Partial Derivative Sensitivity Analysis Research Articles

Generalized structure of the group method of data handling for modeling iceberg drafts

The iceberg draft prediction is vital to mitigate the collision risk of deep keel icebergs with the seafloor-founded infrastructures, including the subsea pipelines, wellheads, hydrocarbon loading equipment, and communication cables crossing the Arctic and subarctic areas since the drifting icebergs may gouge the ocean floor and the physical and operational integrity of the submarine structures would be threatened. In this study, the iceberg drafts were simulated using the generalized structure of the group method of data handling (GS-GMDH) algorithm for the first time. The parameters affecting the iceberg drafts were determined, and five GS-GMDH models comprising GS-GMDH 1 to GS-GMDH 5 were developed utilizing those parameters governing. A dataset comprising 161 distinct case studies measured in the most significant field investigations of iceberg characteristics was generated, and the GS-GMDH models were trained through 60 % of the data, the rest of the data (i.e., 40 %) were considered for the GS-GMDH models’ validation. By defining different scenarios, the most accurate GS-GMDH model and the most important input parameters were identified. The sensitivity analysis demonstrated that the iceberg width ratio (W/H) and the iceberg shape factor (Sf) were identified as the most influencing input parameters. The comparison between the performance of the premium GS-GMDH model and the group method of data handling (GMDH), artificial neural network (ANN) algorithms, and the empirical models proved that the GS-GMDH model simulated the iceberg drafts with the highest level of precision and correlation along with the lowest degree of complexity. Based on the partial derivative sensitivity analysis (PDSA), the magnitude of iceberg drafts grew by increasing the value of the iceberg width and iceberg length. Ultimately, a GS-GMDH-based equation was presented to estimate the iceberg drafts for practical applications, particularly in the early stages of iceberg management projects and engineering designs.

Modeling triangular, rectangular, and parabolic weirs using weighted robust extreme learning machine

In this study, dimensionless parameters influencing the coefficient of discharge (COD) are found and four different WRELM models are developed. After that, a dataset is created for verifying the WRELM models in which 70% of the data are employed to train learning machine models and the remaining 30% to test them. For the mentioned algorithm, the optimal number of hidden layer neurons along with the best activation function is chosen. Additionally, the best value for the regularization parameter of the WRELM algorithm is computed. By analyzing the simulation results, the superior WRELM model and the variables impacting the COD are detected. The superior WRELM model approximates COD values with the minimum error and the highest correlation with laboratory values. For the superior model, the values of the R, MAE and VAF statistical indices are computed to be 0.994, 0.0004 and 98.662, respectively. The analysis of the simulation results indicates that the dimensionless parameters α and T/B are the most influencing input parameters. The superior WRELM model results are compared with the algorithm, and it is concluded that the WRELM model is noticeably more efficient. For the superior WRELM model, a partial derivative sensitivity analysis (PDSA) is conducted in which as the input parameter α increases, the PSDA value increases as well. Finally, an equation is suggested for estimating COD values.

Prognostication of scour around twin and three piers using efficient outlier robust extreme learning machine.

One of the most essential difficulties in the design and management of bridge piers is the estimation and modeling of scouring around the piers. The scour depth downstream of twin and three piers were simulated using a new outlier robust extreme learning machine (ORELM) model in this study. Furthermore, k-fold cross-validation with k = 4 was employed to validate the outcomes of numerical models. Four ORELM models with effective scouring parameters were first created to simulate scour depth. After then, the number of hidden layer neurons increased from two to thirty. The number of ideal hidden neurons was determined by examining the modeling results. The sigmoid activation function was also introduced as the best function. Furthermore, a sensitivity analysis was used to identify the superior model. The best model predicted scour depth as a function of the Froude number (Fr), the pier diameter to flow depth ratio (D/h), and the distance between the piers to flow depth ratio (d/h). The values of the objective function were accurately approximated by this model. As a result, using the ORELM model, the R2, scatter index, and Nash-Sutcliffe efficiency coefficient were calculated to be 0.953, 0.146, and 0.949, respectively. The most efficient parameters for simulating the scour depth were Fr and D/h, according to the modeling results. It is worth noting that nearly half of the superior model's simulated outputs had an inaccuracy of less than 10%. The superior model's performance has been underestimated, according to uncertainty analysis. After that, a simple and practical equation for calculating the scour depth was established for the superior model. Additionally, the influence of each input parameter on the objective function was assessed using a partial derivative sensitivity analysis.

