Embedded fuzzy-based models in hydraulic jump prediction

= model output

= observational value of a phenomenon

B

= flume width

C₁

= cognitive acceleration (related to the PSO method)

C₂

= social acceleration (related to the PSO method)

CRMS

= centered root-mean-square difference

d₅₀

= median diameter of the base material

Fr

= the Froude number

g

= the gravitational acceleration

h₁

= upstream supercritical flow depth

h₂

= downstream flow depth

h₂^*

= downstream flow depth calculated from the RANFIS model

h_c

= critical depth

I₁, I₂, … ,I_k

= independent input parameters in RANFIS

IA

= index of agreement

K_s

= roughness heights

L_j

= hydraulic jump length

L_r

= length roller in the hydraulic jump

MAE

= mean absolute error

N

= number of data pieces

NSE

= the Nash–Sutcliffe efficiency

Q

= flow discharge

r

= correlation coefficient

R

= hydraulic radius

R²

= coefficient of determination

RMSE

= root mean square error

S₀

= slope of the bed

SD

= standard deviation

u*

= shear velocity

υ

= kinematic viscosity

In open channels, depends on the ratio of inertial and gravitational forces, free-surface flow is divided into three groups, including subcritical, critical, and supercritical. Subsequently, the flow in open channels (such as rivers and spillways) can be converted from subcritical to supercritical, or from supercritical to subcritical. The depth of the flow from subcritical to supercritical often changes smoothly by passing the critical depth. A sudden transition from supercritical flow to subcritical flow is associated with a high level of turbulence and energy loss and occurs in a relatively short interval; a phenomenon called the hydraulic jump.

A hydraulic jump is a complex hydraulic phenomenon in which knowing its characteristics such as the length (L_j) and secondary depth (h₂) is a necessity for appropriate planning of different types of hydraulic structures such as stilling basins (Hager 1992; Montano & Felder 2020). In the current study, soft computing techniques are applied to predict L_j and h₂ parameters. In the following subsections, the relevant studies are mentioned and, thereafter, the objectives and contribution of the research are pointed out clearly.

Many studies have addressed modeling hydraulic jump phenomenon using different approaches such as laboratory studies, hard computing simulations (e.g., numerical simulation), and soft computing techniques (Pagliara et al. 2008; Gerami Moghadam et al. 2019; Gu et al. 2019). In terms of experimental and numerical perspective, hydraulic jumps have been broadly investigated and studied over the last decades. In Table 1, a summary of some pertinent experimental and numerical studies is provided.

In accordance with the topic of this paper, pertinent published studies regarding the estimation/prediction of hydraulic jump characteristics (e.g., the jump length) using soft computing methods are mentioned. Abbaspour et al. (2013) employed artificial neural networks (ANNs) and genetic programming (GP) for the estimation of hydraulic jump characteristics such as the free-surface location and energy dissipation. The findings of the study showed that the estimations of the ANN and GP models were in good agreement with the measured data. Additionally, the results of the ANN and GP models were compared. It was found that the ANNs gave better results than GP models.

Karbasi & Azamathulla (2016) attempted to model the properties of hydraulic jumps over rough beds with the aid of gene expression programming (GEP) models. The performances of the GEP model were compared to empirical equations, ANN, and support vector regression (SVR). A comparison of the obtained results showed that the ANN and SVR models outperformed the GEP. Azimi et al. (2018a) predicted the length of hydraulic jump on rough beds utilizing an integrated model of adaptive neuro-fuzzy inference system (ANFIS) and firefly algorithm (ANFIS-FA). Kumar et al. (2019) applied the ANN to predict the sequent depth ratio of hydraulic jump on the sloping floor with rounded and crushed aggregates. The results indicated that the ANN model has the capability to predict the sequent depth ratio with the appropriate accuracy. For the sake of the building ANIFS structure, several effective parameters, including the relative roughness of the jump, Froude number (Fr), and sequent depth were analyzed. Outcomes of the study revealed that the applied integrated ANFIS-FA models were superior to the standard ANFIS model for estimating the length of the hydraulic jump. In Table 2, a summary of the published reports regarding soft computing methods in simulating hydraulic jump is given.

