.485 | 5.85 | 0.2550 | 0.465 | 0.454 | 5.92 | 0.2655 | 0.445 | |

GEP | 0.575 | 5.25 | 0.1885 | 0.500 | 0.545 | 4.84 | 0.1854 | 0.484 | |

M5 | 0.495 | 5.72 | 0.2684 | 0.471 | 0.458 | 5.87 | 0.2595 | 0.450 | |

MARS | 0.584 | 4.80 | 0.1718 | 0.484 | 0.584 | 4.84 | 0.1725 | 0.588 | |

4 | SVM | 0.992 | 1.41 | 0.0114 | 0.982 | 0.994 | 1.33 | 0.0101 | 0.980 |

ANN | 0.998 | 0.65 | 0.0057 | 0.997 | 0.997 | 0.63 | 0.0069 | 0.995 | |

RF | 0.966 | 2.58 | 0.0242 | 0.867 | 0.964 | 3.32 | 0.0270 | 0.859 | |

GEP | 0.992 | 1.28 | 0.0117 | 0.984 | 0.996 | 0.80 | 0.0088 | 0.973 | |

M5 | 0.971 | 2.30 | 0.0212 | 0.968 | 0.978 | 2.25 | 0.0190 | 0.926 | |

MARS | 0.997 | 0.71 | 0.0062 | 0.996 | 0.997 | 0.70 | 0.0074 | 0.995 |

Model no. . | method . | Train . | Test . | ||||||
---|---|---|---|---|---|---|---|---|---|

R (–) . | Avg. RE (%) . | RMSE (–) . | KGE (–) . | R (–) . | Avg. RE (%) . | RMSE (–) . | KGE (–) . | ||

1 | SVM | 0.795 | 3.54 | 0.0982 | 0.794 | 0.814 | 3.52 | 0.0980 | 0.815 |

ANN | 0.802 | 2.94 | 0.0854 | 0.800 | 0.818 | 2.57 | 0.0842 | 0.824 | |

RF | 0.741 | 4.14 | 0.0992 | 0.755 | 0.752 | 4.05 | 0.0991 | 0.755 | |

GEP | 0.800 | 2.98 | 0.0868 | 0.810 | 0.810 | 2.58 | 0.0867 | 0.807 | |

M5 | 0.748 | 4.01 | 0.0972 | 0.757 | 0.757 | 4.00 | 0.0945 | 0.750 | |

MARS | 0.814 | 2.45 | 0.0792 | 0.814 | 0.825 | 2.24 | 0.0795 | 0.821 | |

2 | SVM | 0.868 | 2.55 | 0.0765 | 0.868 | 0.870 | 2.82 | 0.0700 | 0.872 |

ANN | 0.894 | 1.48 | 0.0545 | 0.884 | 0.892 | 1.54 | 0.0514 | 0.892 | |

RF | 0.825 | 2.92 | 0.0923 | 0.825 | 0.826 | 2.85 | 0.0852 | 0.814 | |

GEP | 0.884 | 1.46 | 0.0548 | 0.868 | 0.892 | 1.84 | 0.0714 | 0.884 | |

M5 | 0.840 | 2.85 | 0.0892 | 0.845 | 0.825 | 2.70 | 0.0801 | 0.825 | |

MARS | 0.892 | 1.35 | 0.0546 | 0.887 | 0.892 | 1.58 | 0.0500 | 0.898 | |

3 | SVM | 0.558 | 4.95 | 0.1455 | 0.560 | 0.545 | 5.25 | 0.1580 | 0.545 |

ANN | 0.585 | 5.14 | 0.1865 | 0.465 | 0.582 | 4.88 | 0.1784 | 0.485 | |

RF | 0.485 | 5.85 | 0.2550 | 0.465 | 0.454 | 5.92 | 0.2655 | 0.445 | |

GEP | 0.575 | 5.25 | 0.1885 | 0.500 | 0.545 | 4.84 | 0.1854 | 0.484 | |

M5 | 0.495 | 5.72 | 0.2684 | 0.471 | 0.458 | 5.87 | 0.2595 | 0.450 | |

MARS | 0.584 | 4.80 | 0.1718 | 0.484 | 0.584 | 4.84 | 0.1725 | 0.588 | |

4 | SVM | 0.992 | 1.41 | 0.0114 | 0.982 | 0.994 | 1.33 | 0.0101 | 0.980 |

ANN | 0.998 | 0.65 | 0.0057 | 0.997 | 0.997 | 0.63 | 0.0069 | 0.995 | |

RF | 0.966 | 2.58 | 0.0242 | 0.867 | 0.964 | 3.32 | 0.0270 | 0.859 | |

GEP | 0.992 | 1.28 | 0.0117 | 0.984 | 0.996 | 0.80 | 0.0088 | 0.973 | |

M5 | 0.971 | 2.30 | 0.0212 | 0.968 | 0.978 | 2.25 | 0.0190 | 0.926 | |

MARS | 0.997 | 0.71 | 0.0062 | 0.996 | 0.997 | 0.70 | 0.0074 | 0.995 |

*R*= 0.971, RMSE = 0.0212, average RE% = 2.30, BIAS = −0.0016, DR = 0.998, SI = 0.0304, KGE = 0.968) are noteworthy. These results persist in the testing phase as well, with values of (

*R*= 0.978, RMSE = 0.0190, average RE% = 2.25, BIAS = 0.0007, DR = 1.0023, KGE = 0.926).