Evaluation of discharge coefficient of triangular side orifices by using regularized extreme learning machine

The present paper attempts to reproduce the discharge coefficient (DC) of triangular side orifices by a new training approach entitled “Regularized Extreme Learning Machine (RELM).” To this end, all parameters influencing the DC of triangular side orifices are initially detected, and then six models are extended by them. For training the RELMs, about 70% of the laboratory measurements are implemented and the remaining (i.e., 30%) are utilized for testing them. In the next steps, the optimal hidden layer neurons number, the best activation function and the most accurate regularization parameter are chosen for the RELM model. As a result of a sensitivity analysis, we figure out that the most important RELM model simulates coefficient values with high exactness. The best RELM model estimates coefficients of discharge using all input factors. The efficiency of the best RELM model is compared with ELM, and it is demonstrated that the former has a lower error and better correlation with the experimental measurements. The error and uncertainty examinations are executed for the RELM and ELM models to indicate that RELM is noticeably stronger. At the final stage, an equation is proposed for computing this coefficient for triangular side orifices and a partial derivative sensitivity analysis is also carried out on it.

Ice-seabed interaction modeling in clay by using evolutionary design of generalized group method of data handling

Ice-induced scour is a serious challenge for the subsea pipelines in the Arctic shallow waters. Estimation of the maximum pipeline deformation and its minimum burial depth can guarantee the operational integrity of these structures in the ice-prone regions. The pipeline buried below the ice keel is still threatened by subgouge soil deformation that is extended down the ice tip due to the shear resistance of the seabed soil. Determining the subgouge soil deformations is a challenging process that needs costly experimental and numerical simulations. In this paper, an alternative and cost-effective methodology has been proposed using a robust neural network-based method titled “generalized structure of group method of data handling” (GS-GMDH) for the first time to simulate the horizontal and vertical subgouge soil deformation profiles in clay. Using the parameters governing the subgouge soil deformations, nine GS-GMDH models were defined. The premium GS-GMDH models and the most influencing input parameters comprising the soil depth and the gouge geometry were introduced by performing a sensitivity analysis. Subsequently, results of the best GS-GMDH models were compared with the classical group method of data handling (GMDH), the artificial neural network (ANN), and the empirical approaches. An uncertainty analysis showed that the GS-GMDH slightly overestimated the horizontal and underestimates the vertical subgouge soil deformations. A partial derivative sensitivity analysis (PDSA) was also performed to assess the influence of the input parameters on the subgouge soil deformations. Lastly, a set of GS-GMDH-based equations were proposed for fast estimation of the subgouge soil deformations in clay.

Assessment of discharge coefficient in trapezoidal and rectangular canals through regularized extreme learning machine

Since side weirs efficiently serve in flood management plans, irrigation canals and drainage systems, the estimation or simulation of its discharge coefficient seems to be a necessary task. Moreover, as the determination of this coefficient is perhaps the most important factor for the design of a side weir, there are numerous studies focused on this regard. As for the first case, this paper aims at the utilization of the modern regularized extreme learning machine (RELM) method for simulating the discharge capacity of side weirs within trapezoidal and rectangular flumes. At the first step, the parameters affecting the discharge coefficient are detected, and then 19 RELM models are extended using them. To train and test the RELM models. 70% and 30% of the experimental data are employed, respectively. In addition, the hidden layer neurons and also the activation function belonging to the RELM model are optimized. In other words, the number of neurons is considered as 14 and the activation function is introduced as the best. The superior model predicts the target values using the Froude number, the slope of main conduit walls and ratio of the weir crest to upstream flow depth. The superior model of RELM is very powerful and even shows a better performance in comparison with the ELM model. For instance, the correlation coefficient, Scatter Index and Nash-Sutcliffe Efficiency Coefficient for the RELM superior model are estimated to be 0.982, 0.043 and 0.963. A formula is suggested for approximating the target function for application works. Lately, a partial derivative sensitivity analysis (PDSA) is executed for the provided equation.

