Abstract
This research suggests a robust integration of artificial neural networks (ANN) for predicting swell pressure and the unconfined compression strength of expansive soils (PsUCS-ES). Four novel ANN-based models, namely ANN-PSO (i.e., particle swarm optimization), ANN-GWO (i.e., grey wolf optimization), ANN-SMA (i.e., slime mould algorithm) alongside ANN-MPA (i.e., marine predators’ algorithm) were deployed to assess the PsUCS-ES. The models were trained using the nine most influential parameters affecting PsUCS-ES, collected from a broader range of 145 published papers. The observed results were compared with the predictions made by the ANN-based metaheuristics models. The efficacy of all these formulated models was evaluated by utilizing mean absolute error (MAE), Nash–Sutcliffe (NS) efficiency, performance index ρ, regression coefficient (R2), root mean square error (RMSE), ratio of RMSE to standard deviation of actual observations (RSR), variance account for (VAF), Willmott’s index of agreement (WI), and weighted mean absolute percentage error (WMAPE). All the developed models for Ps-ES had an R significantly > 0.8 for the overall dataset. However, ANN-MPA excelled in yielding high R values for training dataset (TrD), testing dataset (TsD), and validation dataset (VdD). This model also exhibited the lowest MAE of 5.63%, 5.68%, and 5.48% for TrD, TsD, and VdD, respectively. The results of the UCS model’s performance revealed that R exceeded 0.9 in the TrD. However, R decreased for TsD and VdD. Also, the ANN-MPA model yielded higher R values (0.89, 0.93, and 0.94) and comparatively low MAE values (5.11%, 5.67, and 3.61%) in the case of PSO, GWO, and SMA, respectively. The UCS models witnessed an overfitting problem because the aforementioned R values of the metaheuristics were 0.62, 0.56, and 0.58 (TsD), respectively. On the contrary, no significant observation was recorded in the VdD of UCS models. All the ANN-base models were also tested using the a-20 index. For all the formulated models, maximum points were recorded to lie within ± 20% error. The results of sensitivity as well as monotonicity analyses depicted trending results that corroborate the existing literature. Therefore, it can be inferred that the recently built swarm-based ANN models, particularly ANN-MPA, can solve the complexities of tuning the hyperparameters of the ANN-predicted PsUCS-ES that can be replicated in practical scenarios of geoenvironmental engineering.
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Introduction
Expansive behaviour of swelling clay is a complicated process as prominent clay minerals, for instance, kaolinite, illite, montmorillonite etc. are present, which leads to higher swell-shrink as the moisture fluctuates. The physicochemical properties of the expansive soils (ES) are immensely perplexed. Their volume change behaviour is attributed to a typical S-shaped swelling characteristics curve in the form of a three-phase swelling which can be further compartmentalized as preliminary, primary and secondary swelling stages1,2,3,4,5,6,7,8.
Firstly, the larger stresses in the form of swelling pressures (Ps), ASTM D4546, are generated when the volume change is blocked. The swelling pressure of ES (Ps-ES) is a fundamental parameter in estimating the behaviour of soft clays as well as an imperative characteristic of designing geotechnical structures7,9,10. According to Meshram et al.11, it offers comparatively better correlations using mineralogical, geotechnical and microfabric characteristics. Several direct and indirect techniques are available to predict the Ps-ES such that the latter methods are based on experimental results and engineering judgement. Furthermore, Du et al.12 and Yin et al.13 suggested that to characterize the Ps under various conditions, numerous predictive models have been developed, such as Gouy–Chapman diffused double layer models, heat-driven/energy-related models, and data-driven/hybrid models are the three types of existent models14. Secondly, the unconfined compressive strength of ES (UCS-ES) is a desideratum for various parameters used in road design, primarily for highway construction15,16,17. Also, the brittle behaviour of the ES yields low tensile strength thus leading to lesser UCS, and ASTM D2166, which could be improved by soil stabilization18. For instance, the UCS of lime-treated expansive soil increases at higher CaO content for various conditions, and additionally, the other engineering properties are also enhanced15,19,20. The highest UCS of CaO-stabilized ES was recorded for the samples compacted at their optimum moisture content (OMC)15. While evaluating the UCS for various drying-wetting cycles, Wu et al.21 reported that the UCS-ES decreased by around 50% after the first drying-wetting cycle (i.e., UCS is inversely related to the drying-wetting cycles), whereas it perpetually increased at extended curing periods.
