Machine Learning Prediction of Mechanical Properties for Marine Coral Sand–Clay Mixtures Based on Triaxial Shear Testing

Abstract

Marine coral sand–clay mixtures (MCCM) are promising green fill materials in civil engineering projects, where their strength characteristics play a vital role in ensuring structural safety and stability. To investigate these properties, a series of triaxial shear tests were performed under diverse conditions, including variations in asperity spacing, asperity height, the number of reinforcement layers, confining pressure, and axial strain. This experimental campaign yielded a robust strength dataset for MCCM. Utilizing this dataset, several predictive models were developed, including a standard Support Vector Machine (SVM), an SVM optimized via Genetic Algorithm (GA-SVM), an SVM enhanced by Particle Swarm Optimization (PSO-SVM), and a hybrid model incorporating Logical Development Algorithm preprocessing a SVM model (LDA-SVM). Among these models, the LDA-SVM model exhibited the best performance, achieving a test RMSE of 1.67245 and a correlation coefficient (R) of 0.996, demonstrating superior prediction accuracy and strong generalization ability. Sensitivity analyses revealed that asperity spacing, asperity height, and confining pressure are the most influential factors affecting MCCM strength. Moreover, an explicit empirical equation was derived from the LDA-SVM model, allowing practitioners to estimate strength without relying on complex machine learning tools. The results of this study offer practical guidance for the optimized design and safety evaluation of MCCM in civil engineering applications.

1. Introduction

Marine coral sand is a primary material used for reclamation and foundation construction in island building projects [1,2,3]. It is widely distributed in tropical and subtropical marine regions and possesses unique engineering properties [4,5]. The particles vary in shape, including blocky, branched, and spindle-like forms, and contain a high amount of calcium carbonate [6]. This sand also has high porosity and is easily fragmented [7]. With the rapid development of civil engineering, island and reef construction projects involving marine coral sand have grown significantly, particularly with the advancement of cutter suction dredging technology. For example, global dredging volumes have increased by over 30% in the past decade, with marine infrastructure projects in Southeast Asia and the Pacific Islands expanding rapidly [8,9,10]. In line with the concept of environmental protection and sustainable development, the use of locally available materials—such as marine coral sand—has become an essential strategy in marine infrastructure construction. However, during seabed excavation, marine clay is inevitably mixed with the coral sand, resulting in a marine coral sand–clay mixture (MCCM) as the fill material [11]. While this method significantly reduces the environmental footprint by minimizing material transportation and promoting local utilization, the MCCM often exhibits poor mechanical properties. Due to the low strength and large deformability of marine coral sand itself, combined with the effects of marine clay, the strength and stability of island foundations and slopes are greatly reduced [12,13]. Therefore, enhancing the mechanical performance of MCCM through effective reinforcement techniques is a critical challenge in island and reef engineering.

The use of geosynthetics to improve the performance of marine coral sand–clay mixtures (MCCM) has been recognized as an effective solution in geotechnical engineering [14,15,16]. Among the available methods, geogrid reinforcement has attracted considerable attention [17,18]. However, this technique often fails to address the essential requirement of seepage control in marine environments [19,20,21]. A more comprehensive approach is urgently needed—one that not only enhances the strength and stability of MCCM but also prevents the migration of fine coral sand particles, thereby improving the safety and long-term stability of island and reef infrastructure [22,23]. Geomembranes are widely regarded as effective barriers against seepage and offer several advantages, such as long service life, high resistance to seawater corrosion, ease of installation, and cost efficiency [24,25]. Compared to smooth geomembranes, textured geomembranes exhibit superior interface strength [26,27,28] and can provide reinforcement effects comparable to those of geogrids [22,27]. Although textured geomembranes have been applied in reinforcing other types of soil, their potential to enhance the properties of MCCM remains largely uninvestigated. Recent advances in 3D printing technology have now made it possible to develop customized geosynthetic materials that can be specifically tailored to the unique characteristics of MCCM, offering promising opportunities for improved reinforcement [29,30]. This not only facilitates the integration of green and low-carbon materials into marine reinforcement systems, but also enables precision-engineered surface textures and structural configurations [31]. The combination of 3D printing and geosynthetic design offers a promising pathway toward environmentally friendly, high-performance reinforcement solutions for civil geotechnical engineering.

Machine learning has emerged as a powerful tool in civil engineering, particularly for capturing the complex and nonlinear relationships inherent in geotechnical materials [32,33,34,35,36,37]. Among various methods, Support Vector Machine (SVM) has demonstrated strong capabilities in predicting the mechanical properties of marine coral sand–clay mixtures (MCCM), a commonly used foundation material in offshore construction [29,38]. To further improve prediction performance, optimization techniques such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) are often employed to fine-tune model parameters [38,39,40]. However, these traditional optimization algorithms tend to encounter limitations, including low computational efficiency and a tendency to fall into local optima, especially when applied to high-dimensional, nonlinear problems [41]. To address these shortcomings, the Logical Development Algorithm (LDA) has been introduced as an advanced heuristic optimization method [42,43]. By executing similarity and dissimilarity operations in parallel, LDA not only preserves critical data features but also significantly enhances global search ability and convergence speed [44,45,46,47,48]. The integration of LDA with SVM creates a hybrid modeling framework that combines the precision of machine learning with the efficiency of intelligent optimization [49]. This approach improves the overall accuracy, generalization, and robustness of strength prediction models for MCCM. Consequently, it provides a reliable and efficient pathway for the intelligent evaluation and design of civil geotechnical materials, offering substantial value for practical engineering applications.

