## ABSTRACT

The discharge capacity of the piano key weir (PKW) is an important flow feature which ultimately decides the design of PKWs. In the present research work, the different architecture of artificial neural networks (ANNs) was employed to predict the discharge capacity of the trapezoidal piano key weir (TPKW) by varying geometric parameters. Furthermore, adaptive neuro-fuzzy interference system (ANFIS), support vector machines (SVMs) and non-linear regression (RM) techniques were also applied to compare the performance of best ANN models. The performance of each model was evaluated using statistical indices including scatter-index (SI); coefficient of determination (R^{2}) and mean square error (MSE). The prediction capability of all the models was found to be satisfactory. However, results predicted by ANN-22(H-15) [R^{2} = 0.998, MSE= 0.0024, SI = 0.0177] were more accurate than ANFIS (R^{2} = 0.995, MSE = 0.00039, SI=0.0256), SVM (R^{2} = 0.982, MSE = 0.000864, SI =0.0395) and RM (R^{2} = 0.978, MSE = 0.001, SI = 0.0411). It was observed that *S*_{i}*/S*_{o}, *W*_{i}*/W*_{o} and *L/W* ratios have the greatest effect on the discharge performance of TPKW. Furthermore, sensitivity analysis confirmed that h/P is the most influencing ratio which may considerably affect the discharge efficiency of the TPKW and ANN architecture having a single hidden layer and keeping neurons three times of input parameters in hidden layers generated better results.

## HIGHLIGHTS

Application of artificial neural networks in the prediction of discharge capacity of a trapezoidal piano key weir.

Experimental study of the type-A trapezoidal piano key weir.

Prediction of most influencing geometric parameters on the discharge capacity of a trapezoidal piano key weir.

## INTRODUCTION

Parameters . | Definition . |
---|---|

S _{i}, S_{o} | Slope of the inlet and outlet keys (m/m), respectively |

W _{i}, W_{o} | Inlet and outlet keys width, respectively |

B _{i}, B_{o} | Inlet and outlet overhang lengths, respectively |

B _{b} | Weir base length |

L | Total developed crest length |

P | Weir height |

T _{s} | Thickness of the side wall |

W | Total width of the TPKW |

W _{u} | Width of a unit cycle |

α | Side wall angle |

N _{u} | Number of TPKW cycles |

R | Height of parapet wall |

h | Head above the crest of TPKW |

Y | Depth of water at the u/s of TPKW |

σ | Surface tension |

g | Gravitational acceleration |

μ | Viscosity of water |

ρ | Density of water |

Q | Discharge passing over the TPKW |

Parameters . | Definition . |
---|---|

S _{i}, S_{o} | Slope of the inlet and outlet keys (m/m), respectively |

W _{i}, W_{o} | Inlet and outlet keys width, respectively |

B _{i}, B_{o} | Inlet and outlet overhang lengths, respectively |

B _{b} | Weir base length |

L | Total developed crest length |

P | Weir height |

T _{s} | Thickness of the side wall |

W | Total width of the TPKW |

W _{u} | Width of a unit cycle |

α | Side wall angle |

N _{u} | Number of TPKW cycles |

R | Height of parapet wall |

h | Head above the crest of TPKW |

Y | Depth of water at the u/s of TPKW |

σ | Surface tension |

g | Gravitational acceleration |

μ | Viscosity of water |

ρ | Density of water |

Q | Discharge passing over the TPKW |

PKWs can serve as effective side weirs within irrigation canals, serving various functions such as flow rate measurement and water level elevation and they are found to be hydraulically superior to linear weirs (Saghari *et al.* 2019). PKWs can also be used on the crest of a spillway due to their greater discharge passing capacity in a limited spillway width. Foroudi *et al.* (2022) assessed the hydraulic performance of the arched plan stepped spillway by varying the downstream channel width. Roushangar *et al.* (2020) performed an experimental study to investigate the hydraulic performance of ogee spillways by converging its training walls ranging from 0 to 120°. Numerous research has been carried out on the geometric parameters of PKW and their effect on flow characteristics has been investigated. Khassaf *et al.* (2015) executed research on type B rectangular PKW by varying the geometric and hydraulic parameters and examining their effect on the flow efficiency of the weir and found that the *L/W* ratio was found to be a more effective geometric parameter regarding the flow efficiency. Iqbal & Ghani (2024) performed an experimental study to assess the energy dissipation over a type-A piano key weir by varying the inlet to outlet key slope and key width ratios. Idrees & Al-Ameri (2023) investigated a new approach to increase the discharge capacity of labyrinth weirs by modifying their geometric shape.

Recently advanced, multi-dimensional, parametric, and non-parametric prediction machine learning (ML) models have been developed and are being progressively used in almost all areas of sciences (Moghaddas *et al.* 2021; Shekhar *et al.* 2023; Syed *et al.* 2023). Tutsoy & Tanrikulu (2022) successfully applied a parametric model approach in medical sciences for the prediction of future pandemic casualties with pharmacological and non-pharmacological policies. Parametric (mathematical) and non-parametric (statistical and ML) approaches have the ability to develop complex linear and non-linear relationships among variables. Modeling of the discharge coefficient of a weir can be successfully done by using the parametric and non-parametric approaches (Akbari *et al.* 2019; Norouzi *et al.* 2019; Olyaie *et al.* 2019; Seyedian *et al.* 2023). Parametric modeling approaches suit their purpose and are also parameterizable by the available data because they have a certain model structure that represents the mathematical relationships as simply as possible (Tutsoy *et al.* 2018).

A neural network (NN) is a non-parametric ML algorithm which is considered the most successful approach in the prediction of variables. Tutsoy & Polat (2022) applied two ML approaches: recursive neural network and learning non-linear dynamics in the prediction of pandemic outbreaks using non-pharmacological policies. The results indicated that the recursive neural network has superior performance for learning non-linear dynamics.