A rich amount of literature exists on the influence of the ES characteristics, (such as distribution of the grain sizes, consistency limits, compaction characteristics, and swelling, among others) on their mechanical properties. For instance, the plasticity index (PI) increases at higher montmorillonite content which ultimately increases the PsUCS-ES. This cohesive nature can be associated with the low specific surface area (SSA) with higher cation exchange capacity (CEC) value of the smectites in the ES22. Similarly, maximum dry density (MDD) is another major indicator of the compressibility of the ES, and its high value depicts larger UCS and lesser Ps, whereas the OMC behaves vice-versa23. Additionally, the natural water content (wn) also substantially impacts the swell-strength characteristics of various ES. At high values of the wn, more water enters the clay minerals which increases the swelling thereby leading to higher Ps and lesser values of the UCS-ES24,25.
Various machine learning (ML) algorithms approaches have been widely considered in the recent past that are capable of accurately predicting many real-world problems26,27,28,29. The recently developed AI techniques include artificial neural networks (ANNs)30, genetic-based programming31, eXtreme gradient boosting (XGBoost)32,33, multivariate adaptive regression splines (MARS)34, alternate decision trees (ADTs), logistic regression (LR), M5 model trees, genetic algorithm (GA) among others35,36. Giustolisi et al.37 classified the mathematical models, i.e., white, black, and grey box models (WBM, BBM, and GBM, respectively), such that the WBMs exhibit parameters based on physical laws which form accurate physical associations, but their hidden mechanism has not been fully understood. The BBMs incorporate regressive data-driven systems wherein the active associations are not known and require to be predicted. While the GBMs are methodical systems wherein a mathematical framework efficaciously determines the overall behaviour. In this regard, the ANN is classified as ‘BBM’ due to lesser transparency and their inability to form closed-form prediction equations38,39. The ML models are deployed to compute the PsUCS-ES, which are imperative for designing foundations as well as constructing pavements resting on swelling soils. In addition, these laboratory tests are time-consuming, whereas the problematic soils are found in over 40 countries across the globe31. The main advantage of the ANN approach to calculate PsUCS-ES is the capability to model complex, non-linear relationships between input variables and ES characteristics which lead to robust predictions compared to conventional methods The PSO is advantageous because of its rapid convergence ability, requiring only fewer parameters to adjust thus proving to be efficacious in dealing optimization problems40. GWO is advantageous owing to its balanced exploration and exploitation techniques that lead to enhanced convergence speed. Moreover, this algorithm is simple to implement and understand which renders it accessible to researchers and practitioners. SMA exhibits various merits because it is easy to implement, adaptable, and bio-inspired, and it explores the search space efficiently by simulating the growth and foraging behaviour of slime moulds. Finally, inspired by the hunting behaviour of marine predators the MPA is advantageous because of its diversity in maintenance, adaptive strategy, and efficient convergence which makes it suitable for various real-world applications40,41,42,43,44.
ANNs are computer programs which are used to estimate and categorize issues related to the information handling of the data45,46. They are inspired by the biological structure of our brain as well as the nervous system which directly captures the association between inputs and outputs, however, there is no empirical formulation yielded47,48. The formulated ANN model depicted that soil biochar composite having 5% biochar replacement yielded excellent results in lessening soil erosion. The ANN-based model forecasted the soil water characteristics curves reasonably well49. On the contrary, it was found by Das et al.50 that the SVM model outclassed the developed ANN models. In yet another study on Ps-ES and UCS-ES (known as PsUCS-ES), the results of ANN modelling yielded the most satisfactory values in terms of R-value in the case of training as well as testing datasets (TrD and TsD, respectively). The comparison results showed that both the GEP and ANN are efficient and robust methods to determine the PsUCS-ES31,51. Therefore this study incorporates five advanced optimization methods, such as PSO52, GWO53, SMA54, MPA43 alongside the ANN modelling to enhance the predictive capability. Ikizler et al.55 formulated an ANN model to estimate the horizontal and vertical Ps-ES. The ANN formulation decreases the number of laboratory tests thereby attaining cost-effectiveness and robustness. Kumar et al.32,56 used hybrid ANNs and deep learning-based simulation models (on 81 case histories of static pile load tests conducted in various regions of Vietnam) facilitating the safe and economical designs of eco-friendly piles. In a variety of geotechnical engineering systems, a lesser number of easily calculated input factors were used to model the unsaturated ES for the sake of predicting their mechanical behaviour57. In another finding, the modelling results of ANN estimated the mechanical properties of pond ash stabilized ES impressively (with a coefficient of correlation R ≈ 0.96)58. Recently, new empirical prediction models were developed by Jalal et al.31 for the determination of PsUCS-ES by deploying neural networks, i.e., ANN, adaptive neuro-fuzzy inference system (ANFIS), and genetic programming approach, i.e., GEP59,60. The results revealed that both the GEP as well as ANN are efficient methods to accurately compute PsUCS-ES. Furthermore, they suggested reliable and easy-to-use GEP equations for the prediction of PsUCS-ES are given in Eqs. (1) and (2), respectively.
