1. Introduction
Pile foundations, which are extensively utilized in the construction of high-rise buildings, bridges, and various industrial facilities, serve as a crucial form of structural support. In evaluating the safety performance of pile foundations during their operational phase, the ultimate load capacity of the piles emerges as a critical parameter that not only ensures the overall safety and economic viability of the construction but also significantly influences its integrity. However, the availability of field testing methods to assess the ultimate load-carrying capacity of piles is limited, thereby constraining our comprehensive understanding of pile foundations. Consequently, the precise prediction of the bearing capacity of driven piles is of paramount importance and represents a significant area for further investigation within the field of geotechnical engineering.
In the practical application of pile foundations, the initial step involves assessing the ultimate load capacity that the pile is capable of supporting. However, the intricacies associated with the pile–soil interaction mechanism, along with the dynamic changes in stress states during the pile driving process, significantly complicate the precise evaluation of ultimate bearing capacity. At present, a thorough investigation into the ultimate bearing capacity is impeded by various limitations, including site-specific conditions and the theoretical models utilized for calculations. As a result, evaluations predominantly depend on semi-empirical predictive modeling and direct validation through pile loading tests. Unfortunately, a significant discrepancy exists between the predicted values derived from conventional design methodologies and the outcomes obtained from field tests, as noted in the pertinent literature [
1,
2,
3,
4]. While pile load tests can provide accurate results, their implementation requires substantial time and financial investment, which presents a considerable obstacle to their widespread use. In contrast, the semi-empirical approach is relatively straightforward and efficient for computational predictions, being less time-consuming and more cost-effective. However, the predictive accuracy of this method is constrained, and its applicability is largely limited to specific soil types and structural configurations, as discussed in the literature [
5,
6]. In light of existing constraints, numerous researchers have increasingly turned to machine learning techniques to predict the ultimate bearing capacity of both closed and open-ended driven piles in cohesive and non-cohesive soils, thereby aiming to reduce time and financial expenditures [
7,
8,
9]. Ülker M B C et al. [
10] employed a machine learning approach known as the extreme learning machine (ELM) technique to create a predictive model for estimating the lateral bearing capacity of piles in clay soils, subsequently comparing the efficacy of the ELM model with other artificial intelligence frameworks and established empirical equations. Similarly, Muduli P K et al. [
11] utilized machine learning methodologies to assess the bearing capacity of piles in saline soils located in cold regions by developing a predictive model. Deng et al. [
12] implemented kernel ridge regression (KRR) and multilayer perceptron (MLP) algorithms to formulate predictive models for vertical and horizontal load-bearing capacities, respectively, and utilized the Shapley Additive Explanations (SHAP) method for explanatory analysis of these models. Wang et al. [
13] devised two enhanced multitask learning models for evaluating pile driving capacity, which were optimized using multi-output least squares support vector regression (MLSSVR) and further refined through the application of meta-heuristic algorithms. Borthakur et al. [
14] applied the support vector machine (SVM) regression method to assess the load-carrying capacity of micropile clusters in soft clay soils, utilizing data from 54 large-scale static vertical micropile load tests conducted in a test pit, along with a database derived from load-settlement maps obtained through cone penetration tests (CPTs) performed in the same test pit for the SVM model. Conversely, Moayedi et al. [
15] implemented optimization algorithms, specifically a genetic algorithm (GA) and particle swarm optimization (PSO), to enhance the adaptive neuro-fuzzy inference system (ANFIS) for accurately determining the friction capacity ratio (α) of the follower shaft in piles, thereby achieving precise calculations of the friction capacity. Ren et al. [
16] employed an adaptive genetic algorithm (AGA) and adaptive particle swarm optimization (APSO) strategies to optimize a backpropagation (BP) neural network for evaluating the ultimate bearing capacity of piles. Harandizadeh et al. [
17] introduced two novel adaptive neuro-fuzzy inference system (ANFIS) techniques to assess the ultimate bearing capacity of piles based on the widely utilized cone penetration test (CPT) in pile foundation analysis.