Artificial neural networks (ANNs) have gained prominence as a valuable resource for predicting parameters related to water resources (Dawson & Wilby 2001). Many engineering tasks, including rainfall simulation, groundwater level fluctuations, determining discharge coefficients for weirs, predicting traffic flow variations, assessing bridge pier scouring, estimating pile bearing capacities, characterizing concrete mechanical properties, and forecasting damage to offshore wind turbines, involve complex non-linear relationships among variables that can be successfully modeled through ANN (Qiu *et al.* 2020). Norouzi *et al.* (2019) predicted discharge capacity of trapezoidal labyrinth weir by using ANN and SVM. Numerous past studies have recommended the use of ANNs as a superior approach for simulating PKW discharge when compared to empirical relationships. Karami *et al.* (2018) performed a study to analyze the *C _{d}* of a triangular labyrinth weir by applying three artificial intelligence models: ANN, GP, and extreme learning machine (ELM). The ELM produced outstanding results compared to other tested models. Zaji

*et al.*(2016) applied the support vector regression (SVR) approach in the prediction of

*C*of oblique side weirs. Their results revealed that the SVR-RBF model performed better than the SVR-poly model. Najafzadeh & Azamathulla (2015) applied a neuro-fuzzy GMDH to predict scouring around pile groups. Najafzadeh & Azamathulla (2013) used a group method of data handling for the prediction of scouring around the bridge piers.

_{d}Kashkaki *et al.* (2018) successfully employed the ANN technique to estimate coefficient of discharge of a circular PKW spillway. Haghiabi *et al.* (2018) applied adaptive Neuro-Fuzzy Inference System (ANFIS) and MLP approaches for the prediction of coefficient of discharge of a labyrinth weir. The obtained results revealed that the ANFIS technique was found to be more competent than the MLP. Akbari *et al.* (2019) performed a study to evaluate the performance of MLP, GPR, SVM, GRNN, multiple linear, and non-linear regression models (RMs) on the discharge capacity of a piano key weir. The results obtained from the GPR model were found to be more accurate than all other methods. Norouzi *et al.* (2019) assessed the *C _{d}* of the non-linear weir by comparing the performance of ANN and SVM. The results obtained from both techniques were acceptable, but the ANN results were closer to the experimental results than the SVM model.

Dutta *et al.* (2020) conducted research to estimate the discharge capacity of multi-cycle W-form and circular arc labyrinth weirs by applying multiple linear regression, support vector machine, and ANN models. Olyaie *et al.* (2019) estimated the *C _{d}* of piano key weirs by applying high-accuracy ML approaches including least-square SVM, ELM, Bayesian ELM, and logistic regression (LR). The simulated results indicated that the ELM approach achieved better results in comparison with other tested approaches. Seyedian

*et al.*(2023) predicted the discharge coefficient of the triangular labyrinth weir by applying various ML models, including least-square support vector machine (LS-SVM), quantile regression forest (QRF), and Gaussian process regression (GPR). The simulated results show that GPR is a superior approach to other tested approaches.

The study by Haghiabi *et al.* (2018) involved the prediction of discharge efficiency for triangular labyrinth weirs using both ANN and ANFIS models. Olyaie *et al.* (2019) predicted the discharge capacity of PKW under subcritical free flow conditions using high-accuracy ML approaches. Bashiri *et al.* (2016) applied the Levenberg–Marquardt backpropagation algorithm of an ANN to create a novel design equation for estimating the discharge capacity of PKW. Gharehbaghi *et al.* (2023) developed a study in the comparison of artificial intelligence approaches in predicting coefficient of discharge of streamlined weirs.

In the present study, ANN has been adopted and preferred over the other artificial intelligence models because ANN has the capability to realize intricate patterns from complex datasets, adapt to non-linear relationships, and simplify well to new and unseen data. In previous valuable research, ANN successfully predicted the discharge capacity of different types of weirs and the results simulated by ANN were very close to the actual results. As far as uncertainty during the training of ANN models is concerned, it reduces the reliability of the prediction model. Uncertainties in models can be internal, external, parametric, and non-parametric. The model input data, parameters, and structure uncertainty are mainly considered the source of uncertainties in the prediction model. The architecture of the model including the number of input layers, number of neurons, nodes, activation function and training algorithm has often been optimized to improve the model accuracy, but not in terms of model uncertainty. The uncertainty quantification in the ANN model is a recent development in the field, which emerged from the year 2002. Regarding the uncertainties in ANN modeling, there are different methods found in the literature which were used to quantify the uncertainties in ANN models. Some authors (Abbaspour *et al.* 2007; Talebizadeh & Moridnejad 2011) applied *d*-factor and *p*-factor indices in their studies to measure and quantify the uncertainties in different predictive models. Further investigation is still demanded in the literature regarding the effective use of various uncertainty evaluation indices for the meaningful quantification of uncertainties during the training of ANN models.

TPKW has a highly complicated geometry consisting of more than 30 geometric parameters (Pralong *et al.* 2011). Despite previously valuable research on the prediction of the discharge capacity of labyrinth and PKW, designers and researchers are still facing a challenge to assess which parameters have the most significant impact on the discharge efficiency of TPKW. ANNs and RMs are experts in developing some relation between parameters in such types of complex problems. The primary goal of this research was to identify the most influential geometric parameters affecting the discharge efficiency of TPKW and to assess prediction efficacy between the ANN and RMs. In the present research work, soft computing tools including ANN, ANFIS, SVM and regression analysis techniques have been applied to investigate the discharge capacity of experimentally tested TPKW. For that, various ANN models have been trained using different combinations of geometric parameters (*S _{i}/S_{o}, W_{i}/W_{o}, B_{i}/B_{o}, L/W and α*), and different architectures of ANN (number of neurons, number of hidden layers, type of activation function between different layers), keeping same hydraulic parameters (

*h/P*and

*F*).

_{r}## MATERIALS AND METHODS

### Experimental setup and collection of databank

*h*is head above the weir crest) upstream of the TPKW for the calculation of coefficient of discharge () and upstream Froud number () (Bekheet

*et al.*2022). The and was calculated using Equations (1) and (2), respectively:

where *Q* represents the discharge over the TPKW, represents the coefficient of discharge of TPKW, *w* denotes the width of the weir, *g* represents the acceleration due to gravity, *h* indicates the head above the crest of the weir, *v* represents the velocity of flow at the upstream of the weir, *y* is the total depth of water at the upstream of the weir, and denotes the upstream Froude number.