where CF is clay fraction, Gs is specific gravity, MDD is maximum dry density, OMC is optimum moisture content, PI is plasticity index, SP is the swell percent, and wn is the natural moisture content.
The determination of Ps-ES is time-consuming while the prediction of the UCS-ES is also cumbersome from the standpoint of time and cost. Previously, the PsUCS-ES have been determined by developing a variety of correlations using traditional statistical analyses (including GEP and ANN) wherein smaller R-values were recorded and the results were also not optimized31,38. However, Jumaa and Yousif61 found that the ANN outclassed the GEP model by yielding comparatively accurate performance. From the standpoint of these uncertainties, the existing research utilizes ANN in conjunction with PSO, GWO, SMA, and MPA to improve the past models to determine PsUCS-ES. Hyperparameter optimization is critical in ML model development which ensures optimal efficiency by fine-tuning parameters such as learning rates and regularization strengths. It is noteworthy to mention that the hyperparameter optimization review often highlights its role in improving model robustness while addressing issues such as computational complexity and overfitting. They also take into consideration the emerging methods, for example, Bayesian optimization and evolutionary algorithms, that encompass more efficacious exploration of hyperparameter spaces for better model generalization as well as robustness. Furthermore, the ANN-optimized models developed in the current study by using easily determinable geomechanical properties corroborated by past research31,62,63,64. Note that, Ps and UCS were the two output predictor variables. The motive of the research was to optimize the ANN models using recently developed algorithms, and to compare the performance of the developed models, such as (i) ANN-PSO, (ii) ANN-GWO, (iii) ANN-SMA, and (iv) ANN-MPA, for the estimation of PsUCS-ES by deploying simple geotechnical tests.
Methodology
ANN
These are simple yet dependable algorithmic models40,41. To accomplish particular tasks, the ANNs try to mimic how the human nervous system and brain work. Their use has significantly increased in recent years across several technical disciplines. In addition, they have also been applied in evaluating different characteristics of the ES31. Their structure as well as functioning, such that a distinctive ANN structure comprises many processing elements (i.e., nodes) which have been arranged in layers (like input, output, and hidden layer/s) has been previously described. Note that the best-hidden layer can be found through the trial and error method65. The input value of the preceding layer \(({x}_{i})\) over every node is multiplied with the help of varying connection weight \(\left({w}_{ji}\right).\) The addition process of weighted input signals took place on every node alongside the addition of a threshold value \(\left({\phi }_{j}\right)\), too. After that, a non-linear transfer function \(\left(f((.\right))\) is used over the joint input \(\left({I}_{j}\right)\) for generating node output \(\left({y}_{j}\right).\) It is important to state that the transfer functions commonly employed are linear and/or sigmoidal66.
The output of a layer acts as input at nodes in subsequent layers whereas this procedure is iteratively repeated. The entire process is given in Fig. 1 whereas the pertaining formulae are expressed in Eqs. (3) and (4). The data is induced to the input layer after which the system weights must be attuned iteratively according to set guidelines for determining the best combination of weights via a ‘training’ procedure with the help of deploying Levenberg–Marquardt backpropagation approach. Finally, after sufficient training, the model is terminated when the changes in resulting error are minimal. Moreover, the entire data is divided into three distinct sets,
Architecture of developed ANN Model for estimation of PsUCS-ES in this study.
i.e., TrD, TsD and VdD. It is important to state that the ANNs use the training set to identify patterns in the data. Also, the network training evaluates the combination of weights \({w}_{ji}\) among different neurons for yielding a global minimum of the error function by(Eq. 5). Furthermore, the main objective of TsD aims at assessing the robustness of the trained network bt finally evaluating the VdD.
