The complex dynamic equilibrium state of rivers, in which the amount of deposition and erosion are in balance, has been a fundamental research topic in river engineering. In this research, three advanced machine learning approaches, including M5 Model Tree (M5Tree), Multivariate Adaptive Regression Splines (MARS) and Least Square Support Vector Regression (LSSVR), are employed to gain new insights and develop more accurate methods for assessment of the longitudinal slope (S), water-surface width (W) and mean water depth (D) of rivers in regime state. Geometric and hydraulic characteristic of 85 cross-sections of Gamasiab River (located in western of Iran), Kaaj River (located in southwestern of Iran) and Behesht-Abad River (located in southwestern of Iran) are used to train and evaluate the employed methods (M5Tree, MARS, and LSSVR). Seven different models comprising various combinations of effective parameters influencing regime river geometry (the flow discharge (Q), median bed grain size (d₅₀) and Shields parameter (τ*)), are developed to evaluate the effect of each of these variables on the prediction of the geometry of regime rivers (S, W and D). The M5Tree method outperformed the other approaches with respect to correlation coefficient (R) values of 0.872, 0.951, and 0.770 and Mean Absolute Relative Error (MARE) values of 0.484, 0.102, and 0.126 for slope, width, and depth prediction, respectively. Furthermore, the flow discharge Q was the key variable governing regime channel width and depth while the regime channel slope was found to be mainly controlled by the Shields parameter.

This paper presents an extensive and practical study of the estimation of stable channel bank shape and dimensions using the maximum entropy principle. The transverse slope ($S_t$) distribution of threshold channel bank cross-sections satisfies the properties of the probability space. The entropy of $S_t$ is subject to two constraint conditions, and the principle of maximum entropy must be applied to find the least biased probability distribution. Accordingly, the Lagrange multiplier (λ) as a critical parameter in the entropy equation is calculated numerically based on the maximum entropy principle. The main goal of the present paper is the investigation of the hydraulic parameters influence governing the mean transverse slope ($\overline{S_t}$) value comprehensively using a Gene Expression Programming (GEP) by knowing the initial information (discharge (Q) and mean sediment size (d₅₀)) related to the intended problem. An explicit and simple equation of the $\overline{S_t}$ of banks and the geometric and hydraulic parameters of flow is introduced based on the GEP in combination with the previous shape profile equation related to previous researchers. Therefore, a reliable numerical hybrid model is designed, namely Entropy-based Design Model of Threshold Channels (EDMTC) based on entropy theory combined with the evolutionary algorithm of the GEP model, for estimating the bank profile shape and also dimensions of threshold channels. A wide range of laboratory and field data are utilized to verify the proposed EDMTC. The results demonstrate that the used Shannon entropy model is accurate with a lower average value of Mean Absolute Relative Error (MARE) equal to 0.317 than a previous model proposed by Cao and Knight (1997) (MARE = 0.98) in estimating the bank profile shape of threshold channels based on entropy for the first time. Furthermore, the EDMTC proposed in this paper has acceptable accuracy in predicting the shape profile and consequently, the dimensions of threshold channel banks with a wide range of laboratory and field data when only the channel hydraulic characteristics (e.g., Q and d₅₀) are known. Thus, EDMTC can be used in threshold channel design and implementation applications in cases when the channel characteristics are unknown. Furthermore, the uncertainty analysis of the EDMTC supports the model’s high reliability with a Width of Uncertainty Bound (WUB) of ±0.03 and standard deviation ($S_d$) of 0.24.

Statistic indexes have been used to assess the accuracy of the results. The results of the examinations indicate that using MLP model along with simultaneous use of dimensionless parameters for the purposes of estimating discharge coefficient: the ratio of water behind the weir to the channel width (h/b), ratio of weir crest length to weir height (L/W), relative Froude number (F=V/√(2Side weirs are used in open channels to control flood and the flow passing through it. Discharge capacity is one of the crucial hydraulic parameters of side weirs. The aim of this study is to determine the effect of the intended dimensionless parameters on predicting the discharge coefficient of triangular labyrinth side weir. MAPE, RMSE, and Rgy)) and vertex angle (ϴ), offered the best results (MAPE= 0.67, R2= 0.99, RMSE = 0.009) in comparison with other models.