A data bank is very crucial for the training of models, and it serves as the foundation for model training and generation. So, for the collection of data bank a series of laboratory experiments were performed to obtain the of TPKW by varying seven different cases of inlet to outlet key slope ratios (*S _{i}/S_{o}*) and four different key width ratios (

*W*) of TPKW at eight different discharges ranged from 16.01 to 24.88 lit/sec. In the variation of each

_{i}/W_{o}*S*and

_{i}/S_{o}*W*model case, respective changes in every geometric parameter (

_{i}/W_{o}*S*and

_{i}, S_{o}, W_{i}, W_{o}, B_{i}, B_{o}, L,*α*) of TPKW were calculated, accordingly. For the collection of data of hydraulic parameters (

*h/P*,

*C*, and

_{d}*F*), experimental test runs were performed in the laboratory and required necessary measurements were taken upstream of the TPKW and accordingly using experimental readings, the value of

_{r}*C*and

_{d}*F*were calculated using Equations (1) and (2), respectively. After collecting the necessary data bank for the training of models, data was arranged and finally used to achieve the objective of the present study. A total of 224 experimental test runs were performed and the same were the total data points which were used later on for the training of models. In the present study head (

_{r}*h*) above the crest of TPKR varied from 0.03 to 0.0778 m and

*h/P*varied from 0.15 to 0.389. The other geometric parameters including the weir crest height (

*P*), number of cycles (

*),*thickness of the side walls () and the base width ()) of the TPKW models were remained fixed throughout the experimentation. Table 2 shows the geometric parameters that were used in the present study.

TPKW model . | TPKW base width (), cm
. | Height of TPKW (P) cm
. | Thickness of the side wall () cm
. | Number of TPKW cycles () . | Inlet to outlet key slope ratios . | Inlet to outlet key width ratios (W)
. _{i}/W_{o} | Inlet to outlet over hanged length . | Side wall angle (α)
. | Overflowing crest length of TPKW (L) cm
. |
---|---|---|---|---|---|---|---|---|---|

TPKW-1 | 25 | 20 | 1 | 3 | 0.60 | 0.76, 1.0, 1.15, 1.32 | 0.20 | 0.44–0.64° | 355 |

TPKW-2 | 25 | 20 | 1 | 3 | 0.75 | 0.76, 1.0, 1.15, 1.32 | 0.33 | 0.44–0.64° | 295 |

TPKW-3 | 25 | 20 | 1 | 3 | 0.80 | 0.76, 1.0, 1.15, 1.32 | 0.60 | 0.44–0.64° | 415 |

TPKW-4 | 25 | 20 | 1 | 3 | 1.00 | 0.76, 1.0, 1.15, 1.32 | 1.00 | 0.44–0.64° | 355 |

TPKW-5 | 25 | 20 | 1 | 3 | 1.25 | 0.76, 1.0, 1.15, 1.32 | 1.67 | 0.44–0.64° | 415 |

TPKW-6 | 25 | 20 | 1 | 3 | 1.33 | 0.76, 1.0, 1.15, 1.32 | 3.00 | 0.44–0.64° | 295 |

TPKW-7 | 25 | 20 | 1 | 3 | 1.67 | 0.76, 1.0, 1.15, 1.32 | 5.00 | 0.44–0.64° | 355 |

TPKW model . | TPKW base width (), cm
. | Height of TPKW (P) cm
. | Thickness of the side wall () cm
. | Number of TPKW cycles () . | Inlet to outlet key slope ratios . | Inlet to outlet key width ratios (W)
. _{i}/W_{o} | Inlet to outlet over hanged length . | Side wall angle (α)
. | Overflowing crest length of TPKW (L) cm
. |
---|---|---|---|---|---|---|---|---|---|

TPKW-1 | 25 | 20 | 1 | 3 | 0.60 | 0.76, 1.0, 1.15, 1.32 | 0.20 | 0.44–0.64° | 355 |

TPKW-2 | 25 | 20 | 1 | 3 | 0.75 | 0.76, 1.0, 1.15, 1.32 | 0.33 | 0.44–0.64° | 295 |

TPKW-3 | 25 | 20 | 1 | 3 | 0.80 | 0.76, 1.0, 1.15, 1.32 | 0.60 | 0.44–0.64° | 415 |

TPKW-4 | 25 | 20 | 1 | 3 | 1.00 | 0.76, 1.0, 1.15, 1.32 | 1.00 | 0.44–0.64° | 355 |

TPKW-5 | 25 | 20 | 1 | 3 | 1.25 | 0.76, 1.0, 1.15, 1.32 | 1.67 | 0.44–0.64° | 415 |

TPKW-6 | 25 | 20 | 1 | 3 | 1.33 | 0.76, 1.0, 1.15, 1.32 | 3.00 | 0.44–0.64° | 295 |

TPKW-7 | 25 | 20 | 1 | 3 | 1.67 | 0.76, 1.0, 1.15, 1.32 | 5.00 | 0.44–0.64° | 355 |

### Selection of input parameters

*et al.*(2012), the discharge over a TPKW is a function of the following parameters given in Equation (3):

*et al.*(2020). The nappe head over the weir was more than 3 cm for all experimental tests and the flow was entirely turbulent throughout the experiments, so neglecting the effect of and , Equation (5) can be written as Equation (6):

After performing the dimensional analysis, it is found that is the function of the dimensionless parameters given in Equation (6). So, based on the dimensional analysis the geometric ratios were selected as the input parameters of the ANN and regression model, while was selected as the output parameter of the ANN and regression model.

### Normalization of parameters

*et al.*2020). Table 3 shows the statistical analysis of the normalized parameters:where

*X*is the normalized value, value of the parameter before normalization.

Name of parameters . | Minimum value in the data . | Maximum value in the data . | Difference . | Average . | St. Dev . | COV . |
---|---|---|---|---|---|---|

h/P | 0.15 | 0.389 | 0.2475 | 0.24 | 0.06 | 0.25 |

F _{r} | 0.132 | 0.20136 | 0.06929 | 0.17 | 0.02 | 0.12 |

S _{i}/S_{o} | 0.6 | 1.67 | 1.07 | 1.06 | 0.35 | 0.33 |

W _{i}/W_{o} | 0.76 | 1.32 | 0.56 | 1.06 | 0.21 | 0.2 |

B _{i}/B_{o} | 0.2 | 5 | 4.8 | 1.69 | 1.62 | 0.96 |

L/W | 9.516 | 13.3871 | 3.87097 | 11.45 | 1.47 | 0.13 |

α | 0.441 | 0.63659 | 0.19587 | 0.53 | 0.07 | 0.13 |

C _{d} | 1.252 | 3.67314 | 2.42084 | 2.2 | 0.53 | 0.24 |

Name of parameters . | Minimum value in the data . | Maximum value in the data . | Difference . | Average . | St. Dev . | COV . |
---|---|---|---|---|---|---|

h/P | 0.15 | 0.389 | 0.2475 | 0.24 | 0.06 | 0.25 |

F _{r} | 0.132 | 0.20136 | 0.06929 | 0.17 | 0.02 | 0.12 |

S _{i}/S_{o} | 0.6 | 1.67 | 1.07 | 1.06 | 0.35 | 0.33 |

W _{i}/W_{o} | 0.76 | 1.32 | 0.56 | 1.06 | 0.21 | 0.2 |

B _{i}/B_{o} | 0.2 | 5 | 4.8 | 1.69 | 1.62 | 0.96 |

L/W | 9.516 | 13.3871 | 3.87097 | 11.45 | 1.47 | 0.13 |

α | 0.441 | 0.63659 | 0.19587 | 0.53 | 0.07 | 0.13 |

C _{d} | 1.252 | 3.67314 | 2.42084 | 2.2 | 0.53 | 0.24 |

St. Dev, standard deviation; COV, coefficient of variance.

