Abstract

Triangular labyrinth weirs are widely used in hydraulic structures because they can pass higher flow rates under limited water head compared to conventional weirs. However, predicting how much additional discharge they can convey relative to a standard configuration is challenging due to complex flow behaviour and geometric effects. Existing empirical equations are often limited in their applicability to specific conditions.

In this study, four data-driven modeling approaches-Gaussian Process Regression (GPR), Support Vector Machine (SVM), Function Fitting Network (FITNET), and Multi-Layer Perceptron (MLP)-were developed and compared to estimate the relative increase in discharge for triangular labyrinth weirs. The analysis is based on 186 experimental observations covering a range of sidewall angles (15°, 25°, 30°, and 40°) and different numbers of cycles (3 to 5). The models use key geometric and hydraulic characteristics, expressed in normalized form, to capture the influence of flow depth and weir geometry.

Model performance was evaluated using standard statistical indicators of prediction accuracy. Among the tested approaches, GPR provided the most accurate predictions, significantly reducing error compared to the other models.

The results demonstrate that data-driven techniques can effectively represent the complex relationship between flow conditions and discharge performance. The proposed approach offers a reliable tool for the design and optimization of triangular labyrinth weirs, with potential applications in flood control and hydraulic structure safety.

1 Introduction

Dams built in ancient times generally served the purpose of water supply or irrigation. As civilizations developed, expanding water needs emerged, and nowadays, dams are constructed to serve multiple purposes like water supply, irrigation, flood control, navigation, sediment control, hydroelectric power generation, etc. Weirs are constructed as a spillway to safeguard the dams, particularly during floods, when excessive discharge is to be conveyed within a narrow waterway in a short span of time. Labyrinth weir having large crest lengths capable of passing maximum flood discharge with relatively low heads compared to linear weirs serves this purpose very well. Discharge over a labyrinth weir is illustrated in Figure 1, which shows a geometrical sketch of a triangular labyrinth weir.

Figure 1 Triangular labyrinth weir: (a) plan, (b) section at a – a.

The literature described above indicates that parameters affecting discharge coefficient C_d, and discharge magnification, Q/Q_n are h/P (head-to-weir height ratio), h/w (relative overflow depth), w/p (relative weir cycle width), l/w (length magnification), α (side wall angle), and N (number of cycles). Here Q/Q_n is the discharge over a linear weir, h is head over the weir, P is weir height, w is one cycle width, and l is one cycle crest length. Where α is added just to modify the flow magnification ratio in cases where low upstream flow depth requires greater accuracy from the normal weir flow equation.

Gentilini (Hager et al. 2015) conducted the first experimental study on sharp-crested oblique weirs and triangular labyrinth weirs with sidewall angles of 30°, 45°, and 60°. In their study, the discharge coefficient of the labyrinth weir (C_d) and the discharge coefficient of a corresponding normal weir (C_dn) were evaluated separately. The ratio of these two coefficients (C_d/C_dn) was then used to assess the relative discharge performance of the labyrinth weir.

Similarly, the upstream flow depth over the weir crest (h) and the channel width (W) were defined, and their ratio (h/W) was used to represent the relative head condition. The results were presented as plots of C_d/C_dn versus h/W, which showed that the relative discharge coefficient decreases with an increase in the relative head.