Through a series of triaxial shear tests, this study explores the strength characteristics of marine coral sand–clay mixtures (MCCM) under different working conditions. The experiments focused on evaluating the effects of key variables, including the structural characteristics of textured geomembranes (such as asperity spacing and height), the number of reinforcement layers, and confining pressure. A comprehensive set of stress–strain data was obtained, resulting in a high-dimensional dataset that reflects the combined influence of multiple factors. Using this dataset, various machine learning models were developed and compared to identify the most accurate and reliable predictive approach for MCCM strength. Among them, the Support Vector Machine (SVM) model optimized using the Logical Development Algorithm (LDA) achieved the best performance in terms of prediction accuracy and generalization. Based on the LDA-SVM model, sensitivity analysis was performed to determine the dominant input features influencing strength, and an explicit empirical formula was further derived to facilitate practical applications in civil geotechnical design. This study addresses the need for accurate and interpretable models to predict the strength of marine coral sand–clay mixtures (MCCM), a sustainable fill material increasingly used in coastal engineering. By integrating large-scale triaxial tests with advanced machine learning, the research supports the safe and green application of MCCM-reinforced structures.

2. Materials and Methods

2.1. Materials

The materials used in the experiment were marine coral sand and kaolin. The marine coral sand was sourced from an island reef in the South China Sea, and was subjected to a series of processes such as drying and sieving. The portion with a particle size range of 2–4 mm was selected for the experiment. The kaolin had a uniform particle size of 10 μm. The marine coral sand and kaolin used in the experiment are shown in Figure 1. The parameters are listed in

Table 1. Basic physical parameters of marine coral sand.

GS	D50/mm	Cu	Cc	emin	emax
2.81	3.3	1.32	1.08	0.99	1.49

. Specifically, the marine coral sand had a uniformity coefficient of 1.32, a curvature coefficient of 1.08, and poor grading. The calcareous sand used in this study had a particle size range of 0.074–2 mm, exhibited non-plastic behavior with no defined Atterberg limits, a maximum dry density of 1.52 g/cm³, and an optimum moisture content of approximately 9.5%, with strength decreasing as water content increases; its composition was primarily CaCO₃ with minor MgCO₃ and SiO₂. In contrast, the clay component (kaolinite) presented a uniform particle size of approximately 10 μm, a liquid limit of 48%, plastic limit of 24%, maximum dry density of 1.43 g/cm³, and an optimum moisture content of around 17.8%; its main chemical constituents included SiO₂, Al₂O₃, and trace Fe₂O₃.

Three-dimensional printing technology (SLA with photopolymerization) was used to create geomembrane materials with different textured surfaces [50]. The textures consisted of asperity of varying heights (1 mm, 2 mm, 3 mm) and spacing (10 mm, 15 mm, 20 mm). The asperity on the textured surface was conical, and the material used was white resin. To account for the boundary effect during the reinforcement sample preparation, the geomembrane was designed with a diameter of 96 mm and a thickness of 2 mm. The textured geomembrane model and the printed prototype are shown in Figure 2, with the detailed parameters provided in

Table 2. Physical and mechanical properties of the textured geomembrane.

Standard	Tensile Modulus	Tensile Strength	Elongation at Break	Flexural Modulus	Impact Strength	Distortion Temperature
ASTM	2450 Mpa	50 Mpa	10%	2400 Mpa	45 J/m	56 °C

.

2.2. Experimental Procedure

The textured geomembrane could potentially be applied for reinforcement in the shallow surface layer of island reefs, where the shallow marine coral sand–clay mixture (MCCM) remains in a dry state with an extremely low moisture content. To investigate the mechanical and deformation properties of the reinforced MCCM with different geomembrane surface types, confining pressures, and reinforcement layers, and to minimize the impact of moisture content on the experimental results, an unconsolidated undrained test was conducted. This study examines the mechanical properties under various conditions. Since marine coral sand particles are prone to fragmentation, particle sieving tests were conducted before and after the experiment to investigate the particle fragmentation characteristics of the MCCM under different conditions and perform a quantitative analysis. The sample dimensions were Φ100 mm × 200 mm, with a shear rate of 1 mm∙min⁻¹. The experimental design is shown in

Table 3. Experimental parameter combinations used in the tests.