### Description of various models

#### Adaptive neuro-fuzzy interference system

*et al.*2018). Neuro-fuzzy algorithms explain the behavior of complex non-linear relationships between inputs and outputs using fuzzy logic rules within the structure of the neural network. ANFISs are subsets of neuro-fuzzy algorithms (Jang 1993). The architecture of ANFIS consists of five layers. The input values are entered in the first layer and the membership functions are assigned to the input nodes. ANFIS applications are in various fields including control systems, engineering, financial forecasting, pattern recognition etc. Considering an ANFIS model with two inputs, the output of the first layer () can be calculated as follows (Zounemat-Kermani & Mahdavi-Meymand 2019):where and are membership functions and

*x*represents the input at node

*i*.

#### Support vector machines

*et al.*2019; Norouzi

*et al.*2019). Additionally, SVMs could handle linear and non-linear data effectively using kernel functions and the selection of kernel functions is important which can highly influence the performance of models. In the present study, linear, non-linear, and polynomial kernal functions were used for the transformation of information. SVMs can be efficiently designed for regression analysis by modifying the classification algorithms (Seyedian

*et al.*2023). In SVR, the goal is to find a function f(

*x*), that deviates from the actual target values. According to Najafzadeh & Mahmoudi-Rad (2024), the functional relationship of f(

*x*) can be expressed as Equation (9):where f(

*x*) is the function, (

*w*,

*x*) are points in

*X*and

*w*is the weight factor and

*b*is additive noise.

#### Non-linear regression analysis

Regression analysis is a statistical tool which can be used to develop some relation between the variables. Non-linear regression analysis is commonly used in almost every field including engineering, environmental sciences, medicine, and economics to develop relationships between the variables, when the relationship between the variables is complex and non-linear (Akbari *et al.* 2019). Non-linear relationships between the variables could be demonstrated using polynomial, exponential, logarithmic, sigmoidal, or any other non-linear form. In the present study, a logarithmic non-linear relationship was developed between the input and output parameters. It specifies a flexible architecture for modeling complex relationships and making predictions based on observed data. However, it can be more challenging to implement and interpret compared to linear regression due to the complexity of non-linear models and the potential for convergence issues during parameter estimation.

#### Sensitivity analysis

Sensitivity analysis is a technique which can be used to examine the robustness and reliability of model outputs in response to variation of input parameters while holding all other parameters as constant. The alteration in model output due to the change in the input parameter is referred to as the sensitivity (Wan *et al.* 2024). It helps analysts to see how the variation of input parameters affects the model's results and conclusions. This method is applicable to deterministic models and cannot be applied to probabilistic analysis. By applying sensitivity techniques one can identify the most important input parameter which has the greatest effect on the model's outputs. It is extensively used across various fields, including finance, engineering, environmental science, and public policy.

#### Artificial neural networks

An ANN is a computer-developed algorithm and its working mechanism is similar to the human brain. They find widespread application across various scientific disciplines, enabling the prediction or approximation of functions that rely on multiple parameters whose effects are not easily established or quantified (Turhan 1995; Norouzi *et al.* 2019). Due to their adoptive nature and ability to memorize the info given to them during the training process, ANN can easily recognize, generalize, and empirically predict the values (Norouzi *et al.* 2019). ANN model structure generally includes an input layer, one or multiple hidden layers, and an output layer (Iqbal *et al.* 2020). Within each layer, there are neurons responsible for transmitting information from one layer to the next, facilitated by activation functions. This layered arrangement is crucial for the network's capacity to handle complex data and formulate predictions.

*b*is the bias value

Each ANN model given in Table 4 was trained at three different numbers of neurons (one, two and three times of input parameters) in hidden layers and three different numbers of hidden layers (with single, double, and triple hidden layers).

Models . | ANN input parameters . | No. of neurons in hidden layers . | No. of hidden layers . | Allocation of data percentage for training, validation, and testing . | ANN output parameter . |
---|---|---|---|---|---|

ANN-1 | h/P, Fr, B _{i}/B_{o} | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |

ANN-2 | h/P, Fr, S _{i}/S_{o} | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |

ANN-3 | h/P, Fr, W _{i}/W_{o} | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |

ANN-4 | h/P, Fr, L/W | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |

ANN-5 | h/P, Fr, α | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |

ANN-6 | h/P, Fr, B _{i}/B_{o}, S_{i}/S_{o} | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-7 | h/P, Fr, B _{i}/B_{o}, W_{i}/W_{o} | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-8 | h/P, Fr, B _{i}/B_{o}, L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-9 | h/P, Fr, B _{i}/B_{o}, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-10 | h/P, Fr, S _{i}/S_{o}, W_{i}/W_{o} | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-11 | h/P, Fr, S _{i}/S_{o}, L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-12 | h/P, Fr, S_{i}/S_{o}, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-13 | h/P, Fr, W _{i}/W_{o,} L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-14 | h/P, Fr, W _{i}/W_{o}, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-15 | h/P, Fr, L/W, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-16 | h/P, Fr, B _{i}/B_{o}, S_{i}/S_{o}, W_{i}/W_{o} | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-17 | h/P, Fr, B _{i}/B_{o}, S_{i}/S_{o}, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-18 | h/P, Fr, B _{i}/B_{o}, S_{i}/S_{o}, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-19 | h/P, Fr, B _{i}/B_{o}, W_{i}/W_{o}, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-20 | h/P, Fr, B _{i}/B_{o}, W_{i}/W_{o}, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-21 | h/P, Fr, B _{i}/B_{o}, L/W, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-22 | h/P, Fr, S _{i}/S_{o}, W_{i}/W_{o}, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-23 | h/P, Fr, S _{i}/S_{o}, W_{i}/W_{o}, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-24 | h/P, Fr, W _{i}/W_{o}, L/W, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-25 | h/P, Fr, S _{i}/S_{o}, W_{i}/W_{o}, L/W, α | 6, 10, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-26 | h/P, Fr, B _{i}/B_{o} S_{i}/S_{o}, W_{i}/W_{o}, L/W | 6, 12, 18 | 1, 2, 3 | 80, 10, 10% | |

