Abstract:
Bridge pier scour has emerged as a major cause of bridge failure all over the world, and hence, predicting bridge pier scour with high precision and assessing its reliability is of utmost importance. The current research aims at presenting a novel approach using Harris Hawks Optimization-Support Vector Regression (HHO-SVR) and a hybrid Long Short-Term Memory-Convolutional Neural Network (LSTM-CNN) approach with Monte Carlo simulation for assessing the reliability of bridge pier scour prediction. For predicting bridge pier scour, a comprehensive dataset consisting of 153 laboratory-scale clear-water scour data points was compiled using 12 established research studies. Five dimensionless input parameters, V/Vc, y/b, b/d50, Fr, and σg, were used as inputs for predicting normalized scour depth (ds/b). The HHO algorithm was used to optimize SVR hyperparameters, and optimal hyperparameters were found as C = 67.25, ε = 0.047, and γ = 0.326. The HHO-SVR model showed a high level of precision with a high R² value of 0.543 and low RMSE value of 0.213. The LSTM-CNN approach showed high generalization capability with high R² and low RMSE values of 0.757 and 0.156, respectively. Monte Carlo simulation with 100,000 iterations was used as a probabilistic approach to assess the reliability of bridge pier scour prediction using HHO-SVR and LSTM-CNN models. The models showed high precision with high values of reliability index, i.e., β = 3.45 using LSTM-CNN at ds/b = 1.8, whereas HHO-SVR showed high precision with a high value of reliability index, i.e., β = 2.46 at ds/b = 2.0. The sensitivity analysis showed that V/Vc is the most sensitive parameter, contributing around 67-70%, followed by b/d50, contributing around 16-23%. The proposed framework is strictly applicable to single circular bridge piers under clear-water scour conditions within the following parameter ranges: V/Vc = 0.40–0.99, y/b = 0.50–3.30, b/d50 = 18–280, Fr = 0.08–0.32, and σg = 1.10–1.55 (non-cohesive, relatively uniform sediment). Extrapolation to rectangular/wedge/grouped piers, live-bed scour (V/Vc > 1), cohesive sediments, or non-uniform gradation (σg > 2.0) is beyond the validity domain and requires recalibration. To address the limited sample size, 5-fold cross-validation was performed on the entire 153-point dataset, yielding mean R² values of 0.578 ± 0.046 for HHO-SVR and 0.738 ± 0.039 for LSTM-CNN, which confirm the robustness of the reported performance.

Introduction:
It is a widely accepted fact that bridge pier scour is one of the leading causes of bridge failure on a global level. Localized scouring caused by the interaction between flowing water and the pier structures results in the loss of bed materials, thereby causing the formation of a scour hole. Previous studies have indicated that 60% of bridge failures in the United States are caused by hydraulic factors, with the primary cause being scouring itself (Wardhana and Hadipriono 2003). However, the prediction of the depth of the scouring phenomenon is still a complex problem due to the complex interactions between the hydraulic, geotechnical, and geometric factors. In the past, empirical formulas, as proposed by (Melville and Coleman 2000), and the HEC-18 equation, as proposed by (Richardson and Davis 2001), have been commonly used. However, these formulas tend to over- or underpredict the depth due to the underlying assumptions. With the recent developments in the field of machine learning, the prediction of the depth of the scouring phenomenon has been made possible. Support Vector Regression (SVR) has been successfully implemented in the prediction of various hydrological phenomena due to the superior generalization capabilities(Vapnik 1999). However, the performance of the SVR algorithm is highly dependent on the hyperparameters. Harris Hawks Optimization (HHO) algorithm, proposed by (Heidari et al. 2019), is inspired by the social and cooperative hunting behavior of Harris Hawks and has proved its excellent convergence characteristics. Deep learning models, especially hybrid models of CNN and LSTM networks, have shown promising potential in learning spatial and temporal features of problems(LeCun et al. 2015). In (Zounemat-Kermani et al. 2021), an ensemble machine learning model has been proposed for prediction of bridge pier scour depth; however, reliability-based prediction has not been considered in their model. In (Sharafati et al. 2020), soft computing models have been proposed for prediction of bridge pier scour depth in compound channels. In (Bateni et al. 2007), neural network and neuro-fuzzy models have been proposed for prediction of bridge pier scour depth. In (Ebtehaj et al. 2015), genetic programming models have been proposed for prediction of equilibrium scour depth. Monte Carlo simulation is widely used as an effective tool for reliability-based analysis of problems(Rubinstein and Kroese 2016); thus, the current study aims to propose an effective framework of HHO-SVR and LSTM-CNN models coupled with Monte Carlo reliability analysis for prediction of bridge pier scour depth considering 153 experimental data points.