More information about the ANN algorithm and accompanying mechanism can be found in available literature22,41,47,64.
PSO
It is another evolutionary programming approach that is influenced by the flocking habits of birds as well as fish. This concept was given by Kennedy and Eberhart67 for the first time. The algorithm exhibits its roots in social psychology and artificial lifespan as well as engineering. Like other population-based metaheuristics, PSO has a “population of particles” that fly through the hyperspace solution via set velocities. Note that the velocities of each particle can be stochastically updated at each iteration based on the historical best location. A defined fitness function is used to derive both the particle as well as the best positions in the neighbourhood68.
In addition, each particle's motion naturally progresses towards the optimal or nearly optimal solution. At each iteration, the position of an individual particle can be adjusted accordingly. After that, the next generation swarm is produced based on revised particle locations seeing their individual best location (\({L}_{best}\)) and the entire swarm’s best position (\({G}_{best}\)) as depicted in Fig. 2. The positions of the particles and their velocities are computed by Eqs. (6) and (7):
where, \({V}_{i}^{t+1}\) and \({V}_{i}^{t}\) represent the particle \(i\) velocities in the case of iterations t + 1 as well as t, respectively. Similarly, \({Y}_{i}^{t+1}\) and \({Y}_{i}^{t}\) denote the ith positions in the case of iterations t + 1 and t, respectively. The parameters \(w,\) indicates the cognitive social effects, \({m}_{1} and {m}_{2}\) denote the inertial parameters, and \({n}_{1} and {n}_{2}\) correspond to the matrix of arbitrary numbers with range [0,1]. The \({L}_{best}\) and the \({G}_{best}\) in the following generation is obtained using Eqs. (8) and (9):
where \({n}_{s}\) represents the summation of particles in the swarm.
As the exploration for optimum solution progresses, the random and irregular movement of particles (swarm) in search space now closely replicates the swarm of mosquitoes. The main strength of adopting the PSO for complex real-life problems is that it is not largely influenced by non-linearity. Furthermore, PSO can exhibit better and faster convergence to optimum solutions in a variety of scenarios. It is computationally more exhaustive and robust than a variety of exact mathematical methods. However, like other metaheuristics, a key issue in applying PSO is to establish a reasonable trade-off between intensification (exploitation) as well as diversification (exploration). In recent years the algorithm has witnessed widespread applications such as power systems, traffic control71, geotechnical investigation72 and, rainfall-runoff modelling73.
GWO
Mirjalili et al.74 floated the concept of this swarm intelligence optimization approach (metaheuristic algorithm) for the first time. The GWO draws inspiration from the cooperative hunting behaviour observed in grey wolves75. Metaheuristic algorithms are designed to generate high-quality solutions from a random population. The generation takes inspiration from natural system behaviours and continues until a specific termination condition is fulfilled76. GWO is based on three key steps i.e., surrounding prey, hunting, and sand attacking prey. To mathematically simulate wolf leadership order, assume the finest solution is alpha (α), the preceding one is beta (β), and finally it is the delta (δ). All other possible solutions can be assumed as omega (ω).
During the hunt the grey wolves encircle prey; the following equations (Eqs. 10 and 11) are given to numerically simulate grey wolf encircling behaviour.
\(\overrightarrow{A}\) and \(\overrightarrow{\text{C}}\) are the coefficient vectors, t represents existing iterations, and the prey position vector is \({\overrightarrow{\text{X}}}_{\text{prey}}\), and the grey wolf position vector is \({\overrightarrow{\text{X}}}_{\text{wolf}}\). The calculation of vectors \(\overrightarrow{A}\) and \(\overrightarrow{\text{C}}\) is according to Eqs. (12) and (13);
where r1 and r2 are random vectors in the interval [0, 1], whereas a is linearly lowered from 2 to 0 throughout iterations.
Alpha (α) has usually guided the hunt, whereas, β as well as δ may take part in hunting occasionally. To mathematically model grey wolf hunting behaviour77, the first three optimal solutions are preserved, while ω are required to relocate by Eqs. (14) to (20).