ANN-27 | h/P, Fr, B _{i}/B_{o} S_{i}/S_{o}, W_{i}/W_{o}, α | 6, 12, 18 | 1, 2, 3 | 80, 10, 10% | |

ANN-28 | h/P, Fr, B _{i}/B_{o}, S_{i}/S_{o}, W_{i}/W_{o}, L/W, α | 7, 14, 21 | 1, 2, 3 | 80, 10, 10% |

Models . | ANN input parameters . | No. of neurons in hidden layers . | No. of hidden layers . | Allocation of data percentage for training, validation, and testing . | ANN output parameter . |
---|---|---|---|---|---|

ANN-1 | h/P, Fr, B _{i}/B_{o} | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |

ANN-2 | h/P, Fr, S _{i}/S_{o} | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |

ANN-3 | h/P, Fr, W _{i}/W_{o} | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |

ANN-4 | h/P, Fr, L/W | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |

ANN-5 | h/P, Fr, α | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |

ANN-6 | h/P, Fr, B _{i}/B_{o}, S_{i}/S_{o} | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-7 | h/P, Fr, B _{i}/B_{o}, W_{i}/W_{o} | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-8 | h/P, Fr, B _{i}/B_{o}, L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-9 | h/P, Fr, B _{i}/B_{o}, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-10 | h/P, Fr, S _{i}/S_{o}, W_{i}/W_{o} | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-11 | h/P, Fr, S _{i}/S_{o}, L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-12 | h/P, Fr, S_{i}/S_{o}, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-13 | h/P, Fr, W _{i}/W_{o,} L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-14 | h/P, Fr, W _{i}/W_{o}, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-15 | h/P, Fr, L/W, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-16 | h/P, Fr, B _{i}/B_{o}, S_{i}/S_{o}, W_{i}/W_{o} | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-17 | h/P, Fr, B _{i}/B_{o}, S_{i}/S_{o}, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-18 | h/P, Fr, B _{i}/B_{o}, S_{i}/S_{o}, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-19 | h/P, Fr, B _{i}/B_{o}, W_{i}/W_{o}, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-20 | h/P, Fr, B _{i}/B_{o}, W_{i}/W_{o}, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-21 | h/P, Fr, B _{i}/B_{o}, L/W, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-22 | h/P, Fr, S _{i}/S_{o}, W_{i}/W_{o}, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-23 | h/P, Fr, S _{i}/S_{o}, W_{i}/W_{o}, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-24 | h/P, Fr, W _{i}/W_{o}, L/W, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |

ANN-25 | h/P, Fr, S _{i}/S_{o}, W_{i}/W_{o}, L/W, α | 6, 10, 12 | 1, 2, 3 | 80, 10, 10% | |

ANN-26 | h/P, Fr, B _{i}/B_{o} S_{i}/S_{o}, W_{i}/W_{o}, L/W | 6, 12, 18 | 1, 2, 3 | 80, 10, 10% | |

ANN-27 | h/P, Fr, B _{i}/B_{o} S_{i}/S_{o}, W_{i}/W_{o}, α | 6, 12, 18 | 1, 2, 3 | 80, 10, 10% | |

ANN-28 | h/P, Fr, B _{i}/B_{o}, S_{i}/S_{o}, W_{i}/W_{o}, L/W, α | 7, 14, 21 | 1, 2, 3 | 80, 10, 10% |

### ANN architecture adopted for the current work

*et al.*2018). ANN models were trained using different architectures, such as (

*i*) by varying the number of neurons in input layers, (

*ii*) By adjusting the number of hidden layers and the number of neurons within these hidden layers in each ANN training model, aim to identify the most effective ANN prediction model.

The dataset utilized for training, validation, and testing of the ANN models was divided into three portions: 80% for training, 10% for validation, and 10% for testing. In this study, all ANN models achieved a maximum of 100 epochs during the multilayer feed-forward backpropagation (MLFFBP) process. The process concluded when any of the following conditions were met: (*i*) the difference between target and output values reached zero, (*ii*) no further enhancements in ANN performance were observed during iteration, or (*iii*) the minimum performance gradient reached a value of 10^{−10}. The conditions which are currently adopted were proposed by Levenberg–Marquardt backpropagation.

*T*represents the target value,

*O*expressed the output value

*et al.*2012) and this process is started from right to left in the ANN architecture as shown in Figure 4. The purpose of using this approach is to refine the initially randomized weight values in a manner that brings the output values of the ANN into close alignment with the target values provided in the dataset. These refined weights are then applied within the ANN to produce more accurate predictions and to minimize resulting errors. This iterative procedure is reiterated until the resulting error reaches acceptable thresholds, guaranteeing the highest level of precision in the ANN predictions.

## RESULTS AND DISCUSSION

*B*) of TPKW, and different architectural configurations of ANN; including the number of neurons in hidden layers, number of hidden layers, and type of activation functions between different layers, as presented in Table 4. The

_{i}/B_{o}, S_{i}/S_{o}, W_{i}/W_{o}, L/W, α*h/P*and

*F*remained fixed in the input layer of ANN for all models, only geometric parameters were varied in the input layer of ANN to make different combinations of geometric parameters. To evaluate the performance of various trained ANN models, the following statistical indices including

_{r}*MSE*(mean square error),

*R*(coefficient of determination), and

^{2}*MAE*(mean absolute error) were used. Out of 252 trained ANN models, 16 best ANN models were selected based on the maximum

*R*

^{2}value and minimum

*MSE*and

*MAE*values. The best-selected ANN models are presented in Table 5 and terminology to understand the architecture of ANN used in Table 5 is given in Figure 5.

