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 (R2) 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) [R2 = 0.998, MSE= 0.0024, SI = 0.0177] were more accurate than ANFIS (R2 = 0.995, MSE = 0.00039, SI=0.0256), SVM (R2 = 0.982, MSE = 0.000864, SI =0.0395) and RM (R2 = 0.978, MSE = 0.001, SI = 0.0411). It was observed that Si/So, Wi/Wo 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
The geometric and hydraulic parameters of TPKWs
| Parameters | Definition |
|---|---|
| Si, So | Slope of the inlet and outlet keys (m/m), respectively |
| Wi, Wo | Inlet and outlet keys width, respectively |
| Bi, Bo | Inlet and outlet overhang lengths, respectively |
| Bb | Weir base length |
| L | Total developed crest length |
| P | Weir height |
| Ts | Thickness of the side wall |
| W | Total width of the TPKW |
| Wu | Width of a unit cycle |
| α | Side wall angle |
| Nu | 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 |
|---|---|
| Si, So | Slope of the inlet and outlet keys (m/m), respectively |
| Wi, Wo | Inlet and outlet keys width, respectively |
| Bi, Bo | Inlet and outlet overhang lengths, respectively |
| Bb | Weir base length |
| L | Total developed crest length |
| P | Weir height |
| Ts | Thickness of the side wall |
| W | Total width of the TPKW |
| Wu | Width of a unit cycle |
| α | Side wall angle |
| Nu | 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 |
Schematic view of TPKW: (a) plane view and (b) section (A-A) view (Kheir-Abadi et al. 2020).
Schematic view of TPKW: (a) plane view and (b) section (A-A) view (Kheir-Abadi et al. 2020).
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 Cd 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 Cd 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.
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 Cd 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 Cd 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 (Si/So, Wi/Wo, Bi/Bo, 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 Fr).
MATERIALS AND METHODS
Experimental setup and collection of databank
Experimental laboratory setup.
) 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 (Si/So) and four different key width ratios (Wi/Wo) of TPKW at eight different discharges ranged from 16.01 to 24.88 lit/sec. In the variation of each Si/So and Wi/Wo model case, respective changes in every geometric parameter (Si, So, Wi, Wo, Bi, Bo, L, and α) of TPKW were calculated, accordingly. For the collection of data of hydraulic parameters (h/P, Cd, and Fr), 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 Cd and Fr 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 (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.
Geometric parameters of TPKWs used in the present research work
| 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 (Wi/Wo) | 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 (Wi/Wo) | 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
ratio represents the Reynolds number (
), 
ratio represents the weber number (
), 
is the coefficient of discharge of TPKW (
), and 
is the upstream Froud number (
). The effect of surface tension is considered to be insignificant for the case of nappe head more than 3 cm over the weir as recommended by Novák & Čabelka (1981), and similarly, for the case of turbulent flow, the gravitational force is dominant as compared to viscous force as suggested by Karimi 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
value of the parameter before normalization. Statistical analysis of the input and output parameters
| 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 |
| Fr | 0.132 | 0.20136 | 0.06929 | 0.17 | 0.02 | 0.12 |
| Si/So | 0.6 | 1.67 | 1.07 | 1.06 | 0.35 | 0.33 |
| Wi/Wo | 0.76 | 1.32 | 0.56 | 1.06 | 0.21 | 0.2 |
| Bi/Bo | 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 |
| Cd | 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 |
| Fr | 0.132 | 0.20136 | 0.06929 | 0.17 | 0.02 | 0.12 |
| Si/So | 0.6 | 1.67 | 1.07 | 1.06 | 0.35 | 0.33 |
| Wi/Wo | 0.76 | 1.32 | 0.56 | 1.06 | 0.21 | 0.2 |
| Bi/Bo | 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 |
| Cd | 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
) can be calculated as follows (Zounemat-Kermani & Mahdavi-Meymand 2019):
and 
are membership functions and x represents the input at node i.Support vector machines
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.
and 
are the input and weight coefficient respectively, and b is the bias valueEach 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).
