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Article

Enhanced Dung Beetle Optimizer-Optimized KELM for Pile Bearing Capacity Prediction

1
Heilongjiang Province Hydraulic Research Institute, Harbin 150050, China
2
Key Laboratory of Earthquake Engineering and Engineering Vibration of China Earthquake Administration, Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, China
3
College of Civil Engineering and Architecture, Harbin University of Science and Technology, Harbin 150080, China
*
Author to whom correspondence should be addressed.
Buildings 2025, 15(15), 2654; https://doi.org/10.3390/buildings15152654
Submission received: 28 June 2025 / Revised: 19 July 2025 / Accepted: 23 July 2025 / Published: 27 July 2025

Abstract

The safety associated with the bearing capacity of pile foundations is intrinsically linked to the overall safety, stability, and economic viability of structural systems. In response to the need for rapid and precise predictions of pile bearing capacity, this study introduces a kernel extreme learning machine (KELM) prediction model optimized through a multi-strategy improved beetle optimization algorithm (IDBO), referred to as the IDBO-KELM model. The model utilizes the pile length, pile diameter, average effective vertical stress, and undrained shear strength as input variables, with the bearing capacity serving as the output variable. Initially, experimental data on pile bearing capacity was gathered from the existing literature and subsequently normalized to facilitate effective integration into the model training process. A detailed introduction of the multi-strategy improved beetle optimization algorithm (IDBO) is provided, with its superior performance validated through 23 benchmark functions. Furthermore, the Wilcoxon rank sum test was employed to statistically assess the experimental outcomes, confirming the IDBO algorithm’s superiority over other prevalent metaheuristic algorithms. The IDBO algorithm was then utilized to optimize the hyperparameters of the KELM model for predicting pile bearing capacity. In conclusion, the statistical metrics for the IDBO-KELM model demonstrated a root mean square error (RMSE) of 4.7875, a coefficient of determination (R2) of 0.9313, and a mean absolute percentage error (MAPE) of 10.71%. In comparison, the baseline KELM model exhibited an RMSE of 6.7357, an R2 of 0.8639, and an MAPE of 18.47%. This represents an improvement exceeding 35%. These findings suggest that the IDBO-KELM model surpasses the KELM model across all evaluation metrics, thereby confirming its superior accuracy in predicting pile bearing capacity.

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.

2. Principles and Modeling

2.1. Principles of the KELM

The extreme learning machine (ELM) is characterized as a single hidden layer feedforward neural network that requires only the specification of the number of nodes within the hidden layer. This approach facilitates the attainment of a globally optimal solution without necessitating adjustments to the input weights and biases of the hidden layer [24,25]. As a result, the ELM exhibits considerable advantages in terms of both efficiency and generalization capabilities. When kernel functions are integrated with an ELM, the kernel extreme learning machine (KELM) is produced. In contrast to ELMs, the KELM benefits from the application of a kernel mapping strategy, which supersedes the original stochastic mapping method and obviates the requirement to determine a fixed number of neurons in the hidden layer. By judiciously selecting appropriate kernel parameters and a regularization coefficient, the output weights of the network can be derived, thereby enhancing the convergence speed, learning efficiency, and generalization performance of the extreme learning machine.
As shown in Figure 1, the simple structure model of a KELM consists of three layers—an input layer, hidden layer, and output layer—as shown in Formula (1):
f ( x ) = i = 1 L β i g i ( w i , b i , x i ) = i = 1 L β i h i ( x ) = h ( x ) β
where   w i represents the weight vector of the input layer and hidden layer; β i represents the weight vector of the output layer and hidden layer; γ is the weight of output layer; and b i represents the bias of the hidden layer. These represent a hidden node, two or three hidden neurons, and the connection among multiple neural networks, respectively. Meanwhile, x i represents the N-dimension data input; g(-) represents the initial function of the hidden layer; and f ( x ) represents the output function of the ELM.
In the design of the extreme learning machine, the node parameters   w i and b i of the hidden layer are assigned in a randomized manner, and thus we only need to determine the number of nodes in the hidden layer L. β i can be determined based on the input and output data, and the formula for this is
i = 1 L β i g i ( w i , b i , x i ) = t j
where t j is the output value of the training data.
Transforming Equation (2) into a matrix representation yields
H β = T
H = h ( x 1 ) h ( x N ) = g ( w 1 , b 1 , x 1 ) g ( w L , b L , x 1 ) g ( w 1 , b 1 , x N ) g ( w L , b L , x N ) N × L
β = β 1 T   β L T L × m and   T = t 1 T   t N T N × m  
where H is the hidden layer output network matrix; β is the hidden layer weight matrix; and T is the target output matrix, represented by
β = H + T
where H + is the H-generalized pseudo-inverse Jacobi matrix. H + is computed as follows:
H + = H T ( H H T ) - 1
In the KELM, in order to increase the generalization ability of the H H T , the kernel parameter I/C is added to it, and thus the output function of the KELM is
f ( x ) = h ( x ) β = h ( x ) H T ( I C + H H T ) 1 T
where C is the penalty factor and I is the unit matrix. In the KELM, the kernel function is defined by
Ω K E L M = H H T : Ω K E L M = h ( x i ) h ( x j ) = K ( x i , x j )
Thus, the model function of the KELM is
f ( x ) = K ( x , x 1 ) K ( x , x N ) T ( I C + Ω K E L M ) 1 T
This research explicitly employs the radial basis function (RBF) as the primary mapping mechanism within the kernel extreme learning machine (KELM) framework. The RBF facilitates the projection of nonlinear characteristics from the input space into a high-dimensional Hilbert space via a Gaussian transformation, rendering it more appropriate for the attributes of the pile–soil dataset in comparison with alternative polynomial kernel functions. The selection of the penalty coefficient C and the kernel parameter σ plays a crucial role in determining the efficacy of KELM. The coefficient C embodies a balance the minimization of the fitting error and the regulation of input weight standards, while the kernel parameter σ delineates the specificity and nature of the nonlinear mapping from the input space to the high-dimensional feature space. Consequently, a thorough examination of the control parameters of the KELM is essential for achieving optimal performance. The optimization of these parameters is facilitated by the global search mechanism of the multi-strategy improved dung beetle optimization (IDBO) algorithm, which utilizes the root mean square error of predictions from the training set as the fitness function and conducts targeted optimization within a predefined parameter space. Notably, IDBO incorporates innovative elements such as Fuch chaotic mapping initialization and adaptive step size adjustment, yielding Pareto optimal solutions after 20 iterations with a population size of 20. This methodology enhances the optimization process relative to traditional grid search techniques and mitigates the risk of local convergence.