Analyzing the related subjects associated with simulating hydraulic jump using soft computing methods implies that the neural network-based models (ANNs and ANFIS) acted better than other types of soft computing methods (e.g., GEP, SVR, and GP). Besides, in recent years, the adaptive neural fuzzy inference system (ANFIS) method has been successfully used in modeling hydraulic phenomena such as scour depth at piers (Muzzammil & Ayyub 2010), daily streamflow (Li et al. 2018), spillways aerator air demand (Mahdavi-Meymand et al. 2019), and scour depth around pipelines (Sharafati et al. 2020a). Therefore, this method was selected for the purpose of the present study. Indeed, Azimi et al. (2018b) stated that integrating ANFIS with meta-heuristic algorithms would increase the competence of the standard ANFIS model. As a result, the main focus of this study is laid on using different types of integrated ANFIS models in predicting hydraulic jump characteristics over rough beds.

As well, several researchers have proved that the use of recurrent machine learning models would improve the performance of the counterpart standard ones (Bhattacharjee & Tollner 2016; Murali et al. 2020; Wei et al. 2020). In other words, considering a calculated output parameter as an input parameter to the soft computing model (here h₂ as the secondary depth of hydraulic jump) can yield more precise results of the target parameter (here L_j as the length of the hydraulic jump).

Taking into account the above-mentioned issues, the contribution of the present research can be presented into three categories. Firstly, to the best knowledge of the authors, the recurrent soft computing models have not been already applied to model the hydraulic jump phenomenon. To this end, a recurrent intelligent method was employed, and results were compared with those of the standard model. In this paper, RANFIS (recurrent ANFIS) will be applied to model the hydraulic jump length (L_j) for the first time.

Second, the use of five different types of meta-heuristic optimization algorithms in learning ANFIS and RANFIS methods and comparing their performance will be accomplished. Genetic algorithm (GA), firefly algorithm (FA), and particle swarm optimization (PSO) algorithms are well-known methods with their successful applications reported by different researchers in engineering studies, especially hydraulics (Zounemat-Kermani et al. 2019a; Sharafati et al. 2020b; Zanganeh 2020). In this study, the performance of moth-flame optimization (MFO), and whale optimization algorithm (WOA), as two of the more recent intelligent algorithms, is additionally compared to those of other algorithms. So far, the capability of these five algorithms has not been assessed in engineering problems.

The third contribution of this study is associated with performing a detached sensitivity analysis in addition to the standard sensitivity analysis of the effective parameters, including discharge (Q) upstream supercritical flow depth (h₁), roughness heights (K_s), and flume width (B) by detaching the hydraulic jump into four types (weak jump, oscillating jump, steady jump, and strong jump) based on the Froude number. As an illustration, the influence of discharge is on the length of the oscillating wave is far more important than other types of jumps.

The rest of the paper is organized as follows. In the next section (Methodology section), general properties of the hydraulic jump are briefly introduced. Then, the applied empirical equations, as well as the developed soft computing methods, are described. Section ‘Model implication and result’ provides the mathematical and soft computing predicted results for the downstream flow depth (h₂) and hydraulic jump length (L_j) using the statistical measures and sensitivity analysis. Subsequently, in the ‘Discussion’ section, the Kruskal–Wallis test is used to check the significant difference between the predicted results. Finally, the ‘Conclusions’ section summarizes the general findings of the study applied.

The main characteristics of a hydraulic jump can be named as the upstream supercritical flow depth (h₁), the downstream subcritical flow depth (h₂), and the length of the hydraulic jump (the distance between the two parts (L_j)). These parameters are schematically shown in Figure 1.

In Figure 1, K_s is the roughness height of the particles in the channel bed and L_r is the roller length. As noted, in order to have a proper design of hydraulic structures like stilling basins downstream of the hydraulic jump, having the geometric characteristics of the jump is essential. In this regard, preliminary studies were carried out to provide a relationship between hydraulic jump parameters in horizontal channels with a smooth bed (Bakhmeteff 1932).

The phenomenon of hydraulic jump on smooth beds has been widely studied and reported by scholars such as Peterka (1958), Rajaratnam (1967), McCorquodale (1986), and Hager (1992).