Machine learning, a fundamental aspect of artificial intelligence, is experiencing rapid advancements [
18,
19,
20,
21,
22,
23]. The ongoing technological progress is yielding innovative and promising models that offer viable quantitative predictive frameworks for assessing pile bearing capacity. The extreme learning machine (ELM) represents a specific type of feedforward neural network characterized by a single hidden layer. In contrast to traditional neural networks, which require extensive parameter tuning prior to training and often converge to local optima, the ELM necessitates only the specification of the number of nodes in the hidden layer. The weights of the input layer and the biases of the hidden layer do not require configuration, thereby increasing the likelihood of achieving a global optimum. Consequently, the ELM is recognized for its rapid learning capabilities and superior generalization performance. When a kernel function is incorporated into an ELM, it results in the kernel extreme learning machine (KELM). Unlike random mapping, kernel mapping employs a kernel function to configure the hidden layer, eliminating the need to determine the number of nodes in that layer. By judiciously selecting kernel parameters and the regularization coefficient, the KELM can directly compute output weights, thereby expediting convergence and significantly enhancing learning and generalization efficiency [
24,
25]. The KELM has been extensively referenced in the engineering literature, with applications that include predicting multifactor settlement around excavation sites [
26], landslide displacement [
27], and compressive strength [
28]. Traditional optimization algorithms, such as particle swarm optimization (PSO) and genetic algorithms (GAs), frequently encounter challenges in achieving optimal parameter configurations due to their propensity to become ensnared in local optima. Additionally, the penalty factor C and kernel coefficient σ of the kernel extreme learning machine (KELM) are typically determined through empirical methods, which constrains the model’s generalization capabilities. The current literature reveals a deficiency in effective methodologies that integrate the improved dung beetle optimization (IDBO) algorithm with the KELM to address these issues. This study aims to enhance the efficiency of hyperparameter optimization for KELMs by leveraging the global search capabilities of the IDBO algorithm, thereby addressing this identified research gap. The composite advantages of this approach are particularly evident in several aspects. Firstly, in comparison with traditional algorithms such as PSO and GAs, IDBO markedly improves convergence rates and mitigates the risk of local optima entrapment through the implementation of Fuch chaotic initialization and adaptive step size strategies. Secondly, regarding generalization capabilities, the kernel mapping structure of KELMs demonstrates superior nonlinear fitting characteristics compared with extreme learning machines (ELMs), artificial neural networks (ANNs), and adaptive neuro-fuzzy inference systems (ANFISs). Furthermore, the hyperparameters optimized by IDBO facilitate the model’s robust adaptability to various pile types and soil conditions, even when operating with limited datasets.
This paper introduces a predictive model for pile bearing capacity that employs a multi-strategy enhanced dung beetle optimization algorithm to optimize KELMs. Initially, experimental data related to pile bearing capacity are collected from the existing literature and normalized to ensure optimal integration into the model during the training phase. Subsequently, a multi-strategy improved dung beetle optimization algorithm is proposed, and the efficacy of the improved dung beetle optimization (IDBO) algorithm is validated across 23 benchmark functions. Furthermore, the experimental results are subjected to the Wilcoxon rank sum test, demonstrating the superiority of the IDBO algorithm over other conventional metaheuristic algorithms. Finally, the hyperparameters of the KELM model are optimized using the IDBO algorithm, and the resulting pile bearing capacity predictions based on the IDBO-KELM model are presented, showcasing high accuracy and practical applicability.
5. Results and Discussion
The iterative dung beetle optimizer (IDBO) technique was employed to examine the suitable control parameters for the KELM. The finalized control parameters for each method are presented in
Table 6.