Weighted regularized extreme learning machine to model the discharge coefficient of side slots

In the current research, a modern learning machine algorithm named “Weighted Regularized Extreme Learning Machine (WRELM)" is implemented for the first time for the simulation of the coefficient of discharge of side slots. For this purpose, an effective variable on the coefficient of discharge of side slots is firstly introduced, then five distinctive WRELM models are produced by it for the estimation of the coefficient. In the next stage, a database is created for verification of WRELM results. it should be mentioned that 70% of the data are utilized for training the WRELM models, while the rest (i.e. 30%) for testing them. After that, the optimal number of hidden layer neurons as well as the best activation function of the WRELM algorithm are chosen. In addition, the best regularization parameter and also the weight function of the WRELM are achieved. By conducting a sensitivity analysis, the most effective variable for the simulation of the coefficient of discharge along with the WRELM superior model is introduced. The WRELM superior model estimates values of the coefficient of discharge with the maximum exactness and the highest correlation. For instance, the estimations of the correlation coefficient and scatter index for this model are computed to be 0.930 and 0.051, respectively. The sensitivity analysis shows that the ratio of the side slot crest height to its length and the Froude number should be considered as the most important input variables. A comparison between the WRELM with the ELM displays that the former works much better. Furthermore, an uncertainty analysis is executed for both models. Eventually, an equation is suggested for the estimation of the coefficient of discharge and a partial derivative sensitivity analysis is performed on it.

Modeling Length of Hydraulic Jump on Sloping Rough Bed using Gene Expression Programming

Generally, length of hydraulic jump is one the most important parameters to design stilling basin. In this study, the length of hydraulic jump on sloping rough beds was predicted using Gene Expression Programming (GEP) for the first time. The Monte Carlo simulations were used to examine the ability of the GEP model. In addition, k-fold cross validation was employed in order to verify the results of the GEP model. To determine the length of hydraulic jump, five different GEP models were introduced using input parameters. Then by analyzing the GEP models results, the superior model was presented. For the superior model, correlation coefficient (R), Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE) were computed 0.901, 11.517 and 1.664, respectively. According to the sensitivity analysis, the Froude number at upstream of hydraulic jump was identified as the most important parameter to model the length of hydraulic jump. Furthermore, the partial derivative sensitivity analysis (PDSA) was performed. For instance, the PDSA was calculated as positive for all input variables.

Estimation of scour depth around cross-vane structures using a novel non-tuned high-accuracy machine learning approach

Due to the vital role of rivers and canals, the protection of their banks and beds is critically important. There are various methods for protecting beds and banks of rivers and canals in which “cross-vane structures” is one of them. In this paper, the scour hole depth at the downstream of cross-vane structures with different shapes (i.e., J, I, U, and W) is simulated utilizing a modern artificial intelligence method entitled “Outlier Robust Extreme Learning Machine (ORELM)”. The observational data are divided into two groups: training (70%) and test (30%). After that, the most optimal activation function for simulating the scour depth at the downstream of cross-vane structures is selected. Then, using the input parameters including the ratio of the structure length to the channel width (b/B), the densimetric Froude number (Fd), the ratio of the difference between the downstream and upstream depths to the structure height (Δy/hst) and the structure shape factor $$ \left( \phi \right) $$, eleven different ORELM models are developed for estimating the scour depth. Subsequently, the suitable model and also the most effective input parameters are identified through the conduction of an uncertainty analysis. The suitable model simulates the scour values by the dimensionless parameters b/B, Fd, Δy/hst. For this model, the values of the correlation coefficient (R), Variance accounted for (VAF) and the Nash-Sutcliffe efficiency (NSC) for the suitable model in the test mode are obtained 0.956, 91.378 and 0.908, respectively. Also, the dimensionless parameters b/B, Δy/hst. are detected as the most effective input parameters. Furthermore, the results of the suitable model are compared with the extreme learning machine model and it is concluded that the ORELM model is more accurate. Moreover, an uncertainty analysis exhibits that the ORELM model has an overestimated performance. Besides, a partial derivative sensitivity analysis (PDSA) model is performed for the suitable model.