Number	Asperity Spacing (mm)	Asperity Height (mm)	Confining Pressure (kPa)	Geomembrane Layers
T1	-	-	10, 30, 50	0
T2	10, 15, 20	1, 2, 3	10	1, 2
T3	10, 15, 20	1, 2, 3	30	1, 2
T4	10, 15, 20	1, 2, 3	50	1, 2

.

The triaxial testing was conducted using a static-dynamic triaxial test system (VJ Tech Static-Dynamic Triaxial Testing System, VJ Tech Ltd., Berkshire, UK). The system primarily consists of a load frame (with a built-in high-speed servo controller), an automatic confining pressure controller, and an automatic back pressure controller. The accompanying system software offers features such as parameter setting, test control, and data acquisition, providing excellent interactivity.

To minimize the impact of moisture content on the experimental results, dry marine coral sand and kaolin were used to prepare the samples. The dry mass ratio of marine coral sand was 58.8%, while that of kaolin was 41.2% [11,51]. Prior to sample preparation, the marine coral sand and kaolin were mixed in the required proportions. The sample was compacted in six controlled layers to ensure uniform density while avoiding excessive force that could crush the fragile calcareous sand particles, as shown in Figure 3. During the test, the software automatically recorded axial force and axial strain data. The test was stopped when the axial strain reached 12%. After the test, the MCCM was cleaned, separated, dried, and sieved to obtain the particle size distribution curve of the marine coral sand.

2.3. The Influence of Each Variable on the Strength Characteristics of MCCM

Based on the experimental data, the deviatoric stress–strain relationship curves for each test group were obtained. Figure 4 shows the deviatoric stress–strain curves of the unreinforced MCCM samples under different confining pressures. The results indicate that as the confining pressure increases, the strength of the samples continuously increases, with the deviatoric stress–strain curve exhibiting a strain-hardening behavior. This hardening effect becomes more pronounced as the confining pressure rises. The maximum axial deviatoric stresses were 44.84 kPa, 110.44 kPa, and 214.70 kPa, respectively. Figure 4 presents the deviatoric stress–strain curves for the samples under different geomembrane surface types, reinforcement layers, and confining pressures. As both confining pressure and the number of reinforcement layers increase, the deviatoric stress–strain curves shift upward, indicating an enhanced strain-hardening effect and an increase in shear strength. At a confining pressure of 10 kPa, the deviatoric stress increased by up to 92.51%. At low axial strains, the deviatoric stress–strain curves of the reinforced and unreinforced MCCM are nearly identical under the same confining pressure. As the axial strain increases, the deviatoric stress–strain curve of the reinforced samples gradually diverges, showing a progressively stronger reinforcement effect. When the geomembrane surface type was changed, only a slight increase in the deviatoric stress–strain curve was observed for the reinforced MCCM, with minimal variation. The strength changes were not easily detectable from the deviatoric stress–strain curve.

The results indicate that reducing both asperity height and asperity spacing leads to an increase in the strength of MCCM, providing valuable guidance for the optimized design of textured geomembranes.

Therefore, there is an urgent need for an efficient and reliable alternative to address the inherent limitations of traditional experimental approaches. In this context, machine learning has emerged as a powerful tool, owing to its superior data processing capabilities and strong ability to model complex nonlinear relationships.

3. Methodology

3.1. Machine Learning Algorithms

In this study, a machine learning framework based on Support Vector Machine (SVM) is adopted to predict the strength of marine coral sand–clay mixtures. To enhance model performance, several advanced optimization techniques—including Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Logical Development Algorithm (LDA)—are incorporated. The integration of these methods offers distinct advantages, with three core benefits highlighted:

Each algorithm was constructed within a unified and structured modeling framework, ensuring consistency in data preprocessing, parameter selection, and training procedures [52].

These techniques have proven effective in addressing a wide range of complex problems in civil engineering, particularly those involving geotechnical prediction, material behavior analysis, and structural assessment [53,54].

They excel in capturing intricate nonlinear interactions among multiple influencing variables, making them particularly suitable for modeling the multifactorial nature of marine foundation systems [48].

3.1.1. SVM

Support Vector Machine (SVM) is a supervised learning algorithm that maps input data into a high-dimensional feature space using kernel functions to construct an optimal hyperplane for regression or classification tasks [55]. It operates by minimizing a structural risk function, ensuring both fitting accuracy and generalization ability. In this study, the SVM model is developed to predict the stress response of marine coral sand–clay mixtures (MCCM) under varying conditions. The model utilizes five input variables—asperity spacing, asperity height, confining pressure, number of reinforcement layers, and strain—with stress as the output target. A radial basis function (RBF) kernel is employed to capture the nonlinear relationship between inputs and output. Key hyperparameters, including the penalty factor (C) and kernel width (gamma), are optimized using intelligent algorithms. Specifically, C was searched within the range 0.1 to 100, and gamma within 0.001 to 1. These optimizations enhance model performance and ensure robust prediction capability.