Hay and Taylor (1970) extensively investigated the performance of labyrinth weirs of rectangular, triangular, and trapezoidal plan forms. They presented the results in the magnification ratio of labyrinth weir discharge to straight linear weir discharge having the same channel width. They also provided a labyrinth weir design procedure, including criteria for estimating discharge over trapezoidal or triangular labyrinth weirs. Darvas (1971) Labyrinth weir design charts relate the discharge coefficient to the length magnification (l/w). The head ratio (H/P) is considered as a third governing parameter. Lux (1984) related discharge coefficient with a vertical aspect ratio w/p and shape constant k. Amanian (1987) studied triangular labyrinth weirs with half-round crests. He claimed that half-round crest shapes might pass more flow than other weirs at the same head. The study also suggests that the structure exhibits a trapezoidal geometry. Labyrinth weirs are more efficient than triangular labyrinth weirs. The trapezoidal apex affects flow efficiency but is suitable for field construction. Amanian (1987) observed a difference in C_d due to different aeration conditions but he did not define the curves for these conditions. Tullis et al. (1995) developed discharge coefficient equations for (h/P) for different sidewall angles (α) and proposed a procedure for designing a trapezoidal labyrinth weir. Ghodsian (2009) related discharge coefficient with (h/P) and length magnification (l/w). Kumar et al. (2011) performed experiments on a single cycle sharp crested Triangular labyrinth weir having different Sidewall angles (α = 15°, 30°, 45°, 60°, 75°, and 90°) and proposed a generalized equation for C_d as a function (h/P) and (α). Stefano and Ferro (2013) showed that dimensional analysis and an incomplete self-similarity approach can be used to deduce a general head discharge formula for a triangular labyrinth weir. They proposed a generalized formula based on the data of Kumar et al. (2011). Carollo et al. (2012) proposed Equation in terms of h/w and l/w with the two boundary conditions When h/w tends to zero, Q/Q_n tends to length magnification l/w i.e. the behavior of the labyrinth weir tends to be ideal (Hay and Tayler 1970) When h/w tends to infinity, the labyrinth weir behaves as a normal weir. Therefore, Q/Q_n = 1. Gupta et al. (2014) conducted an experimental study on a two-cycle triangular labyrinth weir. They proposed an equation for the discharge coefficient similar to Kumar et al. (2011) Carollo et al. (2012) Proposed stage-discharge relation for triangular labyrinth weirs in Equation 1:

(1)

Where:

b0 and b1	=	coefficients, their values were determined using experimental data as 5.988 and 1.149, respectively; and
Qn	=	discharge over a normal weir placed in the same channel width and operating under similar hydraulic conditions, and the ratio of one cycle's crest length (l) and width (w) is length magnification.

This equation satisfies the two boundary conditions: (i) when head over the weir tends to zero, discharge magnification tends to length magnification, and (ii) when length magnification tends to unity, discharge magnification tends to unity (labyrinth weir reduces to normal weir). Bijankhan and Kouchakzadeh (2017) conducted experiments on a single-cycle triangular labyrinth weir having different sidewall angles (37°, 45°, 60°, and 90°) and proposed a unified formula for calculating C_d in both free and submerged flow conditions.

In recent years, artificial intelligence (AI) and soft computing techniques have been increasingly employed as powerful simulation tools for modeling complex and nonlinear hydraulic problems. These data-driven approaches have demonstrated strong potential in predicting the discharge characteristics of labyrinth weirs, where traditional empirical equations often fall short in generalization accuracy. Researchers have utilized a wide range of methods such as Artificial Neural Networks (ANN) (Mahmoud et al. 2021; Majidi Asl et al. 2024), SVM (Karami et al. 2017; Emami et al. 2022), Gene Expression Programming (GEP) (Roushangar et al. 2017), Adaptive Neuro-Fuzzy Inference Systems (ANFIS) (Izadbakhsh and Hajiabadi 2019; Mahmoud et al. 2021), Extreme Learning Machine (ELM) (Bilhan et al. 2018; Norouzi et al. 2020), and Gaussian Process Regression (GPR) (Seyedian et al. 2023) to predict the discharge coefficient or flow magnification of labyrinth weirs.