2.2. Principles of the Dung Beetle Optimization Algorithm

The dung beetle optimizer (DBO), a novel strategy within the realm of swarm intelligence optimization, was introduced in 2022 [29]. This algorithm draws inspiration from the distinctive behavioral patterns of dung beetles, including ball rolling, dance communication, foraging techniques, strategic theft, and reproductive practices. The primary objective of DBO is to emulate these intricate and effective survival strategies to address optimization challenges. Typically, in the wild, dung beetles are known for two things: (1) being able to roll dung into a ball and (2) using the stars to ensure their path is in a straight line. Dung beetles have a trajectory while traveling which will only become evident if there is no light source. There are also a number of other environmental factors that cause deviations in a dung beetle’s path. Interestingly, dung beetles are able to move in a desired direction despite being off course through a behavior known as dancing. Dung balls are the beetle’s breeding substrate. In the algorithm, there is also a component of competition as some dung beetles exhibit a behavior known as theft, where they steal dung from other beetles. In the DBO algorithm, the position of each individual dung beetle is considered to be a potential solution. The dung beetle’s foraging behavior can be categorized into five actions: rolling, which involves forming the dung into a ball and using the stars to ensure a straight path; dancing, which allows the dung beetle to reorientate itself; foraging, which is when adult dung beetles chew up the ground in search of food; stealing, as some dung beetles which exhibit this behavior, known as thieves, steal the balls of dung from other insects; and reproducing, where male and female dung beetles transport dung balls to a suitable location where they will lay their eggs. As a result, the dung beetle population can be divided into four groups within the algorithm: rolling dung beetles, brood balls, juvenile dung beetles, and thieving dung beetles. Rolling dung beetles adapt their trajectories based on various environmental influences while initially seeking safe foraging locations. Brood balls are deposited in areas deemed secure, while juvenile dung beetles, having matured from larvae, forage within optimal areas. Stealing dung beetles, however, find food by congregating at the locations of other dung beetles and identifying optimal areas to feed. The DBO algorithm is featured for its reliable global searching ability and strong stability. The mathematical equation describing the positional changes of a dung beetle that rolls a dung ball is shown below:
x i ( t ) + 1 = x i ( t ) + α × k × x i ( t 1 ) + b × Δ x
Δ x = x i ( t ) X w
where t represents the current iteration number; x i denotes the position information of the ith dung beetle; k represents the deflection coefficient; α ∈ (0, 1) is a random number; and a is a natural coefficient with a value of −1 or 1. The value of parameter a (−1 or 1) is determined by simulating the random turning behavior of dung beetles when encountering obstacles to achieve environmental adaptability. This design draws on the common paradigm of environment-responsive parameters in swarm intelligence algorithms, where X w is the global worst position.
When an obstacle is encountered, the dung beetle dances to reorient itself, with a position update formula of
x i ( t + 1 ) = x i ( t ) + tan ( θ ) x i ( t ) x i ( t 1 )
where tan(θ) is the deflection system angle.
To model the region where female dung beetles lay their eggs, a boundary selection strategy was used, defined by the equation
L b * = max ( X * × ( 1 R ) , L b )
U b * = min ( X * × ( 1 + R ) , U b )
where X * is the current optimal position; Lb denotes the lower bound; and Ub denotes the upper bound.
During the iteration process, the position of the ovoid is dynamically changing, defined by the equation
B i ( t + 1 ) = X * + b 1 × ( B i ( t ) L b * ) + b 2 × ( B i ( t ) U b * )
where B i denotes the position of the ovoid and b 1 and b 2 are 1 × D random vectors.
In addition, the optimal foraging area needs to be established to guide the little dung beetle to forage and then simulate its foraging behavior, where the optimal foraging area is defined by the equation
L b b = max ( X b × ( 1 R ) , L b )
U b b = min ( X b × ( 1 + R ) , U b )
where X b is the global optimal position. As a result, the position of the small dung beetle is updated as follows:
x i ( t + 1 ) = x i ( t ) + C 1 × ( x i ( t ) L b b ) + C 2 × ( x i ( t ) U b b )
where C 1 is a random number obeying a normal distribution and C 2 ∈ (0, 1) is a random vector.
In addition, it is assumed that X b is the best place to compete for food, and thus the location update for dung beetles with stealing behavior is described as follows:
x i ( t + 1 ) = X b + S × g × ( x i ( t ) X * + x i ( t ) X b )
where g is a random vector obeying a normal distribution with a mean of 0 and variance of 1 and S is a constant.

