Modelling of clear water scour depth around bridge piers using M5 tree and ANN-PSO

b

Pier diameter

d₅₀

Median diameter of sediment

F_r

Froude number

F_d50

Densiometric Froude number

F_rc

Critical Froude number

g

Acceleration due to gravity

U

Approach mean velocity

U_c

Critical velocity

y

Flow depth

σ_g

Geometric standard deviation of bed sediment size

ν

Kinematic viscosity of water

ρ

density of water

E

Nash–Sutcliffe efficiency

I_d

Index of agreement

N

Number of neuron

R²

Coefficient of determination

ANN

Artificial neural network

CWS

Clear-water scouring

GT

Gamma test

PSO

Particle swarm optimisation

RMSE

Root-mean-square error

SDR

Scour depth ratio

In recent years, many artificial intelligence (AI) techniques have been studied in river sedimentation. Sun et al. (2021) developed a new hybrid model of support vector regression (SVR) and fruitfly optimisation algorithm for the prediction of scour geometry at ski-jump spillway. Sharafati et al. (2021) evaluated the performance of the sediment ejector efficiency using hybrid models and found that artificial neural network (ANN)-particle swarm optimisation (PSO) model has more potential to predict better removal efficiency in sediment dischargers. Tao et al. (2021) reviewed and discussed various AI modelling tools to solve the transportation of river sediment problem and discussed its merits and demerits for different flow conditions. For field conditions, laboratory research equations do not usually yield reliable results (Jones 1984; Baranwal et al. 2023a, 2023b). Most studies are conducted with uniform flow, steady flow, and non-cohesive bed materials due to the scale effect, causing laboratory settings to oversimplify or neglect the intricacies of natural rivers. As a result, laboratory flume-based scour depth models overestimate the scour depth reported at bridge piers (Mueller & Wagner 2005). ANNs and other soft computing approaches have been utilised for estimating the scour depth of bridge piers. Several empirical equations have also been used to evaluate the efficacy of these methods. A majority of studies have shown that the neural network technique outperforms empirical relationships. However, for various civil engineering problems, the M5 model tree (MT) approach has been found to work as well as or better than neural network approaches (Bhattacharya & Solomatine 2003; Solomatine & Siek 2004; Pal & Deswal 2009). Najafzadeh & Giuseppe (2021) developed non-linear regression equations using evolutionary polynomial regression (EPR), gene-expression programming (GEP), multivariate adaptive regression spline (MARS), and M5 MT approaches and found that the M5 tree provided better results.

In the present study, a CWS depth prediction model is developed using M5 tree and ANN-PSO techniques. While developing the model, a wide range of datasets, both experimental and field data, were considered. A Gamma test (GT) was used to determine the optimum input combinations.

Scour around bridge piers was first identified by Kothyari et al. (1992), and subsequent research by Dey et al. (1995) confirmed these findings. Using a cylinder-shaped pier installed in regular beds under clear water flows and determining the depth of the pier scour using a sediment transport equation, Mia & Nago (2003) were able to predict the local scour depth with time, obtaining equilibrium local scour depth when the bed-shear stress approached critical bed-shear stress in the scour hole. Lee & Sturm (2009) experimented to determine the effect of relative sediment size on pier scour depth using three uniform sediment sizes and three bridge pier designs at different geometric model scales. The effect of relative sediment size on dimensionless pier scour depth was analysed by filtering the data with a Froude number criterion. Das et al. (2013) utilised an ultrasonic experimental investigation Doppler velocimeter to focus on the equilibrium scour hole at a circular bridge pier. The horseshoe vortex is an important phenomenon to understand for researchers studying scour around circular piers. The circulation around a pier is affected by several factors such as the densiometric Froude number, flow shallowness, and pier Reynolds number. An increase in densiometric Froude number leads to an increase in circulation, whereas an increase in flow shallowness leads to a decrease in circulation. Similarly, a constant densiometric Froude number or constant flow shallowness leads to an increase in circulation with the pier Reynolds number. To learn more about how scour depths change over time near circular piers, scientists have undertaken experimental experiments. To propose a new empirical relationship including flow intensity, relative water depth, and dimensionless time, Aksoy et al. (2017) compared experimental data with those obtained from empirical equations from the literature. Using formulations based on the MT, EPR, and GEP, Najafzadeh et al. (2018) described the current conceptual relationships for determining the local scour depth in rectangular channels. The MT method was found to be superior to GEP, EPR, and conventional equations for predicting scour depth. The scour depth was shown to be larger in circular piers than in rectangular piers in trials conducted by Chavan et al. (2018). A greater wake vortex was also seen downstream of circular piers compared to rectangular piers. According to Pandey et al. (2020), the creation of an armour layer in a non-uniform gravel bed condition is what causes the scour hole. Overall, laboratory and field investigations by many researchers have shown that the shape of the pier, flow depth, approach flow velocity, and bed sediment size have a major impact on the scour depth. Hamidifar et al. (2021) investigated and verified different equations to estimate the critical flow velocity ratio which was further used in estimating scour depth. They proposed better combinations of scour depth predictive models for clear-water scouring (CWS) conditions. The ability of various data-driven models (DDMs) to predict pier scour depth was evaluated by Qaderi et al. (2021). These DDMs included the support vector machine (SVM), the adaptive neuro-fuzzy inference system (ANFIS), the ANN, the GEP, the improved group method of data handling (GMDH1 and GMDH2), and two hybrid forms of Group method of data handling (GMDH) combined with harmony search (HS). By combining the prediction abilities of three well-known machine learning (ML) methods namely, Gaussian process regression (GPR), random forest (RF), and M5 model tree (M5Tree) with a novel least-squares (LS) boosting ensemble committee-based data intelligent technique. Garg et al. (2022) performed experiments around a bridge pier in a tandem arrangement for clear-water conditions and developed the scour depth model using multivariable regression analysis. Nil & Das (2022, 2023) developed the scour predictive model using the SVM technique considering the geometry, flow, and bed sediment parameters for clear water and live-bed scouring conditions.