The new solution appears to be positioned at random within α, β, and δ. It is to say that, the new solution position can be evaluated using these three best solutions. The position updating in GWO is presented in Fig. 3. GWO is advantageous to optimize problems because of its viable properties in contrast to other metaheuristics78. This metaheuristic algorithm is also known for its simplicity, scalability, and special capability to keep the appropriate balance between diversification and intensification. In recent years, GWO has been employed for numerous engineering implications79,80,81,82.
Position updating in Gray wolf optimization GWO, Adopted from42.
SMA
Li et al.54 introduced a modified stochastic optimization method, i.e., SMA, that entirely relies on the oscillating behaviour of slime mould (SM). The SMA independently follows the oscillation method, replicating the Physarum polycephalum activation and morphological changes of SM. This is done during exploration, searching and foraging all without finishing the lifespan. The SMA method incorporates highly customized and adaptive weights for modelling and generating true and false responses to the reaction of the SM. Thus, it creates the optimum path to link food using improved space exploration skills and great exploitation tendencies44,54,83.
The SMA optimization process operates in three distinct phases; (a) searching and approaching food using smell, (b) try wrapping the food as per the quality and composition of food, and (c) swinging and oscillating to seek a superior location54,84. The comprehensive mathematical explanation of every phase is examined in this section and is given in Fig. 4.
A pseudo-code-driven comprehensive flowchart of the slime mould algorithm (SMA).
1st Phase (Searching and approaching food)
In the first phase, the SM seek and approach food owing to its odour in the atmosphere as mathematically expressed by Eqs. (21) to (22).
When, \(r<q\), then;
When, \(r\ge q\), then;
where, \({Y}_{i}\) refers to the location and orientation of the SM in the current cycle (\(t\)). \({Y}_{A}\) and \({Y}_{B}\) are two arbitrarily selected SM entities with weight (\({W}_{t}\)). \({Y}_{b}\) depicts the position of an entity with maximum saturation and concentration of odour. \({x}_{c}\) is the factor which lowers down linearly from 1 to 0. The other additional parameters such as \(q\), \({x}_{b}\), and \(b\) are specified in Eqs. (23) to (25).
where, \({S}_{i}\) as well as \(F\) indicate fitness of \({Y}_{i}\) and best performance among the total iterations completed, respectively. The \({W}_{t}\) can be explicitly stated in Eq. (26).
where, \(r\) shows the randomized variable between 0 and 1. \(WF\) and \(BF\) indicate the worst and optimum fitness within the latest iteration or cycle. The \(smell index\) shows the arranged collection of best fittest scores, given as Eq. (27).
2nd Phase (Wrapping food as per quality)
In the second phase, the vascular tissues of SM are squeezed. The \({W}_{t}\) of the space is regulated. The exploration and research of additional locations are conducted in this phase. When the bio-oscillator produces stronger and greater waves the cytoplasm starts travelling faster and the thicker and bigger vein receives the heavily saturated, concentrated and healthy food. With the rise of highly concentrated food, the \({W}_{t}\) of the search space rises and it is reduced owing to the low concentration. The algebraic interpretation of this phase is provided in the form of Eqs. (28) to (30).
where, \({V}_{min}\) and \({V}_{max}\) show the searching region from minimum to maximum value, respectively.
3rd Phase (waving and oscillation)
The SM depends completely on the propagation and amplification of waves produced during biological activity, such as changing the cytoplasmic flow in the veins. The \({x}_{b}\) varies in the range [− b, b]. It gradually approaches zero with the progression of the algorithm, as the number of iterations increases. While \({x}_{c}\) oscillates between [− 1,1] and it also eventually approaches zero.
Total net level of complexity of SMA:
The total net level of complexity of SMA comprises the complexity of the initialization process, performance assessment, strength or weight transformation and positioning54. Mathematically it can be provided in Eq. (31).
where, \(m\) and \(d\) denote the maximum cells in the SM and the dimensionality of features, respectively.
The absence of an acceleration and mutation strategy may limit the wide-scale adoption of the SMA85. Furthermore, it also lacks in offering feature extraction when performed in the binary versions of algorithms.
MPA
Faramarzi et al.43 presented a novel marine predator algorithm (MPA), which works on the effective swarm-inspired metaheuristic. Unlike other evolution algorithms, swarm-inspired algorithms adapt and generate new approaches which are differentiated mainly by their ability to search across many networks for the best response86. As shown in Fig. 5, MPA pertains to general foraging tactics of aquatic and marine creatures, like Brownian motion and Levy flight of prey and predator (inspired organisms). It is followed by the optimal encounter rate strategy of biological predator–prey interactions43. The predator forages and eats, whilst the prey gets eaten. The ease and simplicity of the velocity-based MPA approach, along with its excellent performance make it a viable substitute for traditional optimization algorithms43.