3.1.2. GA and PSO

Genetic Algorithm (GA) is an optimization technique based on the principles of natural evolution [56]. It operates by simulating processes such as selection, crossover, and mutation to explore a range of possible solutions. The algorithm begins with a randomly generated population, where each individual is assessed using a predefined fitness function. Those with better performance are chosen to reproduce, while genetic variation is introduced to maintain diversity. This cycle repeats until a specified condition—like reaching a certain number of iterations or achieving an acceptable fitness value—is fulfilled. GA is well-suited for tackling complex optimization tasks due to its robust global search ability and adaptability to nonlinear, high-dimensional problems.

Particle Swarm Optimization (PSO) is a heuristic search algorithm inspired by the collective behavior observed in natural swarms, such as flocks of birds or schools of fish [57]. Each solution is treated as a particle that navigates the search space by updating its position and velocity based on both its own best experience and that of the entire group. Through this collaborative mechanism, particles gradually converge toward the optimal solution. PSO is known for its simple design, ease of use, independence from gradient information, and strong capability in finding global optima. It has proven effective in a wide range of optimization scenarios, including function optimization and neural network training.

3.1.3. LDA

Logical Development Algorithm (LDA) is a supervised method for dimensionality reduction that aims to enhance class separability by maximizing the variance between different categories while minimizing the variance within the same category [58]. In regression applications, LDA serves as a preprocessing tool to extract key features that are both informative and less correlated, thereby improving model stability, reducing the risk of overfitting, and boosting predictive performance. When combined with neural networks, LDA helps streamline the input space while retaining critical discriminative information, resulting in more efficient and reliable learning outcomes. In this study, LDA is adapted for regression by identifying key input features and constructing logical expressions to approximate continuous outputs. This reduces input redundancy and improves model stability and accuracy when combined with neural networks, as shown in Figure 5.

3.2. Model Parameter Setting

3.2.1. Hyperparameter Optimization

Hyperparameter optimization is vital for enhancing machine learning model performance, as parameter settings greatly affect training efficiency and predictive accuracy. In this study, Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Logical Development Algorithm (LDA) were employed to optimize the hyperparameters of a Support Vector Machine (SVM), thereby improving its overall effectiveness.

To promote faster convergence and improve prediction accuracy, key hyperparameters for each optimization technique were carefully configured, as outlined in

Table 4. Parameter settings for GA, PSO, and LDA-optimized SVM.

Algorithm	LDA	PSO	GA
Population size (pop_num)	30	30	15
Genetic generations (gen)	-	-	100
Selection function parameter (normGeomSelect)	-	-	0.15
Crossover function parameter (arithXover)	-	-	1.5
Mutation function parameter (nonUnifMutation)	-	-	[0.2, 50, 3]
Optimal solution tolerance (maxGenTerm)	-	1	1 × 10−6
Learning factors (c1, c2)	-	2.1	-
Maximum position (popmax)	-	2.0	-
Minimum position (popmin)	-	−2.0	-
Maximum number of iterations (max_iter)	80	-	-
Mutation factor (F)	0.7	-	-
Crossover probability (CR)	0.9	-	-

. For GA, a population size (pop_num) of 15 was chosen to maintain adequate diversity while controlling computational load; the number of generations (gen) was fixed at 100 to allow sufficient evolutionary progress. The selection probability parameter (normGeomSelect) was set to 0.15 to prioritize selection of high-fitness individuals. The crossover rate (arithXover) was set to 1.5 to encourage genetic variability, while the mutation scheme (nonUnifMutation) was defined as [0.2, 50, 3] to strengthen global search capabilities and prevent premature convergence. The convergence tolerance (maxGenTerm) was set to 10⁻⁶ to allow fine refinement near optimal solutions.

Within the PSO framework, cognitive and social coefficients (c1, c2) were both set at 2.1, balancing personal and collective experience influence. The particle position search space was constrained to [0.001, 1000] for the penalty parameter C, and [1 × 10⁻⁵, 1] for the kernel coefficient gamma, ensuring feasible hyperparameter ranges. A swarm size (sizepop) of 30 and maximum iteration count (maxgen) of 150 were selected to balance exploration depth and computational efficiency. The maximum function evaluations were set at 300 to guarantee robust training.

For the Logical Development Algorithm (LDA), a population size of 30 and a maximum iteration count of 80 were used. The mutation factor (F) was set at 0.7 to facilitate effective global exploration, and the crossover probability (CR) was fixed at 0.9 to maintain population diversity during recombination. This configuration enabled enhanced extraction of discriminative features during the hyperparameter search, improving SVM’s classification accuracy.

Collectively, these optimized hyperparameter settings for GA, PSO, and LDA significantly improved the SVM’s convergence speed, mitigated overfitting, and elevated prediction accuracy. Such improvements proved highly effective in applications including the strength prediction of reinforced marine coral sand–clay mixtures (MCCM).