More recent works have integrated hybrid and optimization-enhanced Machine Learning (ML) frameworks such as Firefly-optimized SVR (Karami et al. 2018), Particle Swarm Optimization coupled with ANFIS (Mahmoud et al. 2021), Walnut-optimized SVR (Emami et al. 2022; Mustafa et al. 2022; Mustafa et al. 2025), and Hybrid Informed Multi-Layer Perceptron (HI-MLP) (Mahmoud et al. 2023), which further improved predictive performance and robustness across different geometric configurations. Similarly, ensemble and tree-based algorithms such as Quantile Regression Forest (QRF) (Salazar and Crookston 2019) and Light Gradient Boosting (LXGB) (Emami et al. 2023) have achieved high accuracy for triangular and trapezoidal labyrinth weirs. Comprehensive comparative analyses by Parsaie et al. (2023) and Seyedian et al. (2023) confirmed that AI methods-particularly GPR and deep neural models-outperform traditional empirical approaches in predicting labyrinth-weir discharge behaviour.

The literature confirms that machine learning (ML) presents a powerful alternative to empirical equations for modeling complex hydraulic structures. Several studies have applied individual ML techniques to predict the discharge coefficient (C_d) of labyrinth weirs. However, a systematic comparison of the predictive performance of advanced models like GPR, FITNET, and SVM specifically for predicting the discharge magnification (Q/Q_n) of triangular labyrinth weirs is lacking. Furthermore, the potential of GPR to provide not only superior accuracy but also inherent uncertainty quantification-a critical feature for risk-informed engineering design-remains underexplored for this application.

The primary objectives of this study are to:

1. Compile a comprehensive experimental dataset on the discharge magnification (Q/Q_n) of triangular labyrinth weirs with varying sidewall angles (15°, 25°, 30°, 40°) and cycle numbers (3, 4, 5).
2. Develop and train four distinct ML models- GPR, SVM, FITNET, and MLP for predicting Q/Q_n using the dimensionless input parameters h/w and l/w.
3. Rigorously evaluate and compare the predictive performance of these models using statistical metrics such as Mean Absolute Percentage Error (MAPE), Root Mean Square Error (RMSE), and the coefficient of determination (R²).
4. Benchmark the accuracy of the best-performing ML model against a leading empirical formulation to quantitatively demonstrate the improvement in predictive capability.
5. Assess the limitations and practical reliability of the optimal model, with a specific discussion on its performance in data-sparse regions of the parameter space.

2 Material and Methods

2.1 Experimental set-up and model fabrication

Experiments for the physical model were conducted within a rectangular cement-plastered masonry channel in the advanced hydraulics laboratory of the Civil Engineering Department at Aligarh Muslim University. This channel served as the primary test environment, measuring 15 m in length, 0.75 m in width, and 0.60 m in depth. Twelve sharp crested triangular labyrinth weirs were made of steel sheets with a thickness of 3 mm and a height of 20 cm. Out of twelve models, four models were of 3 cycles with the sidewall of 15°, 25°, 30°, and 40°, having a length magnification l/w of 3.84, 2.33, 1.99, and 1.53, four models of 4 cycles with the sidewall of 15°, 25°, 30°, and 40°, and l/w of (3.86, 2.35, 1.99, and 1.54) respectively, and four models of 5 cycles with a sidewall angle of 15°, 25°, 30°, and 40°. Figure 2 is a schematic representation of the experimental set-up of triangular labyrinth weir 3, 4, and 5 cycles.

Figure 2 Experimental set-up layout.

Water was supplied through a 17 cm diameter pipe connected to an overhead tank with an overflow mechanism to maintain a constant head. The water collected after flowing through the channel was directed into an underground sump and continuously pumped back up to the overhead tank to ensure a continuous flow. The photographic view of the triangular labyrinth weir with 3, 4, and 5 cycles is shown in Figure 3. Table 1 presents the details of the geometric parameters utilized in the current study. Measurements of the head over the weir were taken 1 m upstream of the labyrinth weir to minimize the influence of curvature, utilizing a pointer gauge with a precision of 0.1 mm The nappe beneath the weir was kept fully ventilated for all the runs. Discharge measurements were obtained using a calibrated sharp crested normal weir provided at the end of the drain.

Table 1 Range of geometric and flow parameters for triangular labyrinth weirs.