Methodology:
A comprehensive dataset consisting of 153 clear water scour data was compiled from twelve studies, including (Ettema 1980), (Chiew 1984), (Melville 1997), (Yanmaz and Altinbilek 1991), (Sheppard et al. 2004), (Chabert 1956), (Shen et al. 1969), (Lança et al. 2013), (Melville and Chiew 1999), (Oliveto and Hager 2002), (Sheppard and Miller Jr 2006), and (Raudkivi and Ettema 1983). Five dimensionless input parameters were chosen: flow intensity (V/Vc), relative flow depth (y/b), relative pier width (b/d50), Froude number (Fr), and sediment gradation coefficient (σg). The output parameter was chosen as the normalized scour depth (ds/b). The data set was split into 70% for training (107 data points), 15% for validation (23 data points), and 15% for testing (23 data points). HHO-SVR used the Harris Hawks Optimizer algorithm with 30 hawks and 100 iterations for tuning SVR model hyperparameters C, ε, and γ within ranges [0.01–1000], [0.001–1.0], and [0.001–100], respectively. The structure of the LSTM-CNN model consisted of Conv1D layers with 64 and 32 filters, batch normalization, ReLU activation, an LSTM layer with 64 units, dropout rate 0.3, and fully connected layers with 32 and 16 units. Adam optimizer was used with 0.001 learning rate, 300 maximum epochs, and 16 batch size for model training. A Monte Carlo simulation with 100,000 trials was performed for propagating uncertainties in the model inputs based on best-fit distributions for V/Vc ~ Normal(μ = 0.689, σ = 0.179), y/b ~ Lognormal, b/d50 ~ Lognormal, Fr ~ Normal(μ = 0.185, σ = 0.087), and σg ~ Normal(μ = 1.280, σ = 0.182). Reliability analysis was performed for seven critical values of scour depth ratios ds/b: 0.5, 0.8, 1.0, 1.2, 1.5, 1.8, and 2.0 in terms of probability of failure (Pf) and reliability index (β = -Φ⁻¹(Pf)).
Results and Discussion:
The convergence trajectory of HHO showed a rapid decrease in RMSE from 0.166 to 0.096 at 100 iterations, thus validating the optimization of hyperparameters with optimal values C = 67.25, ε = 0.047, and γ = 0.326 (Fig. 1). The HHO-SVR model had a training R² of 0.837 and a testing R² of 0.543, while the LSTM-CNN model had a training R² of 0.805 and a superior testing R² of 0.757 (Fig. 2 and Table 1). Taylor diagram analysis showed that the predictions of the LSTM-CNN model had a higher correlation and lower RMSD compared to HHO-SVR (Fig. 3). Analysis of error distribution showed that HHO-SVR had more asymmetrically distributed error with a wider range of -0.3 to +0.8, while LSTM-CNN had a more symmetrical distribution with a narrower range of -0.3 to +0.4 (Fig. 4). Sensitivity analysis showed that the input parameter V/Vc had the highest influence on both models, HHO-SVR (65.9%) and LSTM-CNN (69.5%), followed by b/d50 (HHO-SVR, 23.1%; LSTM-CNN, 16.5%), and then y/b (HHO-SVR, 9.8%; LSTM-CNN, 5.4%) (Fig. 8). Monte Carlo reliability analysis showed that at ds/b = 1.5, the LSTM-CNN model had Pf = 0.071 (β = 1.47) as compared to HHO-SVR, which had Pf = 0.134 (β = 1.11). When ds/b = 1.8, the LSTM-CNN model had β = 3.45, which is much higher than the required index of 3.0 for structural safety, while HHO-SVR had a much lower value of β = 1.99 (Fig. 6 and Table 2). The convergence analysis showed that Monte Carlo estimates were stable beyond 40,000 simulations (Fig. 9). The box plot comparison showed that HHO-SVR had more dispersed predictions with a range of 0.95-1.37, with many outliers, while LSTM-CNN had more concentrated predictions with a narrower range of 0.93-1.28 (Fig. 7).
Conclusion:
In this study, a reliable prediction of bridge pier scour depth was constructed with a detailed framework of the Monte Carlo simulation, the HHO-SVR and the LSTM-CNN model. LSTM-CNN model proved to have better predictive performance (R2=0.757) than HHO-SVR (R2=0.543) on the test data. Monte Carlo reliability simulation (100,000 simulations) demonstrated that LSTM-CNN model has more reliable scour estimates, which had the target index b=3.0 at ds/b=1.8 than HHO-SVR which failed to achieve it within the same analysis range. Flow intensity ratio (V/Vc) was determined as the most dominant governing factor whose relative influence was close to 67-70%. The suggested framework offers an engineer a probabilistic resource to determine the scour risk when there is a hydraulic uncertainty to be quantified so that a better decision can be made in designing the bridge foundation.