Ice-Seabed interaction analysis in sand using a gene expression programming-based approach

Arctic subsea pipelines are usually buried for physical protection against the ice-induced scours. Determination of the maximum horizontal deformations for guaranteed operational integrity and cost-effective design is probably the most challenging aspect of subsea pipelines in ice-prone areas. The large uncertainties associated with the design of ice-protected subsea pipeline using the existing empirical equations, advanced experimental and sophisticated numerical studies with significant time and cost impacts are usually preferred to ensure the sufficiency and the cost-effectiveness of the pipeline design against the ice attack. This has caused the industry to keep looking for more effective and less-costly solutions for modeling the ice impact on buried pipelines. In this study, a Gene Expression Programming (GEP) model representing the Artificial Intelligence (AI) approaches was used for the first time to simulate the subgouge soil deformation in the sand. A database was constructed using some of the published experimental studies identifying the key input parameters including soil depth, bearing pressure, the maximum vertical extent of subgouge deformation, attack angle, and dilation index. Subsequently, six GEP models were developed and validated by using a K-fold cross-validation method. The performance of the GEP method was compared with an Artificial Neural Network (ANN) model, and uncertainty analysis (UA) along with a partial derivative sensitivity analysis (PDSA) was conducted to assess the influence domain of the key parameters. The study showed that the evolutionary numerical methods could be used as an accurate and cost-effective alternative for modeling of the ice-induced subgouge deformations.

Prognostication of discharge coefficient of labyrinth weir using outlier robust extreme learning machine

In this paper, for the first time, a robust artificial intelligence (AI) technique called “outlier robust extreme learning machine (ORELM)” is used to estimate the discharge coefficient of labyrinth weirs. At the beginning, the number of the hidden layer neurons initials from 5 and continues to 45 and the most optimal number of the hidden layer neurons are taken into account equal to 5. Then, the numerical model is verified by the k-fold cross-validation approach in which k is considered equal to 5. After that, different activation functions are evaluated to detect the most accurate one for the numerical model. Subsequently, six different ORELM models are developed using the parameters affecting the discharge coefficient of labyrinth weirs. Also, the superior model and the most effective input parameters are identified through a sensitivity analysis. For example, the values of R2, RMSRE and NSC for the superior model are calculated 0.943, 5.224 and 0.940, respectively. Furthermore, the ratio of the head above the weir to the weir height (HT/P) and the ratio of the width of a single cycle to the weir height (w/P) are introduced as the most important input parameters. Also, the results of the ORELM superior model are compared with the artificial intelligence models including the extreme learning machine, artificial neural network and the support vector machine and it is concluded that ORELM has a better performance. Then, an uncertainty analysis is conducted for the ORELM, ELM, ANN and SVM models and it is proved that the ORELM has an overestimated performance. Eventually, a practical formula is presented to engineers without knowledge of artificial intelligence for approximating the discharge coefficient of labyrinth weirs. Additionally, a partial derivative sensitivity analysis is carried out for this equation.