ANN models and architecture
| 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, Bi/Bo | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-2 | h/P, Fr, Si/So | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-3 | h/P, Fr, Wi/Wo | 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, Bi/Bo, Si/So | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-7 | h/P, Fr, Bi/Bo, Wi/Wo | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-8 | h/P, Fr, Bi/Bo, L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-9 | h/P, Fr, Bi/Bo, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-10 | h/P, Fr, Si/So, Wi/Wo | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-11 | h/P, Fr, Si/So, L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-12 | h/P, Fr, Si/So, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-13 | h/P, Fr, Wi/Wo, L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-14 | h/P, Fr, Wi/Wo, 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, Bi/Bo, Si/So, Wi/Wo | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-17 | h/P, Fr, Bi/Bo, Si/So, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-18 | h/P, Fr, Bi/Bo, Si/So, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-19 | h/P, Fr, Bi/Bo, Wi/Wo, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-20 | h/P, Fr, Bi/Bo, Wi/Wo, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-21 | h/P, Fr, Bi/Bo, L/W, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-22 | h/P, Fr, Si/So, Wi/Wo, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-23 | h/P, Fr, Si/So, Wi/Wo, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-24 | h/P, Fr, Wi/Wo, L/W, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-25 | h/P, Fr, Si/So, Wi/Wo, L/W, α | 6, 10, 12 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-26 | h/P, Fr, Bi/Bo Si/So, Wi/Wo, L/W | 6, 12, 18 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-27 | h/P, Fr, Bi/Bo Si/So, Wi/Wo, α | 6, 12, 18 | 1, 2, 3 | 80, 10, 10% |  |
| ANN-28 | h/P, Fr, Bi/Bo, Si/So, Wi/Wo, 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, Bi/Bo | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |
| ANN-2 | h/P, Fr, Si/So | 3, 6, 9 | 1, 2, 3 | 80, 10, 10% | |
| ANN-3 | h/P, Fr, Wi/Wo | 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, Bi/Bo, Si/So | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |
| ANN-7 | h/P, Fr, Bi/Bo, Wi/Wo | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |
| ANN-8 | h/P, Fr, Bi/Bo, L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |
| ANN-9 | h/P, Fr, Bi/Bo, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |
| ANN-10 | h/P, Fr, Si/So, Wi/Wo | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |
| ANN-11 | h/P, Fr, Si/So, L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |
| ANN-12 | h/P, Fr, Si/So, α | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |
| ANN-13 | h/P, Fr, Wi/Wo, L/W | 4, 8, 12 | 1, 2, 3 | 80, 10, 10% | |
| ANN-14 | h/P, Fr, Wi/Wo, 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, Bi/Bo, Si/So, Wi/Wo | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |
| ANN-17 | h/P, Fr, Bi/Bo, Si/So, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |
| ANN-18 | h/P, Fr, Bi/Bo, Si/So, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |
| ANN-19 | h/P, Fr, Bi/Bo, Wi/Wo, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |
| ANN-20 | h/P, Fr, Bi/Bo, Wi/Wo, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |
| ANN-21 | h/P, Fr, Bi/Bo, L/W, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |
| ANN-22 | h/P, Fr, Si/So, Wi/Wo, L/W | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |
| ANN-23 | h/P, Fr, Si/So, Wi/Wo, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |
| ANN-24 | h/P, Fr, Wi/Wo, L/W, α | 5, 10, 15 | 1, 2, 3 | 80, 10, 10% | |
| ANN-25 | h/P, Fr, Si/So, Wi/Wo, L/W, α | 6, 10, 12 | 1, 2, 3 | 80, 10, 10% | |
| ANN-26 | h/P, Fr, Bi/Bo Si/So, Wi/Wo, L/W | 6, 12, 18 | 1, 2, 3 | 80, 10, 10% | |
| ANN-27 | h/P, Fr, Bi/Bo Si/So, Wi/Wo, α | 6, 12, 18 | 1, 2, 3 | 80, 10, 10% | |
| ANN-28 | h/P, Fr, Bi/Bo, Si/So, Wi/Wo, L/W, α | 7, 14, 21 | 1, 2, 3 | 80, 10, 10% | |
ANN architecture adopted for the current work
(output parameter of ANN) of TPKW with different geometric and hydraulic parameters (input parameters of ANN) of TPKW. The procedure employed for the training of ANN models is described in Figure 3. Multilayer feed-forward neural networks (MLFNNs) were adopted for the current work because it is considered best for such types of complex problems (Turhan 1995; Karami 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. 
Procedure to train ANN models.
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.
Typical structure of feed-forward neural network.
RESULTS AND DISCUSSION
Best-selected ANN models based on the MSE, MAE and R2 value for training, validation, testing, and overall
| 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 | ||
Terminology to comprehend the structure of ANNs.
is the actual value of 
, 
is the predicted value of 
, and N is the number of the data sample.