ANN models . | Type of activation function assigned in various ANN layers . | Training . | Validation . | Testing . | Overall . | |||
---|---|---|---|---|---|---|---|---|

ANN-2, H-9 | Tan-sigmoid | Tan-sigmoid | 0.98043 | 0.97027 | 0.96147 | 0.9803 | ||

ANN-4, H-6 | Tan-sigmoid | Tan-sigmoid | 0.98438 | 0.96199 | 0.98957 | 0.98208 | ||

ANN-8, HHH-4 | Log-sigmoid | Tan-sigmoid | Tan-sigmoid | Tan-sigmoid | 0.98589 | 0.98726 | 0.98421 | 0.98534 |

ANN-10, H-4 | Tan-sigmoid | Tan-sigmoid | 0.9864 | 099046 | 0.98261 | 0.98629 | ||

ANN-11, HHH-4 | Log-sigmoid | Tan-sigmoid | Tan-sigmoid | Tan-sigmoid | 0.98391 | 0.98302 | 0.97561 | 0.98297 |

ANN-12, HHH-4 | Log-sigmoid | Tan-sigmoid | Tan-sigmoid | Tan-sigmoid | 0.98451 | 0.99034 | 0.98937 | 0.98546 |

ANN-13, H-8 | Tan-sigmoid | Tan-sigmoid | 0.98899 | 0.98337 | 0.98977 | 0.9889 | ||

ANN-15, H-12 | Tan-sigmoid | Tan-sigmoid | 0.9889 | 0.98542 | 0.98941 | 0.98816 | ||

ANN-16, H-10 | Tan-sigmoid | Tan-sigmoid | 0.989 | 0.99016 | 0.98667 | 0.98876 | ||

ANN-17, HHH-5 | Log-sigmoid | Tan-sigmoid | Tan-sigmoid | Tan-sigmoid | 0.98787 | 0.97622 | 0.98723 | 0.98621 |

ANN-19, H-5 | Tan-sigmoid | Tan-sigmoid | 0.98946 | 0.98839 | 0.98354 | 0.98879 | ||

ANN-21, H-10 | Tan-sigmoid | Tan-sigmoid | 0.99131 | 0.98763 | 0.98381 | 0.98115 | ||

ANN-22, H-15 | Tan-sigmoid | Tan-sigmoid | 0.99769 | 0.99722 | 0.99739 | 0.99755 | ||

ANN-23, H-15 | Tan-sigmoid | Tan-sigmoid | 0.98513 | 0.99015 | 0.9924 | 0.98438 | ||

ANN-24, H-15 | Tan-sigmoid | Tan-sigmoid | 0.98208 | 0.98254 | 0.9837 | 0.98226 | ||

ANN-25, H-6 | Tan-sigmoid | Tan-sigmoid | 0.98672 | 0.96957 | 0.98135 | 0.98413 |

ANN models . | Type of activation function assigned in various ANN layers . | Training . | Validation . | Testing . | Overall . | |||
---|---|---|---|---|---|---|---|---|

ANN-2, H-9 | Tan-sigmoid | Tan-sigmoid | 0.98043 | 0.97027 | 0.96147 | 0.9803 | ||

ANN-4, H-6 | Tan-sigmoid | Tan-sigmoid | 0.98438 | 0.96199 | 0.98957 | 0.98208 | ||

ANN-8, HHH-4 | Log-sigmoid | Tan-sigmoid | Tan-sigmoid | Tan-sigmoid | 0.98589 | 0.98726 | 0.98421 | 0.98534 |

ANN-10, H-4 | Tan-sigmoid | Tan-sigmoid | 0.9864 | 099046 | 0.98261 | 0.98629 | ||

ANN-11, HHH-4 | Log-sigmoid | Tan-sigmoid | Tan-sigmoid | Tan-sigmoid | 0.98391 | 0.98302 | 0.97561 | 0.98297 |

ANN-12, HHH-4 | Log-sigmoid | Tan-sigmoid | Tan-sigmoid | Tan-sigmoid | 0.98451 | 0.99034 | 0.98937 | 0.98546 |

ANN-13, H-8 | Tan-sigmoid | Tan-sigmoid | 0.98899 | 0.98337 | 0.98977 | 0.9889 | ||

ANN-15, H-12 | Tan-sigmoid | Tan-sigmoid | 0.9889 | 0.98542 | 0.98941 | 0.98816 | ||

ANN-16, H-10 | Tan-sigmoid | Tan-sigmoid | 0.989 | 0.99016 | 0.98667 | 0.98876 | ||

ANN-17, HHH-5 | Log-sigmoid | Tan-sigmoid | Tan-sigmoid | Tan-sigmoid | 0.98787 | 0.97622 | 0.98723 | 0.98621 |

ANN-19, H-5 | Tan-sigmoid | Tan-sigmoid | 0.98946 | 0.98839 | 0.98354 | 0.98879 | ||

ANN-21, H-10 | Tan-sigmoid | Tan-sigmoid | 0.99131 | 0.98763 | 0.98381 | 0.98115 | ||

ANN-22, H-15 | Tan-sigmoid | Tan-sigmoid | 0.99769 | 0.99722 | 0.99739 | 0.99755 | ||

ANN-23, H-15 | Tan-sigmoid | Tan-sigmoid | 0.98513 | 0.99015 | 0.9924 | 0.98438 | ||

ANN-24, H-15 | Tan-sigmoid | Tan-sigmoid | 0.98208 | 0.98254 | 0.9837 | 0.98226 | ||

ANN-25, H-6 | Tan-sigmoid | Tan-sigmoid | 0.98672 | 0.96957 | 0.98135 | 0.98413 |

*et al.*(2018), Najafzadeh

*et al.*(2018), Dutta

*et al.*(2020), Saberi-Movahed

*et al.*(2020) and Najafzadeh & Oliveto (2021) the

*MSE*,

*MAE*, coefficient of determination (

*R*

^{2}), scatter index (SI), discrepancy ratio (DR), BAIS and relative absolute error (RAE) can be determined by Equations (12)–(18), respectively to evaluate the performance of various models:

*MSE, MAE*and

*R*values of training, validation, testing and overall, as shown in Figure 6(a)–6(c). The

^{2}*MSE*for the best-selected ANN models ranged from 0.00022 to 0.0018, while the

*R*

^{2}values ranged from 0.95 to 0.99, and the

*MAE*values showed variability within the range of 0.0122–0.0309.

### Optimistic ANN model

*R*

^{2}value. The

*R*

^{2}values of the optimistic ANN model for training (

*R*

^{2}= 0.99769), and testing (

*R*

^{2}= 0.99769) are presented in Figure 7. Observed values of

*C*and predicted values of

_{d}*C*by ANN-22(H-15) are plotted with respect to data points for both training (a) and testing (b) datasets as shown in Figure 7. Based on the observed and predicted results, it can be concluded that ANN is a more powerful soft computing tool in the prediction of the discharge capacity of TPKW, because the values of

_{d}*C*predicted by ANN are very close to the observed values of

_{d}*C*. Figure 7(c) shows the behavior of loss error and gradient graph of ANN-22(H-15). It can be seen that up to an epoch of 100, ANN-22(H-15) converged and produced the required results.