Discharge coefficient of side weirs on converging channels using extreme learning machine modeling method

Generally, side weirs are utilized in main flumes to control and adjust the flow. In this study, for the first time, the discharge coefficient of side weirs installed on converging channels is simulated using proposed artificial intelligence (AI) measurement “Extreme Learning Machine” (ELM). To enhance the efficiency of the numerical model, the Monte Carlo simulations (MCs) are applied and the k-fold cross validation method is used for validating the yielded numerical results. Then, the most significant input parameters are detected for simulating the discharge coefficient. Subsequently, the number of the ELM hidden layer neurons is determined by trial and error process. Next, the most optimized activation function is also chosen. Then, using the input parameters, six ELM models are developed and the superior model and the most effective input parameter are identified through a sensitivity analysis. It is observed that the model estimates the discharge coefficient with acceptable accuracy. For instance, the coefficient of determination (R2) and the mean absolute percent error (MAPE) for this model are surmised to be 0.963 and 5.135, respectively. Also, the Froude number at the downstream of the side weir (Fd) is introduced as the most effective input parameter. Besides, a relationship is provided for calculating the discharge coefficient. Subsequently, through the conduction of an error analysis, it is concluded that the superior model has an overestimated performance. Furthermore, the partial derivative sensitivity analysis (PDSA) is carried out on the input parameters and it is proved that the PDSA value is obtained negative for all Froude numbers.

Estimation of scour depth around submerged weirs using self-adaptive extreme learning machine

Abstract In this study, the equilibrium scour depth downstream of the weir (ds-a), the maximum scour depth downstream of the weir (ds-max), the equilibrium scour depth upstream of the weir (dus-a) and the maximum scour depth upstream of the weir (dus-max) were simulated around the submerged weirs using the self-adaptive extreme learning machine (SAELM) model. In other words, the SAELM was utilized for the simulation of the scour depths around submerged weirs for the first time. In addition, Monte Carlo simulations (MCSs) were used to increase the accuracy of the artificial intelligence model. The results of modeling were validated using k-fold cross validation. At first, all effective parameters on the scour depth were determined and five distinct SAELM models were defined. Then, the optimal activation function of the SAELM model was obtained. By analyzing the results of modeling, the best models were identified to estimate ds-a/ht, ds-max/ht, dus-a/ht, and dus-max/ht, and the ratio of the average inflow velocity to the critical velocity (U0/Uc) was determined as the most effective input parameter. In the following, the results of superior models were compared with the artificial neural network (ANN) and support vector machine (SVM). The results showed that SAELM models were more accurate. The uncertainty analysis was performed for these models, some of them were overestimated and others were underestimated. In addition, some equations were presented for equilibrium models for calculation of scour depth around the submerged weirs, which are used by environmental and hydraulic engineers without previous knowledge about the artificial intelligence models. Finally, a partial derivative sensitivity analysis (PDSA) was performed for all input parameters of the superior models.

Gene expression programming-based approach for predicting the roller length of a hydraulic jump on a rough bed

ABSTRACT Hydraulic jumps are used for dissipating water flow energy after ogee spillways, chlorinating wastewater flow and other hydraulic purposes. This paper presents a concise and practical relationship based on Gene Expression Programming (GEP) to estimate the roller length of a hydraulic jump on a rough bed. K-fold cross validation (k = 10) is applied to validate the numerical results. Various models are developed in terms of the Froude number upstream of the hydraulic jump , ratio of sequent depth and the relative roughness . The model results are analyzed and compared to determine the superior model. The MARE and RMSE obtained for the best model are 0.079 and 1.965, while the correlation coefficient (R), scatter index and BIAS are 0.959, 0.101 and 0.003, respectively. To evaluate each input variable’s effect on the proposed model, Partial Derivative Sensitivity Analysis (PDSA) as well as uncertainty analysis is done for all GEP models. A number of artificial intelligence approaches are also compared. This study shows that the presented equation is simpler and more practical than other equations from prior studies.