Model	Q (m3/s)	α (°)	N	L (m)	h (m)	P (m)	Data
Model–1	0.0049–0.5512	15°	3	2.879	0.0099–0.0628	0.200	15
Model–2	0.0067–0.0056	25°	3	1.750	0.0152–0.0883	0.200	14
Model–3	0.0033–0.0491	30°	3	1.474	0.0114–0.0914	0.200	15
Model–4	0.0053–0.0531	40°	3	1.147	0.020–0.1073	0.200	15
Model–5	0.0037–0.0507	15°	4	2.859	0.0093–0.0718	0.200	15
Model–6	0.0051–0.0499	25°	4	1.762	0.0154–0.0941	0.200	16
Model–7	0.0045–0.5077	30°	4	1.475	0.0138–0.099	0.200	15
Model–8	0.0033–0.0508	40°	4	1.142	0.0149–0.1078	0.200	16
Model–9	0.0048–0.0581	15°	5	2.898	0.0104–0.0796	0.200	15
Model–10	0.0041–0.0486	25°	5	1.761	0.0144–0.0999	0.200	20
Model–11	0.0046–0.0492	30°	5	1.484	0.0148–0.102	0.200	15
Model–12	0.0052–0.1116	40°	5	1.156	0.019–0.1116	0.200	15
TOTAL DATA							186

NOTE: Q = Discharge, α = sidewall angle (degrees), N = number of cycles, L = crest length (m), h = head over weir (m), P = weir height (m).

Figure 3 Experimental set-up with a triangular labyrinth weir: (a) 3-cycle, (b) 4-cycle, and (c) 5-cycle.

2.2 Discharge head relation for labyrinth weir

A labyrinth weir is designed to increase discharge beyond that of a conventional weir by extending the effective crest length. This discharge magnification is a key performance criterion and is primarily influenced by the weir geometry and the operating head. To develop a dimensionless framework for analyzing this relationship, relevant geometric and flow parameters must be considered.

(2)

Where:

Cd	=	discharge coefficient,
L	=	crest length,
g	=	gravitational acceleration, and
HT	=	head over the weir.

Hay and Taylor (1970) proposed that the discharge magnification can be expressed as a function of dimensionless parameters derived from the weir geometry and flow conditions:

(3)

Where:

Q/Qn	=	dimensionless discharge magnification,
Q	=	discharge over the labyrinth weir,
Qn	=	discharge over the normal weir,
χ	=	functional dependence,
h/P	=	head to weir height ratio,
w/P	=	relative crest width,
l/w	=	length magnification,
α	=	sidewall angle, and
N	=	number of weir cycles.

Carollo (2012) derived a modified dimensionless relationship for discharge magnification using the Buckingham π theorem:

(4)

Where:

ψ	=	functional dependence, and
h/w	=	head to one cycle width.

Experimental analysis of both Carollo et al. (2012) data, and Gentilini’s (1941) data revealed that for α = 45° and N = 2, the parameter w/P has an insignificant effect on discharge magnification. Similarly, for α = 30° when N = 1 or 2, both w/P and N have a negligible impact. As a result, Equation 5 can be further simplified to:

(5)

Consistent with the findings of Carollo et al. (2012) and supported by preliminary analysis of our dataset, the parameters h/w and l/w were selected as the primary inputs for the machine learning models, as they were found to be the most dominant factors influencing Q/Q_n for the tested geometries This dimensionless formulation provides a generalized and scalable approach for analyzing labyrinth weir discharge performance, enabling better comparisons across different designs and flow conditions.

2.3 Machine learning methods

This section discusses the machine learning algorithms selected to predict discharge magnification. Using machine learning, GPR, SVM, FITNET and MLP. The principles of each method are explained and why they are suitable for this task. Details are also provided about the configurations and training parameters used for each model. This will help understand how these algorithms are customized to learn the complex relationships affecting discharge magnification for triangular labyrinth weir. GPR, SVM FITNET and MLP models were implemented in MATLAB.