_{d}### Selection of most influencing geometric parameters based on the ANN results

The main objective of this study was to investigate the most influencing geometric parameters on the discharge capacity of TPKW. From the ANN results, it is revealed that all the selected geometric parameters (*B _{i}/B_{o}, S_{i}/S_{o}, W_{i}/W_{o}, L/W, α*) have a significant impact on the discharge capacity of TPKW because the value of

*R*

^{2}for all models was found to be greater than 0.95. However, the ANN-22(H-15) model having the input parameters of

*h/P, Fr, S*was found to be an optimistic ANN model because it gave outstanding results. The

_{i}/S_{o}, W_{i}/W_{o}and L/W*MAE*,

*MSE*and

*R*values of ANN-22(H-15) were found to be 0.0124, 0.00024 and 99%, respectively. The

^{2}*h/P*and

*Fr*input parameters of ANN remained fixed for all developed ANN models. So, it is found that the

*S*, and

_{i}/S_{o}, W_{i}/W_{o}*L/W*ratios of TPKW were found to be the most influencing geometric parameters which have the greatest effect on the discharge capacity of TPKW.

### Regression analysis

*et al.*2016; Saghari

*et al.*2019; Shariq

*et al.*2022). After developing an equation, the comparison is made between the observed and computed from Equation (19) for both datasets: training, and validation as shown in Figure 8. All the computed values of fall within the ±10% of the error line, which represents a good relationship between the observed and predicted values of

### Comparative study of ANN with other models

To compare the results of ANN-22(H-15), models including ANFIS, SVM, and RMs were also trained, and their results were analyzed and compared. The quantitative comparative results of the best performance models (ANN-22(H-15), ANFIS, SVM and RM during the training and testing stages are presented in Table 6. Table 6 shows that ANN model (ANN-22(H-15 *)* indicated the highest level of precision in the prediction of *C _{d}* of TPKW as compared to predicted results of ANFIS (), SVM (), and regression model () during the training stage of models. The results during the testing of ANN-22(H-15), ANFIS, SVM, and RM were found to be (), (), (), and (), respectively. Additionally, the results of RAE and BIAS (ANN-22(H-15) ) confirmed the dominance of the ANN-22(H-15) model compared to other models including ANFIS (), SVM (), and RM () in the training phase. The results during the testing of ANN-22(H-15), ANFIS, SVM, and RM were found to be (), (), (), and (), respectively. The results of RAE close to zero show more accuracy of the model.

Model name . | Training results . | ||||||
---|---|---|---|---|---|---|---|

R^{2}
. | MSE . | MAE . | RAE . | BIAS . | SI . | DR . | |

ANN-22,H-15 | 0.99769 | 0.000251 | 0.0125 | 0.0655 | 0.003 | 0.0193 | 1.00386 |

ANFIS | 0.99513 | 0.000392 | 0.0145 | 0.0995 | 0.015 | 0.0266 | 1.01049 |

SVM | 0.98237 | 0.000864 | 0.0164 | 0.1479 | −0.020 | 0.0395 | 0.99494 |

RM | 0.97829 | 0.000906 | 0.0232 | 0.1623 | −0.016 | 0.0411 | 0.99536 |

. | Testing results
. | ||||||

ANN-22,H-15 | 0.99769 | 0.000221 | 0.0122 | 0.0794 | 0.095 | 0.0177 | 1.00703 |

ANFIS | 0.99441 | 0.000423 | 0.0145 | 0.1054 | 0.015 | 0.0256 | 1.00769 |

SVM | 0.98127 | 0.000923 | 0.0166 | 0.1579 | −0.029 | 0.0404 | 0.99060 |

RM | 0.97341 | 0.001134 | 0.0266 | 0.1731 | −0.018 | 0.0437 | 0.99280 |

Model name . | Training results . | ||||||
---|---|---|---|---|---|---|---|

R^{2}
. | MSE . | MAE . | RAE . | BIAS . | SI . | DR . | |

ANN-22,H-15 | 0.99769 | 0.000251 | 0.0125 | 0.0655 | 0.003 | 0.0193 | 1.00386 |

ANFIS | 0.99513 | 0.000392 | 0.0145 | 0.0995 | 0.015 | 0.0266 | 1.01049 |

SVM | 0.98237 | 0.000864 | 0.0164 | 0.1479 | −0.020 | 0.0395 | 0.99494 |

RM | 0.97829 | 0.000906 | 0.0232 | 0.1623 | −0.016 | 0.0411 | 0.99536 |

. | Testing results
. | ||||||

ANN-22,H-15 | 0.99769 | 0.000221 | 0.0122 | 0.0794 | 0.095 | 0.0177 | 1.00703 |

ANFIS | 0.99441 | 0.000423 | 0.0145 | 0.1054 | 0.015 | 0.0256 | 1.00769 |

SVM | 0.98127 | 0.000923 | 0.0166 | 0.1579 | −0.029 | 0.0404 | 0.99060 |

RM | 0.97341 | 0.001134 | 0.0266 | 0.1731 | −0.018 | 0.0437 | 0.99280 |

*C*of TPKW predicted by ANN-22(H-15) had more accuracy, because value of this model was found to be more than other models in the training as well as testing phase of models.

_{d}*et al.*(2024), performance prediction of various models was also examined by constructing a Taylor diagram, which was also implemented in the present study to compare the prediction performance of various models. Figure 11 demonstrates Taylor diagrams of all comparative models for both training and testing datasets. Taylor diagram is plotted between the predicted and observed standard deviation () value of all the models with their respective

*R*

^{2}value. It was observed that model ANN-22(H-15 ) and ANFIS () had the highest

*R*

^{2}value in the training datasets and their predicted standard deviation values are very close to observed value () than SVM (), and RM (). In the testing phase the value of for the model ANN-22(H-15), ANFIS, SVM and RM were found to be (), (), (), and (), respectively. However, the prediction performance of model ANN-22(H-15) was found to be significantly more satisfactory than other models because of the highest value for both training and testing datasets.

### Scatter plot error distribution comparison between ANN and other models

*et al.*2020). So, it can be concluded that the results predicted from ANN-22(H-15) are more accurate as compared to results predicted from other models.