The preliminary researches on hydraulic jump on the rough beds were carried out by Rajaratnam (1968). The overall results of the researches indicated that the hydraulic jump on the rough bed creates a smaller downstream subcritical flow depth than that in which the bed is smooth (Carollo et al. 2007). Experimental studies indicated that the downstream depth of the jump on the rough bed is virtually half of the smooth bed (Ead & Rajaratnam 2002).

Several researchers conducted studies for representing empirical relations for calculating sequent depths of hydraulic jump. Carollo et al. (2007) studied the hydraulic jump on several rough horizontal beds and presented a novel relation for the momentum equation for the ratio of sequent depths and the roller lengths. Pagliara et al. (2008) tested and analyzed the hydraulic jump properties on a rough bed and provided experimental relationships for these characteristics. In this section, the five empirical relations for predicting the downstream subcritical flow depth (h₂) and one empirical relation for estimating the length of the jump are presented in Table 3.

In the aforementioned equations, is the upstream Froude number, h₁ and h₂ are upstream supercritical and downstream subcritical flow depth, denotes the roughness height, d₅₀ is the median diameter of the base material, h_c is the critical depth, and S₀ is the channel bed slope.

In this study, the methods employed to predict the secondary depth are divided into two groups: soft computing based (ANFIS models) and empirical ones (see Table 3). Moreover, the methods for predicting the length of the jump are divided into two groups: soft computing based (ANFIS and RANFIS models) and one empirical relation (see Table 3). Today, many optimization algorithms have been developed to determine the relative and absolute extrema of complex functions. In the present work, five optimization algorithms were employed for training ANFIS and RANFIS, i.e., three frequently used algorithms (GA, PSO, and FA) and two novel ones (MFO and WOA). In the following, these methods are briefly explained.

The ANFIS is an artificial intelligence learning algorithm that was first introduced by Jang (1993). ANFIS is a soft computing model generated by the learning ability of ANNs and the fuzzy inference system (FIS). By applying the fuzzy rules and sets of membership function (MFs), ANFIS is trained in accordance with the input–output data pairs of the available problem. The structure of ANFIS consists of five consequent layers, including the (1) fuzzification, (2) applying the fuzzy rules, (3) normalizing the MFs, (4) executing the consequent part of the fuzzy rules, and (5) defuzzification. Figure 2 shows the general procedure for setting up the developed ANFIS models.

Although the high capability and robustness of feed-forward ANFIS models have been proven by numerous researchers (Rajaee et al. 2009; Zounemat-Kermani & Scholz 2013); however, as a result of their feed-forward structure, a major deficiency of ANFIS models is that their effectiveness is limited to static problems. Thus, ANFISs are less efficient for representing dynamic processes in comparison with recurrent networks such as NARX models (Zounemat-Kermani et al. 2019b). Hence, it would be reasonable to upgrade the typical feed-forward ANFIS model to a recurrent version (RANFIS) for promoting its performance in modeling recurrent problems. In this paper, as well as employing the standard feed-forward ANFIS model for predicting the downstream subcritical flow depth of jump (h₂) and the jump length (L_j), the jump length is also predicted with the predicted values for h₂ using an RANFIS model. Figure 2 shows the constructed RANFIS structure with four independent inputs and one recurrent input in this study.

In this study, I₁, I₂, … , I_k denote the independent input parameters such as Q, K_s, B, and h₁.

In this study, for converting the feed-forward ANFIS architecture to the recurrent ANFIS (RANFIS) type in predicting the jump length, the output of typical ANFIS will be changed to for the RANFIS. As shown in Figure 2, the output of the feed-forward ANFIS (h₂) is fed-back to the input vector of the network.

The PSO proposed by Kennedy & Eberhart (1995) is a general random (stochastic) optimization method based on swarms such as the social behavior of birds, herds, and insects. In this method, a group of animals/insects randomly search for food in a space in which only one particle of food exists. None of the birds knows the location of the food. One of the best strategies to solve such problems is to follow birds that have the shortest distance from the food. This strategy is, in fact, the basis of the PSO algorithm. The main advantage of this method over other optimization strategies is the presence of a large number of swarming particles in it that leads to its flexibility against the problem of local optima entrapment. Instead of using mutation, this method exchanges information among the members of the population, i.e., particles in the group. In fact, each particle regulates its path with respect to its previous best location, and the previous best location obtained by its neighbors (Hang et al. 2016).