2.3. Improvements in DBO

Xue and Shen [29] carefully selected a wide-coverage test function set, which encompassed 23 classical benchmark test functions and 29 complex test functions derived from CEC-2017. After a series of rigorous experimental validations, the results revealed that the DBO algorithm exhibited obvious competitive advantages over the current widely used mainstream optimization algorithms in terms of core performance metrics such as the convergence rate, solution accuracy, and algorithmic operation robustness. However, if we apply the DBO algorithm to solve complex optimization problems, how to verify that the above results are achieved is still a big problem. Generally speaking, the disadvantages of the DBO algorithm are as follows. First, the initial position of the population in the search process of the current DBO is randomly selected, and thus the initial population is unevenly distributed in the search space. Second, the current DBO has less flexibility in the population position update, which makes it easy to adjust the search range according to the current iteration stage before maturity. Finally, when dealing with some complex optimization problems, such as high dimensionality, multiple constraints, or dynamic changes, the performance of the existing DBO is not satisfactory, and the convergence speed is relatively slow. With the continuous evolution and expansion of practical application scenarios, more stringent requirements on the convergence accuracy, stability, and applicability of optimization algorithms have been put forward, and their performance needs to be improved to meet a wider range of application requirements. In the R&D of optimization algorithms, in addition to the preliminary verification of the effectiveness of the proposed new method, we also need to conduct a deep evaluation of the new method. This can not only verify the effectiveness of the algorithm but also provide a good theoretical basis for the later improvement and extension. Based on the work of Li et al. [30], this paper further verifies the reliability of IDBO model. Within the basic dung beetle optimization algorithm, the global search ability, rapidity, and stability of the dung beetle optimization algorithm are enhanced by the following three improvement strategies: the strategy of optimizing population initialization based on Fuch chaos and the inverse learning strategy, the strategy of optimizing population initialization based on the adaptive step size and convex lens inverse imaging strategy, and the strategy of enhancing population diversity based on stochastic differential variation strategy, respectively. The important effect of the abovementioned strategies on the algorithm is emphasized, the basis for subsequent combined applications with the advanced model of the extreme learning machine (KELM) is also prepared, and the broad prospect is reserved.

3. IDBO Algorithm Performance Test

3.1. Test Functions and Parameter Settings

This section shows the analysis and validation of improvements of the IDBO algorithm via employing 23 classical test functions from the CEC2005 function set. To understand the effectiveness of the improvements, the IDBO algorithm was compared with other heuristic algorithms: DBO [29], POA [31], GWO [32], PSO [33], and RIME [34]. This research identified six meta-heuristic algorithms—IDBO, DBO, POA, GWO, PSO, and RIME—as benchmarks for comparative analysis. The rationale for this selection was grounded in three stringent criteria. Firstly, regarding the diversity of algorithm types, DBO exemplifies bio-inspired optimizers, while POA and GWO illustrate collaborative group mechanisms (specifically, pelican predation and wolf pack hunting). Additionally, PSO is recognized as a classic benchmark in swarm intelligence, and RIME is categorized under physically inspired algorithms. This strategic selection encompassed the three predominant branches of meta-heuristic algorithms. For a fair comparison, the population sizes of these algorithms were set to 30 and 500 iterations for the population in total. To reduce the impact of randomness, each simulation experiment was implemented independently 30 times. Finally, the optimal value, standard deviation, mean, worst value, and median of each algorithm were calculated. These values were expanded from 5 to 23, namely 5 × 23 in total, with the results of the rank sum test.

3.2. Optimization Capabilities of IDBO

Table 1 presents detailed information regarding the simulation outcomes, including the optimal value, standard deviation, mean, worst value, and median. These metrics were utilized to assess the efficacy of the algorithm in conducting the search.
The single-peak test functions set (F1~F7) had only one optimal solution characteristic, and it could be used to verify the effectiveness of the development ability of an algorithm. From the data in Table 1, it can be found that the proposed IDBO algorithm had an advantage over other algorithms compared with six of the test function sets (F1~F6). For the F7 function, the IDBO algorithm was still in the lead, and its fitness value was not the best. Subsequently, we used the group of multimodal functions (F8~F23) to verify the exploration effectiveness of the algorithm. Functions F8~F13 had a large number of local minima, and the number of local minima was exponential with the dimension count. For functions F9~F23, the search intensity of the IDBO algorithm was stronger than that of other algorithms. This shows that the optimization results of IDBO algorithm were highly competitive.

3.3. Comparative Analysis of Algorithm Convergence Curves

To illustrate the convergence performance of different algorithms applied to the test functions, we meticulously plotted the convergence trends derived from the collected data. In particular, Figure 2 provides a detailed representation of the adaptation curves for the algorithms, including IDBO, during the optimization process of specific benchmark functions. A thorough examination revealed that the IDBO algorithm demonstrated considerable superiority over its counterparts regarding both the optimization rate and convergence accuracy.