A pseudo-code-driven comprehensive flowchart of the marine predator algorithm (MPA).
Similar to the vast number of population-based metaheuristic algorithms, MPA is initialized with uniform allocation of the objective function and initial response in a search space, as expressed in Eq. (32).
\(R\) is the evenly distributed vector on a random basis with a value ranging from 0 to 1 with \({V}_{min,j}\) and \({V}_{max,j}\) representing the minimum and maximum limits of the variable value to be assessed, respectively. In a search space, \(d\) and \(m\) indicate the highest dimension and total agents, respectively. \({Y}_{i,j}\) indicates the randomized matrix of the solution set PIcked randomly having \(m\times d\) dimensional space.
As per the existence of the fittest concept, the best predators who are better at exploring, foraging and searching for prey are permitted to assemble an elite matrix to record cost-function data, as shown in Eq. (33).
Both prey and predators are working as search agents, simultaneously. When the predators explore their prey, the prey simultaneously looks for its feed. Thus, the \({E}_{lite}\) is revised in the end stage of each loop if a leading predator is substituted with a healthier one.
Prey is a separate distinct matrix, equal in dimension to the Elite, that predators have access to change their positions. In short, the initiation of the algorithm produces the first prey, with the finest (predator) evolving into the Elite. Thus, another Eq. (34) is used to describe the prey matrix.
The optimization of MPA includes three phases for revising, modifying and updating the original response with the search space, which are closely linked to the two foregoing matrices. All three phases are evaluated by the predator–prey velocity ratio. The first, second and third phase refers to a high, unit as well as low-velocity ratio, respectively. The comprehensive mathematical explanation of each stage is given below:
1st Phase (exploration with high velocity)
After the completion of one-third of the total iterations, the predators explore and switch locations quicker than the prey with a high-velocity ratio. Following Eq. (35), the mathematical expression for the exploration can be written as Eqs. (36) and (37).
When;
Then;
\({R}_{b}\) is the randomized vector for representing the normally distributed Brownian motion. While \(I\) and \({I}_{max}\) describes the present and maximum possible iteration, respectively.
2nd Phase (evolution from exploration to exploitation with unit velocity)
In this phase, the space exploration is transitorily converted to exploitation and both the prey and predator alter location at similar velocity (with velocity-ratio ≈ 1.0). It occurs between one-third and two-thirds of the total iterations. However, if the prey is adopting Levy flight, then the most appropriate motion for the predator is Brownian motion, thus, the population is separated into two. Following Eq. (38), for a first and second half part, the step size and the position of prey can be mathematically expressed as Eqs. (39) to (40) and Eqs. (41) to (43), respectively.
When;
For the first semi-population;
For other semi-populations;
\({R}_{l}\) is the randomized vector for representing the normally distributed Levy flight and \(F\) is the adaptable variable governing the Brownian movement of predators.
3rd Phase (exploitation with low velocity)
In the final stage of optimization, when the current iteration surpasses two-thirds of the total iterations, the perfect exploitation occurs. Unlike, the first phase, the predators switch their locations considerably more gradually than the prey with lower velocity-ratio. By Eq. (44); the completely altered position of predators adopting Levy flight is mathematically expressed as Eqs. (45) to (46).
if;
then;
Eddy's formation with possible impact
MPA incorporates the formation of eddy's and uses Fish Aggregating Devices (FADs) to find an alternative response to the influence of natural and environmental variables and, as a result, modify the predator behaviour87,88, as can be seen in Eqs. (47) to (50);
if;
then;
if;
then;
\(p\) denotes the FADs probability and \(X\) is the binary vector response. The subscripts \(r1\) and \(r2\) are representing the random locations of the prey matrix (\({Y}_{i}\)).
Marine memory
Marine predators are extremely proficient in recognizing the region of productive foraging. As a result, the marine working memory function is also assessed in the MPA optimization process43. The ultimate focus of this new function is to eliminate local points and recall the previous finest position to assist agents in increasing uniform convergence89,90.