### Uncertainty analysis

*d*-factor is used to measure and quantify uncertainty related to the performance of ANN-22(H-15), ANFIS, SVM, and RM models on a similar pattern as used by (Abbaspour

*et al.*2007). Calculation of the

*d*-factor can be achieved by the Equations (20) and (21) (Talebizadeh & Moridnejad 2011):where is the average distance between the lower and the upper limits of the confidence interval and is the standard deviation of the observed data. In the present study, a 95% prediction interval was chosen. The average distance was calculated between the upper and lower limits and was divided with the observed standard deviation value. Table 7 shows the value of

*d*-factor which has been quantified for each model. As indicated, the values of the

*d*-factor for both the model ANN-22(H-15) and ANFIS are satisfactory and the difference of predicted

*C*value of TPKW is enclosed by the 95% prediction interval was narrower. The value of the

_{d}*d*-factor close to zero shows more accuracy of the predicted results. Furthermore, the distance between the upper and lower limit of 95% confidence interval for the results of ANN-22(H-15) model was much narrower and its value was almost very close to the observed

*C*of TPKW.

_{d}Models . | ANN-22 (H-15) . | ANFIS . | SVM . | RM . |
---|---|---|---|---|

d-factor | 0.27 | 0.34 | 0.68 | 0.98 |

Models . | ANN-22 (H-15) . | ANFIS . | SVM . | RM . |
---|---|---|---|---|

d-factor | 0.27 | 0.34 | 0.68 | 0.98 |

### Sensitivity analysis

To find the most optimal input parameters that have the greatest influence on , sensitivity analysis was also performed among the given input parameters () and the output parameter (). All the dataset of the present study was used to perform the sensitivity analysis, and one by one sensitivity of all seven input parameters was assessed. To perform the sensitivity analysis, values of the each of seven input parameters were varied individually by 10%, and the corresponding change in was calculated using Equation (19). This methodology was applied one by one for all the input parameters for both increment and reduction by 10% and keeping the remaining input parameters constant.

*MSE*, Equation (12);

*MAE*, Equation (13); coefficient of determination (

*R*), Equation (14); and absolute relative error () were used to gauge the performance of the sensitivity analysis. After performing sensitivity analysis, the obtained results are presented in Figure 13.

^{2}Based on the results of *MSE*, *MAE*, *R ^{2}*, and

*ARE*, it is observed that

*h/P*is the most sensitive hydraulic parameter for the estimation of than

*F*, while from geometric parameters; were found to be more sensitive input parameters than . The results obtained from sensitivity analysis increased the authenticity of ANN models because sensitivity analysis determined the same most influencing geometric parameters as determined by ANN, which reflects a good relationship between sensitivity analysis and the ANN models.

_{r}### Comparison of the outcome of the present study with other studies

The application of soft computing tools such as ANN, ANFIS, SVM, etc. have been used in previous studies (Bashiri *et al.* 2016; Kashkaki *et al.* 2018; Akbari *et al.* 2019; Norouzi *et al.* 2019; Zounemat-Kermani & Mahdavi-Meymand 2019; Gharehbaghi *et al.* 2023) for the prediction of discharge capacity of a piano key weir. The performance of all the previously investigated intelligence models was satisfactory. Table 8 shows the performance comparison of best models used in the present study with other research studies. Based on the results as provided in Table 8, it can be stated that the prediction accuracy of the ANN model is more satisfactory than ANFIS, SVM and RM. The *C _{d}* of TPKW predicted by ANN has better performance in comparison with other research. For example, the

*R*

^{2}values of ANN obtained from the present study are higher than those mentioned in previous research work and the RMSE value of ANN from the current study is also lower than other research studies. So, it can be concluded that the prediction efficacy of the ANN model in the prediction of

*C*of TPKW is better than ANFIS, SVM, and RM.

_{d}Statistical indices . | Predicted model used in the present study . | (Zounemat-Kermani & Mahdavi-Meymand 2019) . | (Bashiri, et al. 2016)
. | (Kashkaki et al. 2018)
. | (Gharehbaghi et al. 2023)
. | (Norouzi et al. 2019). | (Akbari et al. 2019)
. | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

ANN . | ANFIS . | SVM . | RM . | ANFIS . | ANN . | ANN . | ANFIS . | ANN . | SVM . | SVM . | |

R^{2} | 0.998 | 0.995 | 0.982 | 0.978 | 0.995 | 0.994 | 0.999 | 0.91 | 0.985 | 0.978 | 0.982 |

RMSE | 0.0158 | 0.0198 | 0.0294 | 0.0301 | 1.444 | 0.03 | 0.5963 | 0.0306 | 0.019 | 0.027 | 0.015 |

Statistical indices . | Predicted model used in the present study . | (Zounemat-Kermani & Mahdavi-Meymand 2019) . | (Bashiri, et al. 2016)
. | (Kashkaki et al. 2018)
. | (Gharehbaghi et al. 2023)
. | (Norouzi et al. 2019). | (Akbari et al. 2019)
. | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

ANN . | ANFIS . | SVM . | RM . | ANFIS . | ANN . | ANN . | ANFIS . | ANN . | SVM . | SVM . | |

R^{2} | 0.998 | 0.995 | 0.982 | 0.978 | 0.995 | 0.994 | 0.999 | 0.91 | 0.985 | 0.978 | 0.982 |

RMSE | 0.0158 | 0.0198 | 0.0294 | 0.0301 | 1.444 | 0.03 | 0.5963 | 0.0306 | 0.019 | 0.027 | 0.015 |

## CONCLUSIONS

In this paper, the influence of geometric parameters of TPKW on its discharge performance has been investigated by using ANN, ANFIS, SVM and RM. The present study concluded that based on the results of dimensional analysis between the geometric and hydraulic parameters of TPKW, it has been determined that the coefficient of discharge of TPKW is dependent on the following set of geometric and hydraulic parameters: . Further, based on the predicted results of ANN-22(H-15), ANFIS, SVM, and RM, it is concluded that all the geometric parameters have a considerable influence on the discharge capacity of TPKW. However, the results predicted by the ANN-22(H-15 ) model were found to be more accurate than ANFIS (), SVM (), and RM (). Moreover, based on the predicted results of various models, it is concluded that the discharge capacity of TPKW has a greater influence on the *S _{i}/S_{o}*,

*W*, and

_{i}/W_{o}*L/W*ratios than

*B*and

_{i}/B_{o}*α*. Furthermore, sensitivity analysis results confirmed that

*h/P*is the highest influencing hydraulic parameter in the estimation of the discharge capacity of TPKW. Additionally, it was also found that keeping a single hidden layer in the architecture of ANN models generated better results as compared to selecting double or triple hidden layers. Furthermore, the models generated by choosing neurons in hidden layers equal to three times the input parameters produced better results as compared to one or two times of input parameters.

## FUNDING

The authors affirm that they did not receive any financial assistance, grants, or other forms of support while preparing this manuscript.

## 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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