3.4. Statistical Analysis: Wilcoxon Rank Sum Test

The Wilcoxon rank sum test fulfilled two primary functions. Firstly, it assessed the performance disparity between the IDBO algorithm and various meta-heuristic algorithms across 23 benchmark functions. Secondly, it acted as a preliminary verification tool to confirm the global optimization capabilities of the IDBO algorithm, thereby establishing a basis for its application in engineering contexts. In the comparative analysis of the IDBO algorithm against other optimization heuristics, we employed an applied version of the Wilcoxon rank sum test for evaluation purposes [35]. This statistical test primarily focused on assessing the significance of the results, specifically the p value. A p value below 0.05 indicated a statistically significant difference between the two algorithms under consideration. The pertinent values are presented in Table 2. A detailed examination of Table 2 indicates that the optimization outcomes produced by the IDBO algorithm exhibited substantial variability when compared with their counterparts. This finding suggests that the IDBO algorithm possesses a distinctly different performance profile in relation to other metaheuristic algorithms when applied to optimization of the 23 benchmark functions.

4. IDBO-KELM Pile Bearing Capacity Prediction Modeling Study

4.1. Data Set Construction

An adequate dataset is crucial for effective modeling. In this research, a total of 65 experimental datasets sourced from the pertinent literature [36] were utilized to examine four input parameters: the pile length, pile diameter, average effective vertical stress, and undrained shear strength. The data utilized in this study were sourced from load test records of driven piles in cohesive soils, as compiled by Vijayvergiya and Focht (1972) [37], Flaate and Seines (1977) [38], and Semple and Rigden (1986) [39]. The dataset provided by Flaate and Seines (1977) [38] is predominantly focused on timber piles, whereas the other datasets concentrate on steel pipe piles. To mitigate the potential bias introduced by repetitive results from a single site (as noted by Semple and Rigden (1986) [39]), certain data points represent the average values derived from multiple test piles located at the same site. The embedded lengths of the piles ranged from 4.7 m to 96 m, with undrained shear strengths varying between 9 kPa and 335 kPa and friction coefficients spanning from 0.42 to 1.73. Most of these values were obtained from compressive load tests conducted on the piles after adjusting for end-bearing resistance, with the unit end-bearing resistance at the pile tip assumed to be eight times the undrained shear strength. The undrained shear strength was primarily assessed through unconfined compression tests; however, for extremely soft and soft cohesive soils, it was predominantly determined using vane shear tests. A random selection of the gathered data is presented in Table 3. Furthermore, the relationship between the input and output variables is illustrated in Table 4. Table 5 presents the distribution characteristics and dispersion of the five variables under consideration. Notably, variable x 3 exhibited the most extensive dynamic range, with its standard deviation and variance significantly surpassing those of the other variables, thereby indicating considerable fluctuations within this parameter. Conversely, variable x 2 demonstrated the lowest degree of dispersion. In terms of distribution shape, all variables displayed right skewness (skewness > 1) and positive kurtosis (kurtosis > 0), with x 3 exhibiting the most pronounced levels of both skewness and kurtosis. This suggests the presence of numerous low-value clusters alongside high-value outliers within the dataset. It is important to highlight that, despite the large range of the y variable, the difference between its median and mean was less pronounced those that observed for variables x3 and x 4 , indicating a relatively balanced distribution. Variables x 1 and x 4 revealed similar statistical patterns, with their means being approximately 1.4 times their medians. This observation, in conjunction with their high skewness values, further corroborates the right-skewed nature of the data.

4.2. Model Development

The IDBO algorithm was employed to optimize the hyperparameters of the kernel extreme learning machine (KELM). Prior to the optimization process, it was essential to calibrate the hyperparameters of the optimization algorithm itself. Preliminary calculations indicated that an excessively large population size significantly extends the execution time. Conversely, a population size that is too small may hinder the model’s ability to achieve optimal fitting, resulting in convergence to a local optimum. Consequently, this study emphasizes the significance of determining an appropriate population size. Based on experimental trials, a population size of 20 was established, with the number of iterations set at 20. The IDBO-KELM integrated model requires IDBO to optimize the KELM’s hyperparameters (C, σ). To meet real-time prediction requirements in engineering applications, we ultimately selected N = 20 and T = 20 to ensure reduced training times while keeping the accuracy loss under control.
Before the optimization process, it was necessary to normalize the initial data collected. The formula for this is as follows:
X Norm = 2 ( X N O R X M I N ) X M A X X M I N 1
Here, X MAX and X MIN are the maximum and minimum values of the variable X I , and X Norm is its normalized value.
Seventy percent of the dataset was allocated for training purposes, while thirty percent was designated for testing. The model was configured with a population size of 20 and subjected to 20 iterations. Figure 3 depicts the closed-loop optimization process employed by the IDBO-KELM model. Initially, 65 groups of original samples were partitioned into a training set (70%) and a test set (30%) through a random data partitioning method, utilizing a stratified approach to mitigate sample bias. The training set was subsequently introduced into the IDBO optimization framework, wherein the IDBO algorithm dynamically modifies the control parameters of the kernel extreme learning machine (KELM)—specifically, the penalty factor C and the kernel coefficient σ—based on the prediction error observed in the training set. During the training phase, this process generates candidate KELM models. Following this, the model evaluation module assesses the efficacy of the parameters by employing the root mean square error (RMSE) as the fitness function. Should the termination criteria not be satisfied, feedback is relayed back to the IDBO for additional optimization, ultimately yielding the optimal control parameters. This closed-loop design facilitates automatic hyperparameter optimization, demonstrating greater efficiency compared with traditional manual parameter tuning methods.