Support Vector Regression models

Support vector machine (SVM) includes support vector classification (SVC) and support vector regression (SVR) algorithms (Cortes et al. 1995). The SVM maps the original data into a high-dimensional feature space. This is achieved through transformation by nonlinear mapping employing kernel functions such as sigmoid, polynomial, and radial basis functions. SVR is used for modeling and prediction wherein the aim is to find a function whose prediction errors are less than ε for all training data. The non-linear model of the SVR is generally used in the field of science and engineering.

SVM is a classification and regression algorithm known as SVC and SVR. SVM maps the original data into a feature space of high dimensionality using kernel functions like sigmoid, polynomial, and radial basis functions.

The linear model of SVR is given by Equation 6:

(6)

Where:

x	=	input feature vector,
y	=	predicted output (discharge magnification),
Φ(x)	=	nonlinear mapping function,
ω	=	weight vector in feature space, and
b	=	bias term.

The goal is to reduce the difference between the model and actual outputs. For this purpose, the objective function mentioned below is minimized using the second-order optimization method:

(7)

Subject to:

(8)

(9)

(10)

Where:

C	=	regularization parameter,
	=	slack variables representing upper and lower deviations,
ε	=	insensitive loss function parameter,
n	=	number of training samples,
w	=	weight vector, and
⟨⋅⟩	=	inner product in feature space.

The parameters C, γ, and ε are hyperparameters of the SVR model and are tuned externally (using a grid search approach) to achieve optimal predictive performance.

Gaussian Process Regression (GPR)

Gaussian Process Regression, a non-parametric Bayesian regression model, has gained attention for engineering applications. Unlike parametric models like neural networks, GPR assumes that adjacent observations share information, creating a prior directly over function space (Rasmussen and Williams 2006). It provides a probabilistic framework for predictions.

A GPR is defined as a collection of random variables where any finite subset has a joint Gaussian distribution. For GPR, the relationship between inputs X and outputs Y is modeled using Equation 11:

(11)

Where:

ξ	=	observational error and has equivalent zero mean value distribution (μ(x) = 0), variance σ2, and
f(x)	=	GPR function values.

The kernel function is used to determine the joint distribution in Equation 12:

(12)

Where:

	=	noise variance, and
Iₙ	=	identity matrix.

 is defined using Equation 13:

(13)

Where:

	=	training input matrix,
	=	training output vector,
x*	=	test input,
y*	=	test output dataset,
σn	=	noise variance, and
In	=	identity matrix of size n×n.

The predictor distribution over y_* is expressed in Equation 14 as

(14)

Where:

(15)

(16)

In the present study, squared exponential kernels were used that can be expressed in Equation 17 as:

(17)

Where:

	=	signal variance (controls the magnitude of function variation),
γ	=	length-scale parameter controlling smoothness of the function, and
xi and xj	=	input feature vectors.

The set of hyperparameters is defined as  (where γ is the length scale parameter,  is the signal variance, and  is the noise variance. These hyperparameters are estimated by maximizing the log marginal likelihood function as:

(18)

To terminate a training program, the specified conditions must be satisfied.

Function Fitting Network (FITNET)

FITNET is a specialized type of neural network designed for regression and curve-fitting tasks. It operates as a MLP model, which maps input features to continuous target outputs through a process known as function fitting. The architecture of FITNET includes an input layer that accepts the features, a single hidden layer with a configurable number of neurons, and an output layer with a linear activation function. The hidden layer employs nonlinear activation functions, such as sigmoid, tanh, or ReLU, to capture complex, nonlinear relationships within the data. The linear activation in the output layer ensures that the network is suitable for generating continuous predictions. FITNET is trained using algorithms such as Levenberg-Marquardt (trainlm) or scaled conjugate gradient (trainscg), which optimize the network's weights by minimizing a specified loss function, typically the Mean Squared Error (MSE). This iterative training process involves forward propagation of inputs, error calculation, and backpropagation to update the weights for improved accuracy. FITNET is particularly well-suited for small to moderately complex datasets and is commonly applied in fields such as scientific modeling, system identification, and financial forecasting. While FITNET is efficient and easy to implement, its architecture-with a single hidden layer-may limit its ability to model highly complex relationships in larger or more intricate datasets.