FA was introduced by Yang (2010) as a multi-factor solution to difficult optimization problems. In this algorithm, the firefly attracts other fireflies using light signals. The artificial firefly defined in this algorithm is unisex, and one firefly can attract all other fireflies. The attraction is proportional to the level of brightness of each firefly; in other words, a firefly with less brightness is less attractive, and the one with more brightness is more attractive. Fireflies with less brightness are attracted by those with higher brightness. However, if there is no firefly brighter than the present firefly, the movement of fireflies will be random. This level of brightness is defined based on the objective function: Any firefly that maximizes/minimizes the objective function based on the problem requirements has further brightness and, thus, higher attraction. On the other hand, any firefly that minimizes/maximizes the problem has less brightness and, therefore, less attraction compared to others (Zhang et al. 2018).

GA was first proposed by Holland (1975). This algorithm belongs to the group of random optimization algorithms that are proper for the optimization of complex problems with unknown search space. The main idea behind GA is Darwin's theory of evolution. This theory proposes that natural attributes that are more compatible with natural rules have a higher survival chance. Since GA does not have constraining assumptions regarding the search space, it is an effective and useful method for solving optimization problems (Cheong et al. 2017). In GA, first, a random set of solutions is generated as the initial population that is replaced by a new candidate in each generation. In each iteration of the algorithm, the population is evaluated by the objective function, and a number of the best candidates are selected for the following generation, creating a new population. In the next step, a number of members of this population are used for generating new children by genetic operators such as crossover and mutation. These steps are iterated until the algorithm reaches the appropriate solution.

MFO algorithm was introduced by Mirjalili (2015) inspired by moths. Moths show an interesting method for flying at night using moonlight. Their orientation mechanism is called the transverse orientation. For flight on a straight path, moths move by maintaining a fixed angle relative to the moon. Because the moon is far from them, the path will be straight. However, it is usually observed that moths move in a spiral form around artificial lights because of the weakness in transverse orientation when the flame is near. Thus, this method is useful only when the flame is located far away. In the MFO algorithm, primary solutions are the population of moths, and the same number of flames is also considered, forming two equal-sized matrices. The fitness of each matrix is determined using the introduced objective function. In this algorithm, moths and flames are both solutions to the problem that are different in terms of the method employed for updating them in each iteration. In fact, moths move in the search space, but flames are the best locations (i.e., solutions) of moths until the performed iteration. Also, the new position of moths is calculated using a spiral.

WOA is inspired by humpback whales. The behavior of these whales while hunting, which is called the bubble-net feeding method, was mathematically modeled by Mirjalili & Lewis (2016), and the WOA algorithm was extracted. Humpback whales prefer to hunt groups of plankton and small fish near the water surface. Hunting is performed by creating bubbles along the diameter of a circle or other forms. WOA considers the best solution from the primary population based on the objective function as the location of the prey or its surroundings. The other whales update their position toward this position. The location of the solution must be updated at the end of each iteration of the algorithm. WOA simulates exploitation and an exploration phase. The exploitation phase consists of creating spiral bubbles around the prey and moving in this path toward the prey. The exploration phase comprises random searching for the prey (Mafarja & Mirjalili 2016). In this algorithm, instead of using a random method for discovering the prey, the location of the best solution is used to update new solutions.

The standard version of ANFIS uses a combination of the gradient descent (GD) and least squares (LS) methods for adjusting the premise parameters (MFs parameters) and consequent parameters (linear parameters) for hybrid training the network in the forward and backward stages. The LS method in the forward stage is employed to adjust the consequent parameters (coefficients of the linear relation in Layer 4; see Figure 2), and then the back-propagation GD method in the backward stage is employed to determine the optimal premise parameter values (e.g., the slope, center, and width of the bell-shaped function of MFs in Layer 1; see Figure 2).