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
MAPE = 1 n i = 1 n η i exp η i p r e d η i exp
2. The root mean square error
RMSE = 1 n i = 1 n ( η i exp η i p r e d ) 2
3. The goodness of fit
R 2 = 1 i = 1 n ( η i exp η i p r e d ) 2 i = 1 n ( η i p r e d η ¯ ) 2
In the formula, the subscripts exp and pred point out the bearing capacity η of the measured and predicted values of the load bearing capacity, respectively, and η ¯ is the η the average of the values, while n denotes the number of points. The most accurate model has the highest R 2 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 (R2) 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 (R2).
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 R2 (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 R2 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.

Author Contributions

Conceptualization, B.C.; methodology, B.C.; software, M.H.; validation, G.D.; formal analysis, G.D.; investigation, B.C.; resources, M.W.; data curation, Q.Z., B.Z.; writing—original draft preparation, Q.Z. and M.H.; writing—review and editing, B.C., Q.Z., M.H., B.Z., and Y.G.; visualization, M.W., B.C., and Y.G.; supervision, M.W., and G.D.; project administration, G.D.; funding acquisition, M.W. All authors have read and agreed to the published version of the manuscript.

Funding

The authors are grateful for financial support from the Heilongjiang Provincial Natural Science Foundation of China (PL2024E032) and Heilongjiang Provincial Research Institutes Scientific Research Business Fund Project (CZKYF2025-1-B012).

Data Availability Statement

The datasets utilized and/or examined in this study can be accessed upon reasonable request directed to the corresponding author.

Acknowledgments

Generative AI and AI-assisted technologies have been utilized exclusively during the writing phase to enhance the manuscript’s readability and linguistic quality. This technology has been employed under human oversight and management, with the authors meticulously reviewing and refining the outcomes prior to the finalization of the paper.

Conflicts of Interest

The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper.