Evolutionary Prediction of Biohydrogen Production by Dark Fermentation

The present work is a study of the performance and effect of operational parameters on biohydrogen production from palm oil mill effluent by dark fermentation in batch mode. The process parameters examined are pH (5, 5.5, and 6), temperature (30, 35, and 40 °C), substrate concentration (5000, 12 500, and 20 000 mg L−1) and inoculum volume (20, 25, and 30 mL). The inoculum concentration prepared was 10 000 mg L−1 volatile suspended solids. The experiments were designed by response surface methodology (RSM). The highest chemical oxygen demand (COD) removal, hydrogen percentage (H2%) and hydrogen yield (HY) obtained were 58.3%, 80%, and 4.83 mol H2/mole of COD consumed, respectively. Based on the experimental data obtained with the RSM design, gene expression programming (GEP) was developed to predict the COD removal, hydrogen production, and hydrogen yield as process responses. The projected models were assessed based on the correlation coefficient (R2), root mean square error, mean absolute relative error, scatter index, and BIAS. The results demonstrate that the GEP model outperformed the RSM model and was superior in predicting the response variables. Partial derivative sensitivity analysis was also employed to assess the effect of each variable on COD%, H2%, and HY prediction. The prediction uncertainty for COD%, H2%, and HY was quantified, and the results were ±0.11, ±0.17, and 0.015, respectively. According to the results, the GEP model is more efficient than the RSM model in predicting the experimental data for biological hydrogen production in the dark fermentation process.

Calculating the energy consumption of electrocoagulation using a generalized structure group method of data handling integrated with a genetic algorithm and singular value decomposition

In this study, a hybrid data mining method for predicting energy consumption is proposed, namely the group method of data handling integrated with a genetic algorithm and singular value decomposition (GMDH-GA/SVD). As the randomness of renewable sources influences prediction methods, prediction model improvements are necessary for further development. Thus, GMDH-GA/SVD is introduced to model energy consumption as the primary criterion for process evaluation in finding the optimum condition to achieve the least energy consumption process. The parameters include the initial pH, the initial dye concentration, the applied voltage, the initial electrolyte concentration and the treatment time. The uncertainty analysis is applied to survey the quantitative performance of the new proposed model compared to existing popular reduced quadratic multiple regression models and two recently published models in the form of a Taylor diagram, indicating the proposed model is the most accurate. Moreover, partial derivative sensitivity analysis was done on the key parameters in the new model to provide insight into the calibration process of the new model.

Estimation of the Darcy–Weisbach friction factor for ungauged streams using Gene Expression Programming and Extreme Learning Machines

The most uncertain parameter in determining the stage-discharge relationship of an ungauged stream is the friction factor, a parameter for which a gauged stream can be back-calculated using hydrometric survey data that has been compiled over a wide range of flow conditions. The existing friction factor models for ungauged streams that do not require hydrometric survey data as input can have very large prediction errors. In this study, 415 streamflow events collected from 52 rivers across three continents were assembled to allow for the use of a suitable global dataset size/range to develop a robust model for the prediction of the friction factor in ungauged streams using two machine learning algorithms (Gene Expression Programing (GEP) and Extreme Learning Machine (ELM)). The new GEP and ELM models outperformed existing models with significantly improved R-squared values of 0.76 and 0.78, respectively. The combination of the two input parameters (relative smoothness and the friction slope) allowed the new models to more accurately predict the friction factor for a wider range of streams and flow conditions. The forecasting uncertainty of the developed GEP and ELM models were compared with past models and found to be the least uncertain, with Width Uncertainty Band (WUB) values of ±0.0536 and ±0.0504, respectively. For the first time in this field of research, partial derivative sensitivity analysis was applied to the trends of the friction factor estimation by GEP and ELM. It was found that GEP and ELM models are extremely sensitive to friction slope in comparison with relative smoothness. The GEP and ELM model results for predicting the Darcy–Weisbach friction factor were compared with different existing relationships, and the superior performance of the proposed models was illustrated.