In this study, in addition to the application of a standard version of ANFIS/RANFIS models, the training parameters are also optimized by using five heuristic algorithms, including PSO, GA, FA, WOA, and MFO. From now on, these integrated models will be named as ANFIS-PSO/RANFIS-PSO, ANFIS-GA/RANFIS-GA, ANFIS-FA/RANFIS-FA, ANFIS-WOA/RANFIS-WOA, and ANFIS-MFO/RANFIS-MFO in this paper. The general procedure of applying integrated ANFIS/RANFIS models is shown in Figure 3.

In total, 574 data series were extracted from three reliable published scientific reports for modeling the hydraulic jump. Hughes & Flack (1984) carried out experiments on artificial rough beds and measured the characteristics of hydraulic jumps. These experiments were conducted on horizontal and rectangular flume having 0.305 m width. Five different artificially roughened beds were tested. In those experiments, the Froude number was varied from 2.34 to 10.5 and a total of 196 data under different conditions were measured. Carollo et al. (2007) conducted experiments on hydraulic jumps with a rough bed in a rectangular horizontal flume and measured the attributes of hydraulic jump. The flume was 14.4 m long, 0.6 m deep, and 0.6 m wide, and the Froude number was varied from 1.87 to 9.89. The number of data pieces used by these researchers was 367. Moreover, Ead & Rajaratnam (2002) performed experiments on hydraulic jump on a corrugated bed. The flume width, depth, and length were 0.446, 0.60, and 7.6 m, respectively. In their experiments, the hydraulic jump characteristics were measured under two roughened beds. The Froude number of the experiments was varied from 4 to 10, and the number of data pieces was 11. Table 4 presents a summary of the data used.

In this study, to predict the length of the hydraulic jump on a rough bed, flow rate (Q), upstream supercritical flow depth (h₁), roughness heights (K_s), and flume width (B) are calculated using the mentioned methods. In RANFIS, the downstream subcritical flow depth (h₂) enters the model as an input after the preliminary prediction. These parameters are given in Figure 1. Table 5 presents the structure of the input vector of the models.

To create a hydraulic jump model, first, the collected data were randomized and then divided into three categories of training (70%), validation (15%), and test (15%) using the cross-validation method. Next, the network was trained using training data, and validation data were used to resolve the over-training problem. Finally, the model was evaluated by comparing the results of the model and observational data in the test category. In this study, in addition to standard models of ANFIS and RANFIS, five optimization algorithms were incorporated to train ANFIS and RANFIS methods. The general characteristics and tuning coefficients of these methods are given in Table 6.

An effective method for identifying the efficiency of the used models is the application of the Taylor diagram, which is a method for depicting the errors of different models on one diagram. In this diagram, three statistical parameters, including centered root-mean-square difference (CRMS), correlation coefficient (r), and standard deviation (SD) of various models, can be observed (Taylor 2001). The closer the representative point of a method is to the point of observational data, the better its performance would be in simulating the phenomenon.

In this study, both empirical and soft computing models were applied to predict the secondary flow depth of hydraulic jump (h₂). Results of training and validation steps for predicting the h₂ using the standard ANFIS are presented in Table 7.

Statistical measures calculated in Table 7 show that the modeling methods used here were properly trained using the laboratory input data. Even for the poorest method, R² was higher than 0.9, RMSE and MAE were close to 0, and NSE and IA were close to 1. Results also demonstrate that FA, GA, and PSO increased the compatibility of ANFIS with laboratory results, whereas WOA and MFO lowered the efficiency of the integrated ANFIS models compared to the standard ANFIS. These results are noticeable for both training and validation data. Among these methods, ANFIS-FA estimated h₂ in the training and validation phases with more precision. During training with this method, RMSE was the minimum (0.0038 m), and NSE was the maximum (0.988). Although the performance of methods matters in training and validation, the results of the testing phase are of special importance, serving as the main criterion for depicting the performance of various methods. Table 8 presents the results of different soft computing models (i.e., the standard ANFIS and its combination with different algorithms) and three empirical relations in predicting h₂ in the test phase.