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Figure 1. Extreme learning machine network structure model.
Figure 1. Extreme learning machine network structure model.
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Figure 2. Algorithm convergence graphs.
Figure 2. Algorithm convergence graphs.
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Figure 3. Algorithm flow chart.
Figure 3. Algorithm flow chart.
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Figure 4. Comparison of the results of different optimization algorithms for predicting models. (a) KELM. (b) IDBO-KELM.
Figure 4. Comparison of the results of different optimization algorithms for predicting models. (a) KELM. (b) IDBO-KELM.
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Figure 5. Sample fitting prediction plots for different optimization algorithms. (a) KELM. (b) IDBO-KELM.
Figure 5. Sample fitting prediction plots for different optimization algorithms. (a) KELM. (b) IDBO-KELM.
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Figure 6. Radar chart of the model.
Figure 6. Radar chart of the model.
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Figure 7. Error analysis of the prediction model with different optimization algorithms. (a) KELM. (b) IDBO-KELM.
Figure 7. Error analysis of the prediction model with different optimization algorithms. (a) KELM. (b) IDBO-KELM.
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Table 1. Simulated results.
Table 1. Simulated results.
IDBODBO POA RIME GWO PSO
F1min07.7402 × 10−1661.6357 × 10−1210.0025175151.01274 × 10−609.75514 × 10−12
F1std01.8291 × 10−1132.6085 × 10−1050.0133254165.98583 × 10−571.69202 × 10−9
F1avg1.0204 × 10−1943.3506 × 10−1146.9821 × 10−1060.0183675811.81951 × 10−577.40012 × 10−10
F1median2.1389 × 10−2548.421 × 10−1363.8085 × 10−1120.0161439712.13492 × 10−582.93984 × 10−10
F1worse2.6725 × 10−1931.002 × 10−1121.1383 × 10−1040.0589799663.23726 × 10−569.27302 × 10−9
F2min3.4163 × 10−1647.37901 × 10−922.31882 × 10−620.0090833381.44959 × 10−351.00343 × 10−7
F2std2.8254 × 10−1031.15948 × 10−526.0188 × 10−520.0122369698.14772 × 10−338.46558 × 10−7
F2avg7.6618 × 10−1042.11708 × 10−531.22332 × 10−520.0265243184.71025 × 10−331.2568 × 10−6
F2median7.6573 × 10−1201.23514 × 10−698.42827 × 10−570.0247199151.46684 × 10−331.07595 × 10−6
F2worse1.3591 × 10−1026.35077 × 10−523.29976 × 10−510.0541215933.90108 × 10−323.90585 × 10−6
F3min01.1225 × 10−1462.5611 × 10−1190.0600786911.67818 × 10−306.81522 × 10−5
F3std6.0785 × 10−1451.21482 × 10−793.8023 × 10−1030.3393860597.0108 × 10−250.007888608
F3avg1.1112 × 10−1452.21795 × 10−801.0207 × 10−1030.4610755842.012 × 10−250.003808562
F3median5.7641 × 10−1622.277 × 10−1173.0017 × 10−1100.4110402547.42946 × 10−270.001475392
F3worse3.3294 × 10−1446.65383 × 10−791.843 × 10−1021.7453467673.51434 × 10−240.043362975
F4min9.8699 × 10−1719.3165 × 10−804.9286 × 10−590.072987634.51028 × 10−200.000141489
F4std9.99649 × 10−985.55602 × 10−543.12882 × 10−520.0700937924.56507 × 10−180.001336683
F4avg1.8251 × 10−981.60099 × 10−549.43095 × 10−530.1743783733.30351 × 10−180.001370955
F4median1.9058 × 10−1541.55998 × 10−672.73189 × 10−550.1675242391.84796 × 10−180.000854689
F4worse5.4753 × 10−972.76051 × 10−531.41393 × 10−510.3128590912.317 × 10−170.006079261
F5min1.53611 × 10−94.81873224.9456610282.9152646365.7570866940.02432321
F5std2.7260653652.3833462140.744185459293.49237290.646475964550.1053812
F5avg2.3572681965.7182724216.472856972139.25589926.725974362109.8718531
F5median0.0030010015.2831048886.27974288456.552784616.4511280435.437807881
F5worse6.03916768118.298532738.9189605881368.4336888.0618612243020.58111
F6min1.3158 × 10−301.72828 × 10−281.10258 × 10−60.0041626741.23082 × 10−66.71796 × 10−13
F6std2.74982 × 10−256.37087 × 10−240.1544128770.0129252540.0458120073.67609 × 10−9
F6avg1.03255 × 10−253.27882 × 10−240.1370501050.0175750810.0083672191.17139 × 10−9
F6median3.20587 × 10−282.39027 × 10−250.0622721260.0137558683.27678 × 10−62.0938 × 10−10
F6worse1.28881 × 10−242.84577 × 10−230.4942086740.054680920.2509258252.00491 × 10−8
F7min9.98066 × 10−63.88727 × 10−58.96934 × 10−60.0008606980.0001356250.000968954
F7std0.0001384510.0006670240.0001558680.0022920640.0002978670.001778493
F7avg0.0001940370.001087740.0002001660.0041639350.0005464740.003343354
F7median0.0001834560.0009861880.0001664250.0037591050.0004779270.003531613
F7worse0.0004528760.0024285910.0006927890.0096464430.0012310120.007519862
F8min−4189.828873−4188.776384−3617.014934−4189.281786−3321.179926−3735.812534
F8std352.7495929564.6124663277.9049948170.3843889273.3378933345.4729817
F8avg−3952.076419−3414.120055−3119.029063−3856.769249−2673.803526−3026.232871
F8median−4114.777704−3476.923276−3123.039803−3834.431619−2668.165606−3005.425127
F8worse−2900.675954−2285.68026−2616.687036−3419.864506−2139.260331−2176.339094
F9min0000.00424171300.994959408
F9std06.83816322801.2423439171.7767228133.671909933
F9avg01.92333244502.7625410340.8672828537.037604591
F9median0002.98879499605.969879763
F9worse028.8537420404.9797589196.56880774514.92438682
F10min4.44089 × 10−164.44089 × 10−164.44089 × 10−160.0205647343.9968 × 10−151.33671 × 10−6