The efficacy of the generated kernel extreme learning machine (KELM) hybrid model was assessed through the application of both statistical and graphical error criteria, with a focus on the statistical metrics that have been established:
1. The mean absolute percentage error
2. The root mean square error
In the formula, the subscripts exp and pred point out the bearing capacity of the measured and predicted values of the load bearing capacity, respectively, and is the the average of the values, while n denotes the number of points. The most accurate model has the highest values, while the RMSE and MAPE have the lowest values.
Figure 4 presents a comparative analysis of the predictive performance between the KELM and IDBO-KELM models, highlighting the correlation between the predicted and actual values. The findings indicate that the IDBO-KELM model exhibited markedly enhanced predictive capabilities relative to the conventional KELM model. In subplot (a) representing the KELM, the blue predicted curve shows considerable divergence from the red actual curve at several data points, with the most pronounced deviation occurring near the 12th group. Conversely, in subplot (b) for the IDBO-KELM model, the blue curve maintains a close alignment with the red curve throughout the dataset, exhibiting only minor discrepancies of approximately 5 kPa at select points. It is particularly noteworthy that within the low-value range of 10–20 kPa, the KELM model demonstrated a consistent tendency to overestimate values, whereas the IDBO-KELM model accurately reflected the fluctuating trends of the true values. The differential responses of the two models to inflection points were especially pronounced; the KELM displayed delayed reactions to abrupt changes in the 6th and 14th groups, resulting in “smoothed” distortions in the prediction curve. In contrast, the IDBO-KELM model effectively tracked the sharp increases and decreases in the actual values in real time. These observations substantiate the assertion that the IDBO optimization algorithm significantly reduces prediction errors by an average of over 60% through the enhancement of kernel function parameters, particularly demonstrating superior adaptability in the context of nonlinear sudden change data.
Figure 5 presents a comparative analysis of the predictive performance of the KELM and IDBO-KELM models, revealing notable disparities in their forecasting capabilities. The overall distribution indicates that the scatter points in the IDBO-KELM subplot are predominantly aligned along the purple diagonal line, particularly within the low-to-medium value range of 0–120, where the average distance between the scatter points and the diagonal line remains below 5 units. Conversely, the scatter points in the KELM subplot exhibit pronounced dispersion, with several groups of outliers deviating by 10–20 units from the diagonal line within the intervals of 40–80 and 120–160. The KELM model tended to overestimate values in the low-value range (less than 60) and systematically underestimate values in the high-value range (greater than 140), resulting in a “two-end divergent” bell-shaped distribution. In contrast, the IDBO-KELM model demonstrated a symmetrical distribution across the entire value range (0–200), with the maximum deviation not exceeding 15 units. It is particularly noteworthy that within the critical interval of 80–120, the dispersion of the KELM data points was 2.7 times greater than that of the IDBO-KELM model. This comparison substantiates the assertion that the enhanced algorithm effectively mitigates the prediction instability characteristic of traditional models in transitional intervals by dynamically adjusting kernel parameters.
The results of the error analysis presented in
Table 7 illustrate the enhancements achieved by the IDBO-KELM model in comparison to the conventional KELM approach. Three critical metrics highlight these improvements: Firstly, regarding prediction accuracy, the IDBO-KELM model reduced the mean absolute percentage error (MAPE) from 0.1847 (18.47%) in the KELM model to 0.1071 (10.71%), representing a relative decrease of 42%, which signifies a nearly 50% reduction in prediction bias. Secondly, in terms of error stability, the root mean square error (RMSE) notably decreased from 6.7357 to 4.7875, reflecting a reduction of approximately 29%, thereby indicating that the enhanced algorithm effectively mitigated the occurrence of extreme prediction errors. Most compellingly, the coefficient of determination (R
2) improved from 0.8639 to 0.9313, nearing the ideal value of 1, which suggests an increase in the model’s capacity to account for data variability of 7.4 percentage points. These findings demonstrate that the IDBO optimization algorithm, through the refinement of the parameter selection mechanism within the kernel extreme learning machine, not only systematically diminishes the magnitude of prediction bias (MAPE) and dispersion (RMSE) but also significantly enhances the intrinsic alignment between the model and actual data (R
2).