In this study, FLUENT software was employed to simulate the flow pattern and water depth changes in a 120° sharp bend at four discharge rates. To verify the numerical model, a 90° sharp bend was first modeled with a three-dimensional numerical model, and the results were compared with available experimental results. Based on the numerical model validation, a 120° bend was simulated. The results show that the rate of increase of the water depth at the cross-section located 40 cm before the bend, compared with the cross-sections located 40 cm and 80 cm after the bend, decreases with the increase of the normal water depth in the 120° curved channel. Moreover, with increasing normal water depth, the dimensionless water depth change decreases at all cross-sections. At the interior cross-sections of the bend, the transverse water depth slope of the inner half-width is always greater than that of the outer half-width of the channel. Hence, the water depth slope is nonlinear at each cross-section in sharp bends. Two equations reflecting the relationships between the maximum and minimum dimensionless water depths and the normal water depth throughout the channel were obtained

In this study, a comparative analysis is done to evaluate the ability of classified radial basis function neural network (CRBFNN) model in estimation of flow variables in sharp open-channel bends with bend angles of 60° and 90°. Accordingly, a RBFNN model is integrated with classification method to design a novel CRBFNN model to predict two velocity and flow depth parameters in a 60° sharp bend. Furthermore, Gholami et al. (Neural Comput Appl 30:1–15, 2018a) pointed out to acceptable ability and more efficiency improvement of hybrid CRBFNN model in prediction of flow variables in 90° sharp open-channel bend compared to simple RBF model. On the other hand, the flow pattern in sharp bends is more complicated than in mild open-channel bends. Moreover, the behavior of flow and its variables is varied in 60° and 90° sharp bends. Therefore, the present paper is aimed to evaluate the performance of RBF and CRBF models in two 60° and 90° (Gholami et al. 2018a) sharp open-channel bends. Available experimental data for velocity and flow depth at six different hydraulic conditions are used to train and test the CRBFNN and simple radial basis function neural network (RBFNN) networks in 60° open-channel bend. Accordingly, efficiency of both RBFNN and CRBFNN models in different bend cross sections is evaluated and compared with each other. The results show that using classified model has improved the simple RBF model performance, as in the CRBFNN model, the error root mean square error and mean absolute error value, 18% and 15.3% for the flow depth prediction and 9% and 5% for the velocity prediction compared to the simple RBFNN model is reduced, respectively. Furthermore, the comparison of model performance in 60° and 90° bends represents that both RBFNN and CRBFNN models in all discharge values in velocity prediction have more ability in 60° bend so that the mean absolute relative error (MARE) value in 60° bend is equal to 0.080 and 0.082 which are lower than MARE values in 90° bend (0.125 and 0.131 for RBFNN and CRBFNN, respectively). Furthermore, both RBFNN and CRBFNN models with lower MARE values equal to 0.015 and 0.012 in 90° bend have more accuracy than the models in 60° bend (0.017 and 0.014). Therefore, the proposed classified models can be used in design and implementation of the curved channels with various bend angles.

This study show the combination of computational fluid dynamics (CFD) and soft computing techniques to make viewpoint for two-phase flow modelling and accuracy evaluation of soft computing methods in the three-dimensional flow variables prediction in curved channels. Therefore, artificial neural network (ANN) and support vectors machines (SVM) models with CFD is designed to estimate velocity and flow depth variable in 60° sharp bend. Experimental results in 6 different flow discharges of 5, 7.8, 13.6, 19.1, 25.3 and 30.8 l/s to train and test, ANN and SVM models is used. The results of numerical models with experimental values are compared and the models accuracy is confirmed. The results evaluation show that all three models ANN, SVM and CFD perform well in flow velocity prediction, with correlation coefficient (R) of 0.952, o.806, and 0.680, and flow depth (R) of 0.999, 0.696, and 0.614 respectively. ANN model to predict both velocity and flow depth variables with mean absolute relative error (MARE) of 0.055 and 0.004 is the best model. Then SVM and CFD models with MARE of 0.069 and 0.089 in velocity prediction and in flow depth prediction CFD and SVM models with MARE of 0.007 and 0.011 are the best models, respectively.