F10std001.59793 × 10−150.0258931021.59793 × 10−156.96486 × 10−6
F10avg4.44089 × 10−164.44089 × 10−163.04941 × 10−150.0617471117.31267 × 10−158.61325 × 10−6
F10median4.44089 × 10−164.44089 × 10−163.9968 × 10−150.0550022427.54952 × 10−156.23604 × 10−6
F10worse4.44089 × 10−164.44089 × 10−163.9968 × 10−150.1477964151.11022 × 10−142.97704 × 10−5
F11min0000.09217155700.025442153
F11std00.05168773400.0521580230.0200701920.038111942
F11avg00.01919057400.1718332010.0203039640.081426019
F11median0000.162402340.0153124910.072546121
F11worse00.21882063100.3568802530.0731050480.201710019
F12min6.84141 × 10−329.53479 × 10−318.79284 × 10−73.78395 × 10−54.57053 × 10−77.00706 × 10−14
F12std1.89412 × 10−241.61921 × 10−100.0163012230.0001479340.0089012434.50907 × 10−9
F12avg4.57389 × 10−252.95627 × 10−110.014228410.0002010980.0052783798.36894 × 10−10
F12median7.90289 × 10−293.86308 × 10−260.0103406780.000139661.79256 × 10−65.36215 × 10−12
F12worse9.94343 × 10−248.86879 × 10−100.0524361210.0006137190.0203432012.47106 × 10−8
F13min1.87744 × 10−305.90867 × 10−292.49244 × 10−60.0002838821.61055 × 10−62.15877 × 10−12
F13std0.0020070740.0570084010.2116918760.0038376440.0241967392.13065 × 10−10
F13avg0.000396250.0347685580.319539170.0027725330.0063476741.02836 × 10−10
F13median1.33978 × 10−280.0074191750.3015422090.0012836856.87475 × 10−62.95316 × 10−11
F13worse0.0109873660.1977397540.9889443930.0148822280.1019781631.06464 × 10−9
F14min0.9980038380.9980038380.9980038380.9980038380.9980038380.998003838
F14std2.47396 × 10−162.08040609207.18258 × 10−124.0547457760.181483682
F14avg0.9980038381.8842992910.9980038380.9980038384.2965531181.031138073
F14median0.9980038380.9980038380.9980038380.9980038382.9821051570.998003838
F14worse0.99800383810.763180670.9980038380.99800383810.763180671.9920309
F15min0.0003074860.0003074860.0003074860.000447320.0003075420.000307496
F15std9.30471 × 10−50.0003421322.13269 × 10−50.0114717250.0093048250.000449855
F15avg0.0003659390.0007196950.0003113810.0046396210.006391170.00066267
F15median0.0003076620.0006845160.0003074860.0007236340.0003763170.000441501
F15worse0.0006913280.0014889640.0004242990.0565429550.0203633480.001655371
F16min−1.031628453−1.031628453−1.031628453−1.031628453−1.031628452−1.031628453
F16std5.97518 × 10−166.25324 × 10−165.29643 × 10−161.88292 × 10−72.92289 × 10−86.25324 × 10−16
F16avg−1.031628453−1.031628453−1.031628453−1.031628343−1.031628427−1.031628453
F16median−1.031628453−1.031628453−1.031628453−1.031628401−1.031628442−1.031628453
F16worse−1.031628453−1.031628453−1.031628453−1.031627432−1.031628356−1.031628453
F17min0.3978873580.3978873580.3978873580.397887360.3978873810.397887358
F17std03.24339 × 10−1608.84171 × 10−71.55815 × 10−60
F17avg0.3978873580.3978873580.3978873580.3978877560.397888750.397887358
F17median0.3978873580.3978873580.3978873580.3978874440.3978881410.397887358
F17worse0.3978873580.3978873580.3978873580.3978916850.3978932640.397887358
F18min3333.0000000083.0000001623
F18std2.76966 × 10−153.26858 × 10−151.78209 × 10−154.92585 × 10−73.43745 × 10−51.52505 × 10−15
F18avg3333.0000003563.000040613
F18median3333.000000223.0000334563
F18worse3333.000001783.0001307723
F19min−3.862782148−3.862782148−3.862782148−3.862782146−3.862780704−3.862782148
F19std0.0032065070.002404882.29277 × 10−153.01709 × 10−70.0030419152.64018 × 10−15
F19avg−3.861205842−3.861993995−3.862782148−3.862781879−3.861040787−3.862782148
F19median−3.862782148−3.862782148−3.862782148−3.862781981−3.862683331−3.862782148
F19worse−3.854900617−3.854900617−3.862782148−3.862781041−3.854896351−3.862782148
F20min−3.321995172−3.321995172−3.321994373−3.321993687−3.32199221−3.321995172
F20std0.0996897410.0915475391.76164 × 10−50.059237510.0867311050.059922957
F20avg−3.181047102−3.245894752−3.321985281−3.274420338−3.221130731−3.270474819
F20median−3.19590753−3.321995172−3.321991639−3.321959018−3.202378411−3.321995172
F20worse−2.840421628−3.020699112−3.321905611−3.203054229−3.077987167−3.20310205
F21min−10.15319968−10.15319968−10.15319968−10.15305446−10.15296969−10.15319968
F21std2.2929611942.4316989361.9323658222.7930462111.7509238443.530259991
F21avg−8.793732492−6.77003396−9.303472845−7.28572641−9.476139679−6.98267156
F21median−10.15319968−5.10077214−10.15312957−5.100754598−10.15102846−10.15319968
F21worse−5.055197729−5.055115367−5.05519466−2.630186031−5.055156781−2.630471668
F22min−10.40294057−10.40294057−10.40294056−10.40288768−10.40245571−10.40294057
F22std1.8377323942.6217522361.3485104973.2652481130.9701358353.50499167
F22avg−9.694238068−8.734350373−10.04852806−7.834630159−10.22417325−6.862983886
F22median−10.40294057−10.40260431−10.40289228−10.40178501−10.40155302−6.428473438
F22worse−5.087671825−2.765897328−5.087670031−2.765707539−5.087644566−2.751933564
F23min−10.53640982−10.53640982−10.53640982−10.53638654−10.53608749−10.53640982
F23std1.71401 × 10−153.0510577249.46753 × 10−51.7919421170.9870329893.796638526
F23avg−10.53640982−8.144709587−10.53633186−9.951026545−10.35442983−7.317832337
F23median−10.53640982−10.53531376−10.53636144−10.53548909−10.53480961−10.53640982