The comparative RMSE trends of GA, PSO, and LDA during the optimization process reveal distinct convergence behaviors. All three algorithms steadily reduce RMSE as iterations progress, demonstrating effective error minimization. PSO achieves the quickest initial convergence, reaching lower RMSE values in fewer iterations, reflecting its efficiency in exploring the solution space. However, after a rapid early drop, its improvement slows and plateaus around an RMSE of 3. GA shows a fast initial RMSE decrease as well, then a gradual decline that stabilizes near 1.5 after approximately 20 iterations. In contrast, LDA starts with a slower RMSE reduction but accelerates later, eventually converging to an RMSE close to 1. Overall, PSO excels in early convergence speed, while LDA attains better final accuracy and stability, surpassing both GA and PSO over the full optimization.

3.2.2. Establishment of Database and Data Processing

A dataset comprising 2436 samples was established to investigate the strength characteristics of MCCM under different conditions. The dataset incorporates five key factors: asperity spacing, asperity height, confining pressure, number of reinforcement layers, and strain. Prior to modeling, all input and output variables were normalized to enhance model performance and maintain uniform scaling:

$$x^{\prime }={\displaystyle \frac{x-x_{\min }}{x_{\max }-x_{\min }}}$$

(1)

where $x$ is the original data value, $x^{\prime }$ is the normalized data value, x_min is the smallest value in the data, and x_max is the largest value in the dataset.

Table 5. Statistical table of factors affecting the stress of the MCCM.

Argument	Type	Min Value	Max Value	Mean Value	Standard Deviation
Asperity spacing	Numerical type	10	20	15	5
Asperity height		1	3	2	1
Confining pressure (Kpa)		10	50	30	20
Number of reinforcement layers		0	2	1	1
Strain (mm)		1	12	7.27	3.26

summarizes the key statistics of the dataset variables. Asperity spacing ranges between 10 and 20 mm, while asperity height is divided into three discrete levels: 1 mm, 2 mm, and 3 mm. The reinforcement layers are classified into three categories: 0, 1, and 2. Confining pressure spans from 50 kPa to 200 kPa, and strain values range from 1% to 12%. To guarantee data quality and suitability for machine learning, preprocessing steps such as cleaning, normalization, and dataset splitting were performed.

3.3. Predictive Performance Assessment Index

Choosing appropriate evaluation metrics is essential for effectively assessing model prediction performance. In this study, two main evaluation criteria were utilized:

1. Root Mean Square Error (RMSE) measures the standard deviation of the differences between predicted and true values. A smaller RMSE indicates better prediction accuracy and greater model reliability [59].

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}}$$

(2)

where n is the number of samples, y_i is the observed value, and fi is the predicted value.

2. Mean Absolute Percentage Error (MAPE) represents the average absolute error between predicted and actual values, expressed as a percentage of the actual measurements. A lower MAPE indicates reduced prediction errors and enhanced model accuracy [60].

$$\mathrm{MAPE}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|\frac{y_i-{\widehat{y}}_i}{y_i}\right|}\times 100\%$$

(3)

where n is the number of samples, y_i is the observed value, and ${\widehat{y}}_i$ is the predicted value.

4. Results and Analysis

4.1. Establishment of Datasets

In this study, dataset construction forms the basis for training the machine learning models. A dataset comprising five input variables and one output variable was developed to train and evaluate the performance of SVM, GA-SVM, PSO-SVM, and LDA-SVM, as shown in Figure 6.

A total of 2436 samples were collected, each containing five key input features: asperity spacing, asperity height, confining pressure, number of reinforcement layers, and strain. To assess the model’s generalization performance, the dataset was divided into training and testing subsets. The model was trained on 80% of the data, while the remaining 20% was used to evaluate its final predictive accuracy. This split ensured a robust assessment of the model’s ability to perform in practical scenarios.

4.2. Machine Learning Predicting Performances

The prediction results of the machine learning model, developed and tested using 2436 samples, are displayed in Figure 7 and Figure 8.

Overall, the predicted results (marked by red hollow circles) from each model align well with the actual measurements (black hollow diamonds), indicating that all models can effectively capture the nonlinear relationships between input variables and the target output to some degree. This demonstrates their capability in fitting the training data. As shown in Figure 7j, the BPNN model exhibits signs of overfitting, with predictions closely matching the training data but deviating noticeably for more complex or extreme samples. This indicates that while BPNN can fit the known data, it lacks the generalization ability required for reliable predictions under varying geotechnical conditions. The overfitting behavior reflects the model’s sensitivity to training noise and its limited capacity to capture essential underlying patterns, suggesting that traditional models like BPNN may no longer be well-suited for modern geotechnical prediction tasks. However, variations in prediction accuracy reveal differences in model learning capacity. Among the methods assessed, the LDA-SVM model (Figure 7h) achieves the highest accuracy, with predictions closely matching observed values. This suggests that LDA-SVM not only fits the data precisely but also robustly represents complex data patterns. Conversely, the conventional SVM model (Figure 7b) shows larger errors, particularly for samples with extreme peak values, implying limited representational power that leads to underfitting in complex regions. Enhanced performance is evident in GA-SVM (Figure 7d) and PSO-SVM (Figure 7f), which incorporate Genetic Algorithm and Particle Swarm Optimization, respectively, to optimize model parameters. These hybrid approaches yield predictions that better track actual trends, reflecting improved parameter tuning and increased robustness. Nonetheless, mild overfitting appears in some cases, especially where data variance is higher, indicating a need for additional regularization or validation to improve generalization.