Statistical analyses on the obtained results (Table 8) reveal that conventional ANFIS and its combination with optimization algorithms predict h₂ with higher accuracy. Mean RMSE was 0.013 m for soft computing models, while it was 0.019 m for the best empirical relations. Based on the calculated statistical parameters, ANFIS-FA showed the highest performance in predicting the secondary depth of the jump. For this method, RMSE and MAE equaled 0.0053 and 0.0016 m, respectively (minimum values among the applied methods). Also, R², NSE, and IA were 0.971, 0.975, and 0.993, respectively. PSO, FA, and GA increased the precision of ANFIS modeling by 30.263, 17.105, and 21.053%, respectively, whereas MFO and WOA reduced this precision by 10.526 and 3.947%, respectively. Another method for the analysis and evaluation of results is the use of scatter plots (Figure 4).

The scatter plots in Figure 4 depict the superior performance of soft computing models compared to experimental ones. Scatter points for these methods are closer to the 45° line compared to experimental methods. Scatter points for soft computing models are not located outside the −20% and +20% lines, while this is true for the points of some experimental methods. Scatter points for soft computing models are similar to each other in terms of dispersion, showing that they have similar performance. Still, the FA outperforms the others.

Another method to evaluate and visualize the results of various methods is plotting the Taylor diagram. Figure 5 illustrates this diagram for the predicting results of the applied models.

Based on Figure 5, the points belonging to ANFIS methods are closer to the points from the observation point, suggesting that the ANFIS and its combination with optimization algorithms outperform the empirical methods. Points belonging to soft computing models are close to each other, and some are overlapping, indicating that their output and performance are close. Still, the ANFIS-FA is closer to the observational point and outperforms the other methods.

Another parameter predicted in this research is the length of the hydraulic jump on the rough bed (L_j). This parameter is modeled using RANFIS, standard feed-forward ANFIS, and their combinations with the meta-heuristic algorithms. Table 9 presents the results of the training and validation phases for both the ANFIS and RANFIS models.

Based on Table 9, in the training and validation phases, both ANFIS and RANFIS managed to model L_j accurately. The statistical parameters show that ANFIS and RANFIS were trained similarly, such that mean RMSE was 0.0569 and 0.0544 m, respectively, for RANFIS and ANFIS. The performance of the optimization algorithms was similar to their performance in the previous problem (i.e., predicting the secondary depth of hydraulic jump). GA, PSO, and FA improved training, while MFO and WOA degraded it. In this step of modeling, ANFIS and RANFIS models show a similar performance. Still, ANFIS was trained better than RANFIS. The results of the testing phase based on the comparison of methods are presented in Table 10.

Based on Table 10, RANFIS has a higher precision in modeling L_j in comparison to ANFIS. Compared to ANFIS, RANFIS has a smaller RMSE and MAE (i.e., 0.0531 and 0.0178 m, respectively) and a higher NSE, R², and IA (0.789, 0.787, and 0.94, respectively). Among the five meta-heuristic algorithms used for training, FA, PSO, and GA improved the performance of both ANFIS models, whereas MFO and WOA weakened it. Figure 6 presents the scatter plot for visualizing and comparing the performance of different methods.

Based on Figure 6, the applied methods show very close performances. The scattering of points shows that recurrent methods (i.e., RANFIS) have less dispersion and, therefore, better performance compared to standard ones (i.e., ANFIS). The Taylor diagram for the length of the hydraulic jump (L_j) predicting is depicted in Figure 7.

Based on Figure 7, although the points are close to each other, RANFIS-GA was closer to the observational point, suggesting its superior performance compared to others. GA, PSO, and FA improved the performance of RANFIS compared to ANFIS, while the performance of MFO and WOA had a slight effect on RANFIS and even worsened it.

In this step, the simple correlation coefficient (SCC) statistical parameter was employed to examine the effect of independent input parameters on modeling and perform sensitivity analysis. Based on the Froude number, the data used for this purpose are divided into four groups (Figure 8).

This classification was performed based on the type of jump. A Froude number between 1.7 and 2.5 is a weak jump, 2.5–4.5 is an oscillating jump, 4.5–9 is a steady jump, and >9 is a strong jump.

Based on Table 4, the size of particles and width of the channel of the collected data have slight diversity; therefore, the results in Table 11 on these two parameters are not reliable. In comparison, the results for Q and h₁ are reliable. Based on Table 11, for the total data, h₁ and Q have almost equal importance in L_j modeling, but Q is more significant than h₁. For weak, oscillating, and steady jumps, the importance of Q is more than that of h₁, while the opposite is true for strong jump (with very large Froude numbers). For the case of L_j, it is found that the importance of Q is reduced by increasing the Froude number while that of h₁ is increased such that Q is more important than h₁ for weak jump, and the opposite is true for steady and strong jumps.