Prediction of wave runup on beaches using Gene-Expression Programming and empirical relationships

This paper assesses the accuracy of seven empirical models and an explicit Gene-Expression Programming (GEP) model to predict wave runup against a large dataset of runup observations. Observations consist of field and laboratory measurements and include a wide array of beach types with varying sediment sizes (from fine sand to cobbles) and bed roughness (from smooth steel to asphalt). We show that the best performing models in the literature are prone to significant errors (minimum RMSE of 1.05 m and NMSE of 0.23) when used with unseen data, i.e., uncalibrated models; however, overall error values and correlations are significantly reduced when models are optimised for the dataset. The best performing empirical models use a Hunt type scaling with an additional parameter for wave induced setup. The predictive ability of the explicit GEP model, which better captures the complex nonlinear effects of the key factors on the wave runup length, resulted in a statistically significant improvement in predictive capacity in comparison to all other empirical models assessed here, even on unseen data. Wave height, wavelength, and beach slope are shown to be the three primary factors influencing wave runup, with grain size/bed roughness having a smaller, but still significant influence on the runup. The r2 of the best optimised existing models (which takes the form of Holman (1986) and Atkinson et al. (2017) their M2 model) was 0.77, with a RMSE of 0.85 m. These were improved to an r2 of 0.82 (6% increase) and RMSE of 0.75 m (12% decrease) in the GEP-based model. The sensitivity of the proposed GEP-based model to each input variable is assessed via a partial derivative sensitivity analysis. The results demonstrate a higher sensitivity in the model to small values of each input and that wave steepness and beach slope are the primary factors influencing wave runup.

Determining the Scour Dimensions Around Submerged Vanes in a 180° Bend with the Gene Expression Programming Technique

Submerged vanes are installed on rivers and channel beds to protect the outer bank bends from scouring. Also, local scouring occurs around the submerged vanes over time, and identifying the effective factors on the scouring phenomena around these submerged vanes is one of the important issues in river engineering. The most important aim of this study is investigation of scour pattern around submerged vanes located in 180° bend experimentally and numerically. Firstly, the effects of various parameters such as the Froude number (Fr), angle of submerged vanes to the flow (α), angle of submerged vane location in the bend (θ), distance between submerged vanes (d), height (H), and length (L) of the vanes on the dimensionless volume of the scour hole were experimentally studied. The submerged vanes were installed on a 180° bend whose central radius and channel width were 2.8 and 0.6 m, respectively. By reducing the Froude number, the scour hole volume decreased. For all Froude numbers, the biggest scour hole formed at θ = 15°. In all models, by increasing the Froude number, the scour hole volume significantly increases. In addition, by increasing the submerged vanes’ length and height, the scour hole dimensions also grow. Secondly, using gene expression programming (GEP), a relationship for determining the scour hole volume around the submerged vanes was provided. For this model, the determination coefficients (R2) for the training and test modes were computed as 0.91 and 0.9, respectively. In addition, this study performed partial derivative sensitivity analysis (PDSA). According to the results, the PDSA was calculated as positive for all input variables.

Evolutionary design of generalized group method of data handling-type neural network for estimating the hydraulic jump roller length

Hydraulic jumps generally occur subsequent to structures such as ogee spillways, control gates, and weirs. The jump roller length is considered one of the main hydraulic jump parameters. In this study, the roller length of a hydraulic jump on a rough channel bed is predicted using a novel, evolutionary, generalized structure design of a group method of data handling (GS-GMDH)-type neural network. The topology of GMDH is designed with a genetic algorithm . Initially, the three most important non-dimensional parameters affecting hydraulic jump roller length, including the Froude number upstream of a hydraulic jump \(\left( {Fr} \right) \), the ratio of sequent depths \(\left( {{h_2 }/{h_1 }} \right) \), and the relative roughness \(\left( {{ks}/{h_1 }} \right) \) were used to generate four different GS-GMDH models, and the most accurate model is identified. The best new GS-GMDH model prediction statistics, including RMSE, MARE, and correlation coefficient are 1.816, 0.081, and 0.966, respectively, while the scatter index and BIAS values are 0.084 and 1.45, respectively. A partial derivative sensitivity analysis of the input parameters for the new model is also performed. The new model predictions are then compared with predictions of a number of other models. The superior performance of the new GS-GMDH over these existing models is illustrated.