*Fr*

_{A},

*B*/

*W*, and

*y*

_{B}/

*y*

_{A}play a crucial role in predicting the relative energy loss. Statistical indices during the training phase (

*R*= 0.997, RMSE = 0.0062, average RE% = 0.71, BIAS = 0, DR = 1, SI = 0.0090, KGE = 0.996) and testing phase (

*R*= 0.997, RMSE = 0.0074, average RE% = 0.70, BIAS = −0.0005, DR = 0.999, SI = 0.0106, KGE = 0.995) demonstrate high performance (Figure 10).

Equation (22) of MARS and Basis Function (BF) . | Coefficient . | Value . | Coefficient . | Value . |
---|---|---|---|---|

ΔE_{AB}/E_{A} = a + b × BF(1) – c × BF(2) + d × BF(3) – e × BF(4) + f × BF(5) – g × BF(6) | a | 7.80 × 10^{−1} | h | 7.45 × 10^{0} |

BF(1) = max(0, Fr_{A}–h) | b | 4.98 × 10^{−2} | i | 7.45 × 10^{0} |

BF(2) = max(0, i–Fr_{A}) | c | 1.90 × 10^{−2} | j | 3.30 × 10^{−1} |

BF(3) = max(0, B/W–j) | d | 3.51 × 10^{−1} | k | 2.35 × 10^{0} |

BF(4) = max(0, k–y_{B}/y_{A}) | e | 2.19 × 10^{−1} | l | 6.15 × 10^{0} |

BF(5) = max(0, Fr_{A}–l) | f | 3.36 × 10^{−2} | m | 4.55 × 10^{0} |

BF(6) = max(0, Fr_{A}–m) | g | 2.37 × 10^{−2} | – | – |

Equation (22) of MARS and Basis Function (BF) . | Coefficient . | Value . | Coefficient . | Value . |
---|---|---|---|---|

ΔE_{AB}/E_{A} = a + b × BF(1) – c × BF(2) + d × BF(3) – e × BF(4) + f × BF(5) – g × BF(6) | a | 7.80 × 10^{−1} | h | 7.45 × 10^{0} |

BF(1) = max(0, Fr_{A}–h) | b | 4.98 × 10^{−2} | i | 7.45 × 10^{0} |

BF(2) = max(0, i–Fr_{A}) | c | 1.90 × 10^{−2} | j | 3.30 × 10^{−1} |

BF(3) = max(0, B/W–j) | d | 3.51 × 10^{−1} | k | 2.35 × 10^{0} |

BF(4) = max(0, k–y_{B}/y_{A}) | e | 2.19 × 10^{−1} | l | 6.15 × 10^{0} |

BF(5) = max(0, Fr_{A}–l) | f | 3.36 × 10^{−2} | m | 4.55 × 10^{0} |

BF(6) = max(0, Fr_{A}–m) | g | 2.37 × 10^{−2} | – | – |

*R*= 0.988, average RE% = 1.108%, and RMSE = 0.0098. Additionally, the KGE for this relationship falls within the ‘very good’ range. These statistical indicators suggest that the provided relationship exhibits high accuracy in predicting the amount of energy loss, with over 99% of the data falling within the relative error range of ±3%. It should be noted that the relationship provided by MARS model is more accurate compared to Equation (23).

## CONCLUSIONS

The current research investigates experimental and data mining methods, including SVM, ANN, RF, GEP, MARS, M5 algorithm and regression equation for predicting the relative energy loss in constrictions along the flow path. Experiments were conducted within the Froude number range of 2.85–8.85. 70% of the data were used for the training phase and 30% for the testing phase for all mentioned models. Experimental results indicate that the arc-shaped constriction leads to a relative energy loss in the range of 0.50–0.89. In the SVM model, the examination of various kernels revealed that the linear kernel outperforms polynomial, RBF, and Sigmoid kernels when compared to experimental results. The statistical indicators of the correlation coefficient (R), RMSE, mean percentage relative error (average RE%), BIAS, DR, SI, and KGE for the SVM-Linear model in the testing phase are 0.994, 0.0101, 1.33%, −0.0002, 0.999, 0.0105 and 0.980, respectively. For the ANN method with MLP and RBF networks, the ANN-MLP approach shows more accurate results compared to other network types. Specifically, the statistical indicators for ANN-MLP in the testing phase are *R* = 0.997, RMSE = 0.0069, average RE = 0.63%, BIAS = 0.0004, DR = 0.999, SI = 0.0098, and KGE = 0.995. In the RF model, results are comparatively weaker than the other models. ANN-MLP outperforms SVM, GEP, M5, MARS and RF models and is closer to experimental results. It should be noted that the MARS model yields results very close to the ANN Model, and the equation provided by the MARS model can confidently be used. The statistical indicators for the MARS model in the testing phase are *R* = 0.997, RMSE = 0.0074, average RE% = 0.71%, BIAS = −0.0005, DR = 0.999, SI = 0.0106, and KGE = 0.995. The non-linear polynomial regression equation, in comparison to the MARS equation, exhibits relatively lower accuracy but can be used with high confidence. The non-linear regression and MARS relationships are presented in the scope of the present research.

## FUNDING

There is no funding source.

## 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.