The maximum scour depth to flow depth ratio was chosen as an output parameter during the development of this model, and five non-dimensional characteristics were chosen as input parameters. In the current investigation, 942 datasets for clear-water bridge pier scouring have been collected from the previously published literature. 75% (708 data) and 25% (234 data) of the datasets are used for training and testing, respectively. The details of datasets with non-dimensional parameter ranges collected from previous work of CWS have been presented in Table 1.

To choose the combination of best input parameters for modelling the scour depth in the M5 tree and ANN-PSO techniques, the GT was performed.

ANN, or networks of artificial neurons, are a type of computational model used in data processing. These methods are used to model non-linear relationships between input and output in statistical data. Without the requirement for specific physics, an ANN can be considered theoretically as a universal approximation that learns from examples. Neural networks are computational models that can easily process data in parallel. The basic building blocks of an ANN are a network of simple processing units (neurons) that exchange information via analogue signals. These impulses are transmitted along weighted connections between neurons. In this layer, neurons process data from the input layer. Connectivity between neurons in neighbouring layers is mediated by weights. For a network to learn, the training patterns' relative weights at each stage must be adjusted. PSO was utilised by Kennedy & Eberhard (1995) to optimise the methods with the population in mind. The flocking habits of birds and the swimming patterns of schooling fish are two examples of social phenomena that have an effect. The algorithm begins with a population of particles, each of which has a position, speed, and fitness value. The swarm or particle as a whole uses individual knowledge and memory to determine the best course of action. During particle movement, each particle chooses the position that will benefit it the most based on its own and the experiences of its neighbours. The social nature of fish schools and bird flocks served as inspiration for the creation of PSO in 1995, a population-based technique used for process optimisation. The particles' fitness values, positions, and velocities are initialised in the PSO algorithm. Each member of the swarm or particle contributes their unique knowledge and memory to the problem-solving process. Each optimal location of a particle is established by its interactions with its neighbours and other particles. When conventional methods have failed, ANN is called upon to find a solution. One of the most useful kinds of neural networks, ANN can handle both non-linear functions and basic structures. ANNs are built from a collection of simple, parallel components. The neural network of the human brain served as a model for these parts. An ANN is built from layers of cells (neurons) connected by weighted communication lines, and it has three main layers: input, hidden, and output. As signals, these data are transmitted through these lines and into the input network. After that, the activation function of each cell is applied to the input signal to generate an output signal, which is then multiplied by the strength of the connections between any two cells. Optimisation methods known as PSOs work by sampling from an extensive collection of possible starting points. The PSO algorithm was developed by Kennedy & Eberhart (1995), Kennedy et al. (2001), and Kennedy (2010), who were themselves motivated by the social behaviours of animals such as flocks of birds and schools of fish. Many similarities could be drawn between this method and genetic algorithms and other forms of evolutionary computing. A PSO differs from a genetic algorithm in that it does not make use of fast-converging evolutionary operations like inversion and crossover. PSOs perform in many applications, such as neural network training, function optimisation, and even fuzzy control systems. Each ‘particle’ in a PSO stands for a possible answer to the optimisation problem. The fitness function of the problem quantifies the merit of each existing solution. The best local position in terms of fitness is the target of this method. Particle motion in the direction of the ideal solution and at the optimal speed is determined by this fitness value. The system generates a random population of solutions and iteratively shifts these solutions through the search space until it reaches the optimal one. The best position each particle has achieved so far is called its ‘local best,’ whereas the best position overall is the one that is the average of all the local bests inertia where rand1 and rand2 are random numbers between 0 and 1; C1 and C2 are the individual and social learning rates, which are typically equal to 2, and w is the inertia weight, which is used to strike a balance between the particles' capacity to explore both the global and local environments. Using the same evolutionary framework as algorithms like Genetic Algorithm (GA) and PSO, the algorithm seeks an optimal solution.