As previously explained, MPA is referred to as solely a velocity-driven method, therefore introducing a binary multi-objective alternative can be a major enhancement43,91. Finally, Fig. 6 illustrates the construction steps of hybrid ANNs deployed in the current research to evaluate the PsUCS-ES. This figure outlines the process of using ANNs combined with swarm intelligence algorithms for optimizing the modelling of ES. It begins with the initialization of the swarm size and ANN parameters, followed by setting the metaheuristic parameters for algorithms such as PSO, GWO, SCA, and MPA approaches. Furthermore, this process includes testing the metaheuristics with ANN and selecting the best fit, which is then utilized to calculate the optimized weights and buses for the ANN model. Each model was ranked based on its performance in training and testing, with the highest scores given to the top performers and the lowest to the underperformers for each metric. The final score for each model was calculated by summing these individual rankings. Ultimately, the combined scores from both phases determined the model's overall ranking32. As a result, this leads to a sustainable construction approach by enhancing the understanding of the swell-strength nature of the problematic soils.
Flowchart for the mechanism of hybridization process of the metaheuristics used in the current study (With slight modifications after92).
Data processing and analysis
Data preprocessing
To formulate the prognostic models, 168 and 145 observations of Ps and UCS, from 61 and 99 internationally published papers (Table 1), respectively, were considered. In addition, nine basic soil characteristics were collected from two separately developed databases. The original database was constructed after an extensive literature study by initially recording 250 datasets (for Ps-ES) and 190 datasets (for UCS-ES). After that, easy-to-determine geotechnical parameters were recorded for developing models to predict the swell-strength properties of ES. After the collection of all data points, numerous ANN trials were run to evaluate the validity. The data points that diverged substantially (around 20% or more) from the general trend were ignored (i.e., 82 records for Ps and 45 records for UCS). Therefore, 168 observation points of Ps-ES and 145 points of UCS-ES were finally deployed to formulate the hybrid models. The important factors affecting the PsUCS-ES were investigated based on a recent literature review. However, the swell percent, MDD and OMC for some cases were absent and correlations were used to determine the missing values. Similarly, an average value of Gs was considered for some of the datasets31. Additionally, the contribution of Gs (between 2.3 and 2.8) on the PsUCS-ES was negligible owing to its small range, however, it was considered by Akan and Keskin93 for predicting the UCS-ES. The information related to several other geotechnical factors was scarcely present in the existing literature for several datasets. As a result, it could significantly reduce the total number of observations. Also, it may affect the generalization capability of the predicted models. As a result, these parameters were omitted in the development of models in the current study.
Descriptive statistics and statistical visualization
Table 2 presents the descriptive statistics of the considered input as well as the two outputs such that these geotechnical indices are observed to affect the PsUCS-ES. It is shown in Table 2, that the PsUCS-ES range between 12.5 and 521 Kpa, and 6.4 and 1060 kPa, respectively. Additionally, wn and sand content values for the Ps have not been included because their impact is lesser for the given range of data. Note that the wn of the ES is different at different temperatures and drying times. However, the motive for selecting the wn as an input parameter is due to its close association with the plastic limit and a variety of environmental factors. According to Patel94, the swelling capacity of ES primarily relies on its mineral composition, as well as the moisture content and density present in its natural environment. In general, clays with PI > 25, LL > 40, and wn near the PL or less may witness higher expansion. Also, the ES are problematic owing to their mechanical behaviour which is largely hydrophilic95. Also, the Pearson correlation coefficient (r) calculated for the wn was − 0.23293 for UCS-ES. It is known that the r-values illustrate a higher share of changes in the engineering characteristics of the ES. Moreover, the values given in Table 2 are suggested for the evaluation of PsUCS-ES using the aforementioned computational intelligence models in the current research study. The efficacy and robustness of the formulated models is significantly affected by the dispersal of various data points47. Moreover, to envisage the association among the ES input factors, graphical plots are given in Figs. 7 and 8 which depict the distribution histograms of various input factors as well as the two outputs (Ps and UCS), respectively The distribution of input data for Ps-ES is shown as a box plot in Fig. 9a which shows the 25% to 75% data distribution alongside the visual interpretation of the mean and median of the given dataset. Similarly, the box plot for the Ps-ES is manifested in Fig. 9b. Most of the data points considered in this study vary between 70 and 200. Secondly, the distribution of input data for UCS-ES is supplemented with a box plot shown in Fig. 10a which shows the 25% to 75% data distribution alongside the visual interpretation of the mean and median of the given dataset. Similarly, for the UCS-ES, the box plot is manifested as Fig. 10b. Most of the data points considered in this study vary between 100 and 300 MPa.