Through k-fold cross-validation, the mean squared error (MSE) values obtained for different models are as follows: BPNN achieved an MSE of approximately 23.9 (training) and 28.01 (testing), traditional SVM about 17.02 (training) and 15.63 (testing), GA-optimized SVM (GA-SVM) around 3.70 (training) and 2.61 (testing), PSO-optimized SVM (PSO-SVM) near 16.84 (training) and 14.59 (testing), and the LDA-optimized SVM (LDA-SVM) obtained the lowest MSE of approximately 1.99 (training) and 2.40 (testing). These results demonstrate that the hybrid optimization algorithms effectively improve the predictive accuracy of the models, with LDA-SVM performing best among the tested methods.

In conclusion, the LDA-SVM model consistently surpasses the other techniques across both the training and testing sets. Its superior fitting precision and reliable generalization to unseen data highlight its strength in modeling complex relationships, making it the most effective and dependable model among those evaluated.

As shown in Figure 8, the LDA-SVM model consistently outperforms the other three machine learning models on the testing dataset. Specifically, Figure 8 indicates that LDA-SVM achieves the best prediction results, with the lowest RMSE values of 1.67245 for testing and 1.41230 for training. It also records the smallest MAPE values—5.93210% on the test set and 7.83025% on the training set—demonstrating strong generalization ability. The coefficient of determination (R²) for the traditional SVM is 0.97012, while the LDA-SVM achieves a notably higher R² of 0.99580, demonstrating that the LDA-based optimization significantly improves the model’s predictive accuracy and fitting performance.

In comparison, the GA-SVM model, benefiting from genetic algorithm optimization, performs better than the base SVM but still lags behind LDA-SVM. The GA-SVM reports RMSE values of 1.61542 (test) and 1.92354 (training), and MAPE values of 6.21087% and 11.20032%, respectively. Its test set correlation coefficient of 0.98234 suggests a moderate-to-strong relationship, though some complex data patterns remain underrepresented.

The PSO-SVM model shows relatively weaker performance, with higher errors and more dispersed residuals, indicating limited improvement from particle swarm optimization. RMSE values are 3.82056 (test) and 4.10321 (training), while MAPE values reach 9.72451% and 11.65124%. The test correlation coefficient (R) is 0.97945, reflecting decreased predictive consistency.

The original SVM model yields the poorest results, with the highest RMSEs of 3.95420 (test) and 4.12450 (training), and MAPE values of 9.86420% and 13.05210%. Its correlation coefficient of 0.97012 along with visible deviations between predicted and actual points indicates insufficient modeling capacity for complex nonlinear relationships.

In summary, the LDA-SVM model outperforms GA-SVM, PSO-SVM, and basic SVM across the training and testing stages. It offers enhanced accuracy, stronger generalization, and greater model stability, particularly on unseen data. Despite similar optimization frameworks, LDA-SVM consistently delivers superior predictive results compared to the other approaches.

4.3. Sensitivity Analysis

Gaining insight into the internal workings of machine learning models is crucial for improving their transparency, interpretability, and overall reliability in practical use. A fundamental aspect of this is understanding the importance of input features, which clarifies how each variable impacts the model’s predictions and sheds light on the decision-making process [61]. Among the various methods developed for interpretability, Shapley Additive Explanations (SHAP) has become a leading approach due to its theoretical foundation and comprehensive assessment of feature contributions. Based on principles from cooperative game theory, SHAP evaluates every possible combination of features to fairly distribute the influence each one has on an individual prediction. It assigns a numerical SHAP value to each feature per prediction, quantifying how much that feature pushes the outcome higher or lower [62]. Positive values indicate an increasing effect on the prediction, while negative values suggest a decreasing impact. By summarizing SHAP values over many samples, it is possible to identify the most influential features at the global level and understand specific variations at the local level. This detailed interpretability is especially useful for complex models like neural networks or ensemble techniques, where simple coefficient-based explanations are inadequate. SHAP thus aids not only in verifying and improving models but also in fostering responsible application of machine learning technologies.

Figure 9 highlights the five most influential features impacting the predictions of the LDA-SVM model, with their respective contributions quantified. The pie chart shows the average SHAP values, where larger values indicate stronger effects on the output. On the left, the swarm plot illustrates the detailed influence of each feature: the x-axis represents SHAP values, reflecting the direction and strength of impact, and the y-axis corresponds to feature values. Generally, higher SHAP values paired with higher feature values suggest a positive correlation, whereas negative SHAP values denote suppressive effects within certain ranges.