In ANN-PSO modelling, the coefficients C₁ and C₂ were set to 1.5 and 2.5, respectively, after performing several trials. To assess the strength of different models and evaluate the performance of existing models, two statistical indices, RMSE and coefficient of determination (R²), as well as two relative indices, E and I_d, were used. Table 4 shows the results of error analysis for the training data, testing data, and the entire dataset for various swarm sizes and numbers of neurons (N).

In Table 4, for N = 5 and swarm size = 20, the values of R², E, I_d, and RMSE were found to be 0.84, 0.87, 0.96, and 0.29, respectively, indicating a superior predictive model for scour depth when using the ANN-PSO approach. It was observed that, as swarm sizes increased with the same values of C₁ and C₂ while maintaining the number of neurons constant, the values of R², E, and I_d decreased, while the value of RMSE increased. Figure 3(b) shows that the predicted scour depth values from the newly developed ANN-PSO model closely match the observed values within the dataset ranges, as indicated by the proximity of the data points to the best-fit line. Figure 3(b) shows that the ANN-PSO model is overpredicting the scour depth (SDR) ratio (ds/y) for some data in the range of 0.5–1.5 and underpredicting SDR for some data in the range of 1.3–2.5.

To determine the effectiveness of the current model with a minimum of 100 instances, a sensitivity analysis was performed using the M5 model tree. Table 5 presents the results in terms of four statistical parameters: R², RMSE, E, and I_d. Model 5, which uses all four parameters except σ as inputs, shows high accuracy with R², RMSE, E, and I_d values of 0.84, 0.86, 0.95, and 0.42, respectively. However, the current model with all five non-dimensional parameters as input combinations shows decreasing values of RMSE and R² as the number of instances increases, resulting in increased error. From Table 5, it can be observed that model 6 has the lowest R² value of 0.64, while other models have R² values close to or greater than 0.8. This indicates that b/y and Fr have a significant influence on the estimation of scour depth ratio.

In the ANN-PSO model, the input parameters b/y and Fr are important for modelling, as the first one represents geometric features and the second one represents flowing behaviour around the bridge pier. As the pier diameter increases, scouring also increases because the width of obstruction increases. A Froude number (Fr) less than 1.0 represents subcritical flow, while a Froude number greater than 1.0 represents supercritical flow. Therefore, these flow patterns significantly affect the scouring process. Table 6 illustrates this relationship in terms of four statistical parameters: R², RMSE, E, and I_d, for the M5 model tree. The values of R², RMSE, E, and I_d are found to be 0.80, 0.33, 0.84, and 0.95 respectively. Model 5, with all four parameters excluding σ as input combinations in a model, shows high accuracy for swarm size = 20, n = 5. In the absence of b/y, Model 6 produces poor results, and Model 4 produces a lower value of R² in the absence of Fr.

• The current research utilised the M5 tree and ANN-PSO, two soft computing models, to create a scour depth model for CWS situations near bridge piers. The investigation uncovered the subsequent findings:

• The GT revealed that the diameter of the pier, the depth of the flow, the Froude number, the approach flow velocity, the critical flow velocity, the geometric standard deviation of the bed sediment, and the mean sediment size were the most important elements for forecasting scour depth at the pier.

• Using a minimum of 100 instances in M5 tree modelling yielded better results than alternative combinations of instances for the given datasets for predicting clear-water scour depth. The RMSE for predicting scour depth was 0.318, and the coefficient of determination was 0.82, using the ANN-PSO model with five neurons and a swarm size of 20. In contrast to the ANN-PSO model, the M5 tree model is proven to have superior overall performance. M5 tree modelling reduced the number of instances used during modelling and provided a higher value of R² producing better results.

• Sensitivity analysis revealed that b/y and Fr were critical in determining to scour depth as they handle the geometric shape of the pier and the flow pattern around it, respectively.

• Compared to existing scour depth predictive models of different researchers, the present M5 model tree for CWS conditions shows good accuracy with observed scour depth with an R² value of 0.836 for the present dataset.

• The limitation of the present study is the variety of datasets employed in the modelling of CWS. If the input parameter value falls between the ranges listed in Table 1 for CWS then the model will produce better results. To incorporate the effect of bridge pier shape, different flow conditions and increase model accuracy, more CWS datasets need to be considered. Furthermore, modelling of scour depth for live-bed scouring and temporal scouring conditions will help the researcher and river engineer to efficiently design the bridge pier to avoid its failure.

The authors disclosed no potential conflicts of interest.

Data are available in the supplementary file.

The authors declare there is no conflict.