In the previous sections, we discussed and analyzed the superior methods and parameters affecting the hydraulic jump phenomenon. Still, the question remains whether the employed methods present statistically different results for various models. To answer this question, the Kruskal–Wallis test was run over the predicted results of different applied models (Table 12).

Based on Table 12, in modeling the hydraulic jump at 95 and 99% confidence levels, no statistically significant difference exists between the results of ANFIS and RANFIS models. Furthermore, at these significance levels (0.05 and 0.01), no significant difference exists between the outputs of different algorithms. Findings on the secondary depth of jump (h₂) and jump length (L_j) show that a significant difference exists between empirical relations (see Table 3) and soft computing models results at 95 and 99% confidence levels, while no significant difference is found between the applied soft computing models.

According to the summary results in Table 2 and the outcomes of this study, soft computing models are evidently superior to experimental relations; proportional to the results of the present study. Moreover, the superiority of the FA in combination with the recurrent ANFIS compared to the standard ANFIS is in line with the results obtained in this study.

Knowing hydraulic jump characteristics is a necessity for appropriate designing and planning of hydraulic structures such as stilling basins. In this study, the most important characteristics of the hydraulic jump (L_j and h₂) on a rough bed were modeled and analyzed via ANFIS and RANFIS. In addition to ANFIS, five optimization algorithms (i.e., FA, PSO, GA, MFO, and WOA) were employed to train both ANFIS methods. To set up the ANFIS model, the input structure, including K_s, Q, h₁, and B parameters, was used. Also, to establish the RANFIS structure, the h₂ parameter as a recurrent factor was introduced to the model in addition to the mentioned inputs. The summary of the major findings of this study is presented below.

Results of h₂ modeling indicated that soft computing models are more precise than experimental ones. The calculation of error measurement parameters showed that GA, PSO, and FA enhanced the efficiency of models. Nevertheless, MFO and WOA failed to improve the performance of models. Among these algorithms, ANFIS-FA had the best performance with RMSE, R², and IA of 0.0053 m, 0.975, and 0.993, respectively. By considering RMSE as the basis, the use of GA, FA, and PSO improved the precision of ANFIS in modeling h₂ by 30.26, 21.05, and 17.10%, respectively. According to the L_j predicted outcomes, even though the prediction efficiency of ANFIS and RANFIS methods are close, the application of recurrent architecture in the RANFISs enhanced the performance of ANFISs by around 3.50% (based on the RMSE values). However, to have a better view and judgment about the higher performance of recurrent neuro-fuzzy models, it is suggested that RANFIS methods be used and evaluated in other engineering phenomena. The comparison of the efficiency of the heuristic algorithms demonstrated that the FA and GA have a superior performance in combination with the RANFISs. The results also indicated that the FA, PSO, and GA improve the performance of the ANFISs and RANFISs. Thus, these three algorithms are suggested to be utilized as embedded learning techniques for integrative soft computing methods in future studies.

The results of sensitivity analysis for h₂ input parameters showed that Q is more important than others in predicting h₂ for weak, oscillatory, and steady jumps (1.7 < Fr < 9), while h₁ is more important for strong jumps (Fr > 9). In terms of L_j, by increasing the Froude number (Fr > 9), the importance of h₁ is increased and that of Q is decreased such that for a weak and oscillating jump, the importance of Q is higher than that of h₁, while the opposite is true for the steady and strong jump. Results of the Kruskal–Wallis test revealed that, at 95 and 99%, a statistically significant difference exists between the intelligent and experimental method in modeling h₂ (p < 0.0001), while no significant difference exists among soft computing models in modeling h₂ and L_j.

The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.

All relevant data are available from an online repository or repositories. (https://doi.org/10.1061/(ASCE)0733-9429(1984)110:12(1755), https://doi.org/10.1061/(ASCE)0733-9429(2007)133:9(989), and https://doi.org/10.1061/(ASCE)0733-9429(2002)128:7(656)).