Among the features, asperity spacing leads with a 35.2% contribution, emphasizing its dominant role in controlling the stress–strain behavior of MCCM. This likely results from its effect on inter-particle friction at contact interfaces, where wider spacing enhances shear resistance. Asperity height follows closely at 29.1%, reflecting its influence on mechanical interlocking, which promotes energy dissipation and strengthens resistance during deformation.

The number of reinforcement layers accounts for 15.3%, highlighting its significance in enhancing composite structural integrity. Confining pressure contributes 13.0%, as it directly affects the compaction and strength of the material. Strain has the smallest effect, with a contribution of 7.4%, indicating its role as more of a dependent response variable in the prediction model.

Overall, the SHAP analysis identifies asperity spacing and height as the key factors influencing model accuracy, underlining the importance of their precise control in MCCM engineering applications. The results suggest that optimization efforts should primarily target these parameters to ensure the durability and safety of marine structures subjected to complex environmental forces.

4.4. Empirical Formulas

The motivation behind integrating LDA with SVM lies in the need to enhance both the interpretability and predictive accuracy of strength modeling for MCCM, a material characterized by complex, nonlinear, and moisture-sensitive behavior; this novel combination enables effective dimensionality reduction while preserving essential mechanical features. Previous modeling outcomes indicate that the developed LDA-SVM model effectively predicts the strength of MCCM. However, the complexity of machine learning algorithms often creates barriers to practical use, especially for engineers without a background in artificial intelligence. To improve accessibility and promote wider adoption, this section presents an analytical empirical formula that closely approximates the predictive performance of the LDA-SVM model. This formula offers a simple and efficient approach for estimating MCCM strength in engineering applications. The SVM model employed utilizes a standard kernel-based method, and its output can be expressed mathematically based on the optimized support vectors and their corresponding coefficients, as detailed below [63]:

$$Y_n=f_{\mathrm{s}\mathrm{i}\mathrm{g}}\left(b_0+\sum _{k=1}^h w_k\cdot f_{\mathrm{s}\mathrm{i}\mathrm{g}}\left(b_k+\sum _{i=1}^m w_{ik}{X}_i\right)\right)$$

(4)

The normalized predicted output Yn, ranging from −1 to 1, is calculated based on the normalized input variables X_i, which includes asperity spacing (mm), asperity height (mm), number of reinforcement layers, confining pressure (kPa), and strain (%), through the connection weights W_ik between the ith input node and the kth hidden node, hidden layer biases bk, weights W_k connecting the hidden nodes to the output node, output layer bias b₀, and the hyperbolic tangent sigmoid transfer function $f_{\mathrm{s}\mathrm{i}\mathrm{g}}(x)={\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}}}$, where h and m denote the numbers of hidden nodes and input variables, respectively.

The normalized output Yn can be converted into the actual predicted strength τ using the following denormalization formula:

$$\tau =0.5\left(Y_n+1\right)\left({\tau }_{max}-{\tau }_{min}\right)+{\tau }_{min}$$

(5)

Here, τ_max and τ_min represent the maximum and minimum MCCM strength values in the dataset, respectively.

To facilitate engineering applications, the neural network structure described above can be further expressed in the following simplified form:

$$Y_n=\tanh \left(C_1\right)$$

(6)

$$C_1=b_0+\sum _{k=1}^h w_k\cdot \tanh \left(A_k\right)$$

(7)

$$A_k=b_k+\sum _{i=1}^m w_{ik}{X}_i$$

(8)

All connection weights (W_ik, W_k) and bias parameters (b_k, b₀) in the model are automatically optimized during training using the Differential Evolution algorithm. The complete set of parameters, which can be directly applied in engineering calculations, is provided in

Table 6. Connected weights and biases for the constructed LDA-SVM algorithm.

Hidden Layer	Weight						Bias
	Input					Output	Hidden Layer	Output
	S	H	P	L	Y	H
1	0.42	−0.44	0.21	0.44	−0.27	0.23	1.05	0.51
2	1.18	1.16	−0.14	1.03	0.15	−0.18	−0.72
3	0.91	−1.32	1.05	−1.15	0.57	0.12	0.45
4	−1.12	0.34	−1.17	1.78	−0.93	0.03	0.84
5	1.24	1.45	0.92	−1.67	1.23	−0.01	1.09

.

This empirical analytical model provides clear interpretability and allows for quick estimation of MCCM strength without the need for machine learning software, making it highly suitable for engineering applications where transparency and computational speed are essential.

The input parameters S, H, P, L, and Y represent asperity spacing, asperity height, confining pressure, number of reinforcement layers, and strain, respectively. The output variable H denotes the MCCM strength, as summarized in Table 6.

4.5. Experimental Verifications

To evaluate the accuracy and practical applicability of both the machine learning model and the analytical formula, the proposed empirical model was applied to predict MCCM strength across the 30 representative test cases listed in

Table 7. Thirty representative datasets selected for model validation.