Multi-layer Perceptron (MLP)

In this study, MLP is employed, a class of feedforward artificial neural networks, to model the relationship between input features and target outputs. The MLP consists of an input layer, one or more hidden layers, and an output layer. Each layer is composed of multiple neurons, which apply a non-linear activation function to the weighted sum of their inputs. The input layer receives the feature vectors, while the hidden layers transform the data through a series of non-linear mappings. The output layer produces the final predictions, such as class labels or continuous values, depending on the task. The model is trained using backpropagation with gradient descent to minimize a loss function, such as cross-entropy for classification or mean squared error for regression. To prevent overfitting, regularization techniques such as dropout or L2 regularization are applied. The MLP's ability to approximate complex, non-linear functions make it a suitable choice for capturing intricate patterns in the data.

2.4 Data preprocessing and model training

All machine learning models were developed and trained in MATLAB R2023a using the Statistics and Machine Learning Toolbox and the Neural Network Toolbox. Hyperparameters for each model were randomly divided into 80% for training and 20% for testing and grid search approach to minimize the MAPE on the validation dataset. The optimal parameters were as follows: for the GPR model, a squared exponential (RBF) kernel with a length scale of 0.43 and a noise level (σₙ) of 0.012 were used; for the SVR model, the RBF kernel was adopted with a regularization parameter C=15, epsilon ε=0.001, and kernel scale γ=0.25; and for the FITNET, a (2–10–1) architecture with a tangent-sigmoid activation in the hidden layer, linear activation in the output layer, Levenberg-Marquardt training algorithm, learning rate of 0.01, and 1000 epochs was employed. The dataset and MATLAB scripts used to produce the results are available from the corresponding author upon reasonable request, ensuring reproducibility of the analysis.

2.5 Criteria evaluations

The accuracy of the prediction models in the present study has been examined using three different statistical indices Mean Absolute Percentage Error (MAPE), coefficient of determination R², and Root Mean Square Error (RMSE), as defined in Table 2. The best model is one where MAPE and RMSE tends to zero and R² tends to one. The metrics mentioned above are expressed by the following Equations (19), (20), and (21):

Table 2 Statistical indices.

Performance parameters	Expressions for performance parameters	Equations
RMSE		(19)
R2		(20)
MAPE		(21)

NOTE:  = Observed Discharge Magnification, and
= Predicted Discharge Magnification

3 Results and Discussion

The present study investigated the hydraulic performance of triangular labyrinth weirs with varying sidewall angles (15°, 25°, 30°, and 40°) and cycle numbers (3, 4, 5) by evaluating the discharge magnification ratio (Q/Q_n). The experimental data were utilized to develop predictive models using SVM and GPR techniques. Dimensionless parameters such as l/w and h/w were considered as input variables, while Q/Q_n was the output variable. The dataset was divided into training and testing phases with different ratios (60/40, 70/30, 75/25, and 80/20) to identify the optimal split for model performance. It was observed that an 80% training and 20% testing split yielded the best results across most machine learning techniques, minimizing overfitting and underfitting issues.