F23worse−10.53640982−2.4273352−10.53603507−3.835307307−5.128438525−2.421734027
Table 2. Wilcoxon rank sum test.
Table 2. Wilcoxon rank sum test.
DBOPOARIMEGWOPSO
F12.8646 × 10−112.8646 × 10−112.8646 × 10−112.8646 × 10−112.8646 × 10−11
F23.01986 × 10−113.01986 × 10−113.01986 × 10−113.01986 × 10−113.01986 × 10−11
F33.68973 × 10−113.01986 × 10−113.01986 × 10−113.01986 × 10−113.01986 × 10−11
F43.01986 × 10−113.01986 × 10−113.01986 × 10−113.01986 × 10−113.01986 × 10−11
F50.0022657811.77691 × 10−105.07231 × 10−104.07716 × 10−110.000110577
F62.67842 × 10−63.01986 × 10−113.01986 × 10−113.01986 × 10−113.01986 × 10−11
F72.19474 × 10−80.9823070533.01986 × 10−113.80526 × 10−73.01986 × 10−11
F81.70786 × 10−54.38323 × 10−90.0038059276.35515 × 10−111.62079 × 10−9
F90.08152297211.21178 × 10−120.0006614021.21178 × 10−12
F1015.35908 × 10−91.21178 × 10−121.58364 × 10−131.21178 × 10−12
F110.02157719211.21178 × 10−126.24703 × 10−101.21178 × 10−12
F120.0002388483.01986 × 10−113.01986 × 10−113.01986 × 10−113.01986 × 10−11
F137.35182 × 10−83.68973 × 10−111.17374 × 10−95.96731 × 10−98.48477 × 10−9
F140.0674920970.0006511241.23791 × 10−111.23791 × 10−110.170226616
F152.15403 × 10−61.60621 × 10−62.1544 × 10−100.0004981820.000189162
F160.2813794320.0210339251.33688 × 10−111.33688 × 10−110.281379432
F170.33371069611.21178 × 10−121.21178 × 10−121
F180.9288592330.0021724042.90637 × 10−112.90637 × 10−110.040131936
F190.7870772790.0720110785.29239 × 10−51.07105 × 10−50.025472598
F200.0101397036.68183 × 10−50.0034770630.4115662712.56739 × 10−5
F210.0032107420.0002514967.5914 × 10−50.0011887490.042685963
F220.0079990892.49472 × 10−72.32321 × 10−83.45143 × 10−70.000250769
F237.21977 × 10−51.47423 × 10−111.3315 × 10−111.3315 × 10−115.67825 × 10−6
Table 3. Sample dataset.
Table 3. Sample dataset.
Test ID x 1 x 2 x 3 x 4 y
114.115962624.26
213151021521.11
311.720542319.44
417.514.3872320.78
515.915491714.56
68.113.5371313.92
77.716.5321514.56
81013.5331011.64
91215.5391212.71
1010.222191512.44
1124.2151461925.28
1217.1151095736.41
1312.723.2381915.66
141017823628.42
1514.326892223.48
1622.547604524.5
175.530.5443023.94
1819.2611423140.21
1915.235.6448104106.91
2012.235.6718162164.05
2143.930.51623827.8
2296613548046.75
2373.8612736744.32
2422.676.7651170190.33
2530.532.51534535.04
2645.732.51485226.45
2713.732.51124536.28
285.516.951.6129.574.83
2929331053926.1
3012.216.8331613.67
311435.1593023
3239.627.429716576.79
3330.561915228.26
3425.932.5996133.14
3513.127.48011059.14
3620.46110520892.34
379.1455414474.65
3816.8618710051.42
3913.732.511213773.89
4018.376.2115335153.65
414.616.943120.570.2
4233.632.5121.435.425.65
4333.632.510848.826.39
4420.332.5158.2112.863.12
4530.551102.824.424.02
46813.527911.11
479.429.3522923.08
4814.616672921.96
4911.617.5572721.28
509.619.2421515.3
5121.645.71473137.22
5236.930.6149.628.227.39
5366.432.52236027.43
5411.611.4442116.86
5522.932.5915230.92
5613.819212113.78
5725.327.424418593.59
5814.952.8665332.57
5918.332.5513320.83
6048.2611526431.93
613221.414111549.98
6213.427812222.76
6324.2151471925.42
6415.517.5807240.6
6512.832.51109656.55
Table 4. Variable relationships between model inputs and outputs.
Table 4. Variable relationships between model inputs and outputs.
Inputs and OutputsParameterVariant
importationPile length (m) x 1
Pile diameter (m) x 2
Average effective vertical stress (kPa) x 3
Undrained shear strength (kPa) x 4
exportsLoad capacity (kPa)y
Table 5. Overall descriptive results.
Table 5. Overall descriptive results.
Variable Maximum ValuesMinimum ValueAverage ValueStandard DeviationMedianVariance KurtosisSkewness
x 1 964.621.55416.3715.5267.9677.3912.436
x 2 76.711.431.3616.64530.5277.0520.3531.049
x 3 71819124.486127.7869116,329.24410.8523.078
x 4 335962.16360.03383603.6315.842.114
y190.3311.1141.10936.01827.391297.3276.3712.415
Table 6. Statistics of optimization results.
Table 6. Statistics of optimization results.
ParametersIDBO
C20
σ1.4632
Table 7. Error analysis table.
Table 7. Error analysis table.
Forecasting MethodologyMAPERMSER2
KELM0.18476.73570.8639
IDBO-KELM0.10714.78750.9313
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Chen, B.; Hai, M.; Di, G.; Zhou, B.; Zhang, Q.; Wang, M.; Guo, Y. Enhanced Dung Beetle Optimizer-Optimized KELM for Pile Bearing Capacity Prediction. Buildings 2025, 15, 2654. https://doi.org/10.3390/buildings15152654

AMA Style

Chen B, Hai M, Di G, Zhou B, Zhang Q, Wang M, Guo Y. Enhanced Dung Beetle Optimizer-Optimized KELM for Pile Bearing Capacity Prediction. Buildings. 2025; 15(15):2654. https://doi.org/10.3390/buildings15152654

Chicago/Turabian Style

Chen, Bohang, Mingwei Hai, Gaojian Di, Bin Zhou, Qi Zhang, Miao Wang, and Yanxiu Guo. 2025. "Enhanced Dung Beetle Optimizer-Optimized KELM for Pile Bearing Capacity Prediction" Buildings 15, no. 15: 2654. https://doi.org/10.3390/buildings15152654

APA Style

Chen, B., Hai, M., Di, G., Zhou, B., Zhang, Q., Wang, M., & Guo, Y. (2025). Enhanced Dung Beetle Optimizer-Optimized KELM for Pile Bearing Capacity Prediction. Buildings, 15(15), 2654. https://doi.org/10.3390/buildings15152654

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