Asperity Spacing	Asperity Height	Confining Pressure (Kpa)	Reinforced Layers	Strain (%)
0	0	10	0	3
		30	0	6
		50	0	10
10	1	10	1	2
	3	30	2	3
	2	50	1	6
10	1	50	2	9
	3	10	1	10
	2	30	2	11
10	1	30	1	8
	3	50	2	7
	2	10	1	1
15	1	10	1	2
	3	30	2	3
	2	50	1	6
15	1	50	2	9
	3	10	1	10
	2	30	2	11
15	1	30	1	8
	3	50	2	7
	2	10	1	1
20	1	10	1	2
	3	30	2	3
	2	50	1	6
20	1	50	2	9
	3	10	1	10
	2	30	2	11
20	1	30	1	8
	3	50	2	7
	2	10	1	1

. Using the experimental conditions, Equation (4) estimated the MCCM strength for each scenario. The predicted outcomes were then compared with experimental results from earlier studies, as shown in Figure 10, providing a comprehensive assessment of the model’s predictive capability and real-world usefulness under diverse conditions.

Figure 10 demonstrates that the proposed empirical formula effectively predicts the strength of marine coral sand–clay mixtures (MCCM) under diverse conditions. The model achieves a root mean square error (RMSE) of 1.234, a mean absolute percentage error (MAPE) of 6.12%, and a coefficient of determination (R²) of 0.9945, reflecting strong predictive capability and reliability. These findings validate the model’s ability to accurately represent the strength characteristics of MCCM across varying scenarios.

Compared with more complex machine learning-based approaches, the proposed formula offers enhanced practicality and accessibility, particularly for engineering professionals without backgrounds in programming or algorithm development. Its transparent structure and interpretable parameters make it a straightforward and efficient alternative for rapid strength estimation. This approach not only reduces technical barriers in modeling and analysis but also serves as a reliable tool for engineering design, construction planning, and safety evaluation. Overall, it holds significant promise for broader application and dissemination in geotechnical engineering practice.

Although the proposed model demonstrates strong predictive performance, its current validation is limited to 30 randomly selected test sets. These samples represent a range of typical MCCM conditions but do not include field-scale verification. Future work will focus on extending validation to real-world island construction scenarios, which will further assess the model’s applicability and enhance its engineering relevance.

5. Conclusions

This study conducted a series of laboratory experiments to explore the strength characteristics of marine coral sand–clay mixtures (MCCM) under various conditions, establishing a comprehensive experimental dataset. Utilizing this data, an LDA-SVM machine learning model was developed to predict MCCM strength. The model integrates several critical input parameters influencing strength, including asperity spacing, asperity height, confining pressure, the number of reinforcement layers, and strain. To assess its effectiveness, three comparative models—SVM, GA-SVM, and PSO-SVM—were also constructed. Moreover, sensitivity analysis was performed to quantify each input variable’s influence on the strength prediction. An empirical formula was further derived to enable practical application by engineers without extensive machine learning expertise. The key conclusions of this work are as follows:

(1)

Experimental findings show that asperity spacing, asperity height, confining pressure, reinforcement layer count, and strain are the dominant factors affecting MCCM strength. Among these, asperity spacing and height notably enhance strength. These insights can assist in optimizing material composition and design parameters in engineering practice.

(2)

The LDA-SVM model demonstrated superior accuracy and generalization in predicting MCCM strength, outperforming benchmark models such as SVM, GA-SVM, and PSO-SVM by achieving lower RMSE values. This highlights the benefit of combining the Logic Development Algorithm (LDA) with Support Vector Machines (SVM) in complex marine geotechnical modeling.

(3)

Sensitivity analysis identified asperity spacing as the most influential factor on MCCM strength, followed by asperity height and confining pressure. This knowledge supports prioritizing critical design parameters for efficient engineering management.

(4)

The empirical formula derived provides clear interpretability and practical value, enabling accurate strength prediction without requiring advanced machine learning tools. This approach benefits engineers with limited ML background and supports decision-making when computational resources are limited.

6. Limitations

Predicting MCCM strength remains challenging due to complex nonlinear interactions between multiple factors. The Logic Development Algorithm (LDA) has some limitations, including sensitivity to initial parameter settings that may impact convergence and performance, challenges in handling very high-dimensional data due to computational complexity, limited capability in capturing highly nonlinear relationships compared to deep learning methods, and a risk of overfitting when training data are limited or noisy. However, the LDA-SVM model developed here effectively addresses these challenges by integrating the Logic Development Algorithm with machine learning, capturing key variable effects accurately. In this study, MCCM samples were tested in a dry state to simplify conditions, which limits the direct applicability of results to saturated marine environments; future research will consider wet and cyclic loading to better simulate in situ behavior. The derived empirical formula provides a straightforward and practical approach for strength estimation, especially valuable in engineering scenarios with limited resources. Future research could focus on extending the model by incorporating additional MCCM characteristics such as particle morphology, fabric, and mineralogy, as well as expanding the experimental dataset to cover a wider range of loading and environmental conditions. These improvements are expected to enhance the model’s robustness and applicability, thereby contributing to safer and more efficient marine infrastructure design.