The performance of the GPR, FITNET, SVM, and MLP models was evaluated using statistical indices such as MAPE, RMSE, and R². Table 3 shows the results for training and testing of model. The GPR model showed superior performance in both training and testing phases. In the training, the MAPE and RMSE were 2.03% and 0.0449, respectively, while in the testing, these were 2.22% and 0.0701. The R² of 0.99 in both phases indicates a strong correlation between the predicted and observed results. GPR model showed the lowest RMSE values in both phases. In the training phase, the RMSE of GPR (0.0449) which was 23.0% lower than that of SVM (0.0524) and 22.9% lower than that of FITNET (0.0582). In the testing phase, the RMSE of GPR (0.0701) was 13.1% lower than that of SVM (0.06196) and 37.2% lower than that of FITNET (0.0511). These reductions show the GPR model's ability to minimize prediction errors more effectively than that of other models. The FITNET model also performed well, with MAPE values of 2.25% and 2.31% in the training and testing phases, respectively. The RMSE values were 0.0582 and 0.0511, with an R² of 0.99 in both training and testing phases. While FITNET showed competitive performance, it was outperformed by GPR in terms of error reduction. SVM model exhibited strong performance, with MAPE values of 1.93% and 2.42% in the training and testing phases, respectively. The RMSE were 0.0524 and 0.06196, with an R² of 0.99 in both training and testing phases, respectively. Despite its high accuracy, the SVM model was less effective than GPR in minimizing errors. The MLP model had the highest MAPE and RMSE values among the four models, with MAPE of 3.23% and 4.10% in the training and testing phases. The RMSE values were 0.0711 and 0.0843, with an R² of 0.99 in both phases. While the MLP model provided reasonable predictions, its performance was inferior to the other models, particularly in error minimization.

Table 3 Performance metrics of soft computing techniques.

Model	Training			Testing
	MAPE	RMSE	R2	MAPE	RMSE	R2
GPR	2.03	0.0449	0.99	2.22	0.0701	0.99
FITNET	2.25	0.0582	0.99	2.31	0.0511	0.99
SVM	1.93	0.0524	0.99	2.42	0.0620	0.99
MLP	3.23	0.0711	0.99	4.10	0.0843	0.99

Figure 4 shows the scatter diagrams for the training and testing, the closer that the data points lie to the 1:1 line, the better the model's prediction accuracy. GPR model exhibited the best alignment with the 1:1 line, followed by SVM and FITNET, while MLP showed slightly more dispersion.

Figure 4 Predicted discharge magnification of: (a) GPR, (b) FITNET, (c) SVR, and (d) MLP.

The experimental data distribution (Figure 4) shows concentration in the 1.0 < Q/Q_n < 2.5 range, with sparse representation at higher magnification ratios (>2.5). This distribution has important implications for model reliability: All models demonstrate high accuracy (R² > 0.99) within the well-sampled range, with GPR showing particular robustness. For Q/Q_n values exceeding 2.5, model predictions should be treated with caution due to limited training examples. The GPR model provides inherent uncertainty quantification through prediction variance, which serves as a reliability indicator. Models are validated for Q/Q_n ≥ 1, with performance degradation expected outside the trained parameter space. For practical applications, we recommend using the models primarily for interpolation within the demonstrated range.

The results highlight the effectiveness of GPR in predicting the discharge magnification for triangular labyrinth weirs. The model minimizes errors and maintains high accuracy in both training and testing phases, which underscores its robustness. The superior performance of GPR can be attributed to its capability to handle non-linear relationships and provide probabilistic predictions, making it well-suited for hydraulic engineering applications. FITNET and SVM models also demonstrated good performance. However, their slightly higher MAPE and RMSE values compared to GPR indicate that they are less effective in capturing the complex relationships within the data. The MLP model, while providing reasonable predictions, was the least accurate, likely due to its sensitivity to hyperparameter tuning and dataset size.

4 Conclusion

This study developed and compared four machine learning models for predicting the discharge magnification (Q/Q_n) of triangular labyrinth weirs. Based on 186 experimental data points, the GPR model emerged as the most accurate and robust, achieving a test MAPE of 2.22% and RMSE of 0.0701. This performance signifies an error reduction of 13% against SVM and 37% against FITNET. The primary limitation of the current study is the sparsity of data at very high Q/Q_n values, which suggests that the models are most reliable for interpolation within the trained range. The GPR framework presented herein offers hydraulic engineers a powerful, data-driven tool for the effective design and analysis of labyrinth weirs, with its inherent uncertainty quantification being a key asset for risk-informed decision-making in flood management applications.

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