The radar chart presented in
Figure 6 offers a clear visual comparison of the performance of the KELM and IDBO-KELM models across three fundamental metrics. The graphical representation indicates that the two models exhibited comparable performance in terms of root mean square error (RMSE). However, notable disparities were observed for other metrics. The IDBO-KELM model achieved an R
2 (coefficient of determination) value of 0.9313, which significantly surpassed that of the KELM model, suggesting that the enhanced model demonstrated a superior capacity for data interpretation. Furthermore, the polygon area corresponding to the IDBO-KELM model was generally larger than that of the KELM, particularly along the R
2 axis, which underscores its overall superior performance.
Figure 7 provides a visual representation of the enhancements attained through algorithm optimization by contrasting the prediction error distributions of the KELM and IDBO-KELM models. In subplot (a) depicting the KELM model, the red error points display a broad distribution range, with notable extreme negative errors at sample 1 and considerable fluctuations observed near samples 8 and 16. This pattern suggests a lack of stability in the prediction outcomes of the base model. Conversely, in subplot (b), representing the IDBO-KELM model, the error points are predominantly clustered within a restricted range from −10 to 5. These visual representations effectively illustrate that the IDBO optimization algorithm, through its enhanced mechanism for selecting kernel function parameters, mitigates the extremes of prediction errors and diminishes the amplitude of fluctuations. Consequently, this optimization significantly improves the stability and reliability of the prediction results.
6. Main Conclusions
The load-bearing capacity of piles serves as a fundamental indicator of their operational safety, significantly influencing the stability, safety, and economic viability of structures. This research focuses on the integration of relevant datasets pertaining to pile load-bearing performance, leveraging the notable advantages of the kernel extreme learning machine (KELM) in terms of generalization capabilities and learning efficiency. To enhance the KELM model’s structure, the improved dung beetle optimization (IDBO) algorithm was introduced, which facilitates superior convergence accuracy, an accelerated convergence rate, and enhanced robustness. Consequently, we innovatively developed a pile bearing performance prediction model utilizing the IDBO-KELM framework. To thoroughly assess the model’s performance, we employed a range of machine learning evaluation metrics for comprehensive analysis. The principal findings and conclusions of this study are summarized as follows:
1. In constructing the sample dataset for the prediction model, we compiled 65 datasets of pile bearing capacity influenced by various factors from the existing literature. The model’s input variables included the pile length, pile diameter, average effective vertical stress, and undrained shear strength, while the output variable was defined as the pile bearing capacity.
2. We introduced the multi-strategy improved dung beetle optimization (IDBO) algorithm and selected 23 benchmark functions for a comprehensive performance evaluation of the IDBO. The experimental results were rigorously analyzed using the Wilcoxon rank sum test. The findings indicate that the IDBO algorithm demonstrated superior performance compared with other prevalent meta-heuristic algorithms, thereby providing a theoretical foundation for its application in predicting pile load-carrying capacity.
3. This study established a novel pile bearing capacity prediction model based on the multi-strategy improved dung beetle (IDBO) algorithm-optimized kernel extreme learning machine (KELM). This model not only employs advanced machine learning techniques but also finely calibrates model parameters through the IDBO algorithm, significantly enhancing prediction accuracy. In our experiments, 70% of the sample data were utilized as the training set, while the remaining 30% served as the test set. A comparative analysis of the prediction results between the IDBO-KELM model and the original KELM model revealed an R2 value of 0.9313 and a mean absolute percentage error (MAPE) of 0.1071, confirming the high accuracy and feasibility of the proposed method for predicting pile bearing capacity. Future research could explore the application of this model to larger datasets, compare it with other optimization algorithms, and adapt it to different types of pile foundations and soil conditions to further enhance its practical value in engineering.