(a) MSE values of best-selected ANN models for training, validation, testing and overall. (b) MAE values of best-selected ANN models for training, validation, testing, and overall. (c) R2 values of best-selected ANN models for training, validation, testing, and overall.
(a) MSE values of best-selected ANN models for training, validation, testing and overall. (b) MAE values of best-selected ANN models for training, validation, testing, and overall. (c) R2 values of best-selected ANN models for training, validation, testing, and overall.
Optimistic ANN model
Performance of optimistic ANN model (ANN-22,H-15): (a) training data points; (b) testing data points; (c) displaying of loss error and gradient graph of ANN-22(H-15).
Performance of optimistic ANN model (ANN-22,H-15): (a) training data points; (b) testing data points; (c) displaying of loss error and gradient graph of ANN-22(H-15).
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 (Bi/Bo, Si/So, Wi/Wo, L/W, α) have a significant impact on the discharge capacity of TPKW because the value of R2 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, Si/So, Wi/Wo and L/W was found to be an optimistic ANN model because it gave outstanding results. The MAE, MSE and R2 values of ANN-22(H-15) were found to be 0.0124, 0.00024 and 99%, respectively. The h/P and Fr input parameters of ANN remained fixed for all developed ANN models. So, it is found that the Si/So, Wi/Wo, and 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
for TPKW as presented in Equation (19) by applying non-linear regression analysis between the input parameters (
) and output parameter (
) based on the dimensionless parameters obtained during the dimensional analysis. Eighty per cent of the dataset of the present study was used for the development of the equation and the remaining 20% of the dataset was used for its validation. Generally, 70–80% of the data is recommended for developing an equation and the remaining 20–30% of the data can be used for the validation of the equation (Hussain 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 
Comparison between the observed and computed 
for the training and validation dataset based on the regression analysis.
Comparison between the observed and computed for the training and validation dataset based on the regression analysis.
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 Cd 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.
Performance of various models in the prediction of Cd of TPKW during the training and testing phases
| Model name | Training results | ||||||
|---|---|---|---|---|---|---|---|
| R2 | 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 | ||||||
|---|---|---|---|---|---|---|---|
| R2 | 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 |
value of this model was found to be more than other models in the training as well as testing phase of models. 
Comparative performance of ANN-22,H-15, ANFIS, SVM, and RM in the training and testing of models.
Comparative performance of ANN-22,H-15, ANFIS, SVM, and RM in the training and testing of models.
Comparison of the performance of models by Violin plot illustration.
) value of all the models with their respective R2 value. It was observed that model ANN-22(H-15 
) and ANFIS (
) had the highest R2 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. 
Taylor diagram of the various models used for the prediction of Cd of TPKWs: training results and testing results.
Taylor diagram of the various models used for the prediction of Cd of TPKWs: training results and testing results.
Scatter plot error distribution comparison between ANN and other models
results predicted from ANN-22(H-15) and other trained models including, ANFIS, SVM, and RM, scatter plots have been plotted between the ratio of observed 
and computed 
with each input parameter as shown in Figure 12. In these plots, it was observed that most of the data points fall within the smaller error range (near about one) for the case ANN-22(H-15) predicted results as compared to other models, ANFIS, SVM, and RM. Data points that fall near about one show more accuracy as compared to other points that fall out of the one (Iqbal 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. 
Model error distribution comparison between 
(observed/computed) with the input parameters.
Model error distribution comparison between (observed/computed) with the input parameters.
Uncertainty analysis
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 Cd value of TPKW is enclosed by the 95% prediction interval was narrower. The value of the 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 Cd of TPKW. Uncertainty measuring parameter of models
| 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.
) were used to gauge the performance of the sensitivity analysis. After performing sensitivity analysis, the obtained results are presented in Figure 13. 
Graphical representation between input parameters and MSE, MAE, R2, and ARE based on the results obtained from sensitivity analysis.
Graphical representation between input parameters and MSE, MAE, R2, and ARE based on the results obtained from sensitivity analysis.
Based on the results of MSE, MAE, R2, and ARE, it is observed that h/P is the most sensitive hydraulic parameter for the estimation of 
than Fr, 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.
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 Cd of TPKW predicted by ANN has better performance in comparison with other research. For example, the R2 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 Cd of TPKW is better than ANFIS, SVM, and RM.
A comparison of outcomes of present study with previous research work
| 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 | |
| R2 | 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 | |
| R2 | 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 Si/So, Wi/Wo, and L/W ratios than Bi/Bo and 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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