Distribution histogram of (a) clay fraction CF, (b) liquid limit LL, (c) plasticity index PI, (d) specific gravity Gs, (e) maximum dry density MDD, (f) optimum moisture content OMC, (g) swell percent SP, (h) natural water content wn, (i) sand, (j) silt, and (k) swell pressure Ps.
Distribution histogram of (a) clay fraction CF, (b) liquid limit LL, (c) plasticity index PI, (d) specific gravity Gs, (e) maximum dry density MDD, (f) optimum moisture content OMC, (g) swell percent SP, (h) natural water content wn, (i) sand, (j) silt, and (k) unconfined compression strength UCS.
Box plots of input parameters and the output value (i.e., Swell Pressure Ps).
Box plots of input parameters and the output value (i.e., Unconfined compressive strength UCS).
The Spearman rank correlation coefficient for Ps UCS-ES has been plotted in Fig. 11a,b, respectively. One of the most widely employed measures of relationship is Pearson's correlation coefficient, which is generally given by r31,96. In the current research, nine parameters were selected to model Ps UCS-ES to avoid further complexity of the developed models. Note that, the Ps-ES is largely governed by all parameters especially CF (r = 0.64), OMC (r = − 0.60) and PI (r = 0.45), while, UCS-ES is significantly influenced by sand content (r = 0.58), MDD (r = 0.47) and OMC (r = − 0.39), respectively. By and large, a high correlation prevails in the PsUCS-ES in the case of all input factors here.
Correlation coefficient matrix results plotted for swell pressure as well as UCS of the expansive soils (PsUCS-ES).
AI-based analysis
The collected databases (168 instances for Ps, and 145 instances for UCS) were distinctly distributed as TrD and TsD. Note that the testing was performed to check the accuracy and robustness of the trained model using unseen data. Therefore, 70% of the dataset was selected randomly as the TrD, while the remaining 30% dataset was employed to test and validate the formulated models. Taherdangkoo et al.97 developed an efficient neural network model to determine the maximum Ps of clayey soils by partitioning the dataset into ratios of 70:30. Several other studies in the same field follow the same partitioning ratio31,98,99.
To evaluate the performance of the formulated models, commonly employed performance indices such as MAE, NSE efficiency, Pi, R2, RMSE, RSR, VAF, WI, and WMAPE were determined100,101,102. The formulae of these indices can be expressed as Eqs. (51) to (59), respectively:
where \({y}_{i}\) and \({\widehat{y}}_{i}\) refer to actual and predicted ith values, \(n\) means data samples in a dataset, \({y}_{mean}\) refers to the average of the actual values whereas \(p\) means the total input parameters.
Results and discussion
This section presents the detailed results of the developed models to predict the PsUCS-ES. For both the target variables, similar nine attributes, namely, clay fraction CF, liquid limit LL, plasticity index PI, maximum dry density MDD, optimum moisture content OMC, swell percent SP, natural water content wn, sand and silt acted to be the input parameters, as mentioned earlier. As a result, 168 experimental results for Ps-ES and 145 records of UCS-ES were employed. Initially, 70% of the data was utilized as the TrD, whereas the remaining data was separated into validation dataset (VdD) and training dataset (TsD). Subsequently, the performance of the formulated models was validated and tested with the help of the aforementioned performance indices. Moreover, the comparison of robustness as well as the general performance of the formulated models is also described. Finally, statistical testing and uncertainty analysis (UA) were performed to determine the overall performance of the ANN-based models.
Configuration of ANN hybrid models
It is a desideratum to initially determine the optimum hyperparameters for the development of ANN-based models which is generally established using a trial-and-error procedure103. The optimum number of neurons achieved from trials for both Ps-ES and UCS-ES models varied from 8 to 14, as listed in Table 3. The maximum number of iterations (k), as well as swarm size (ns), were kept constant during modelling at 500 and 50, respectively, to compare the developed models.
For developing ANN-PSO hybrid models, first of all, the ANN was initialized using RMSE as a fitness function, and then the PSO algorithm was deployed for optimizing hyperparameters of the ANN. After that, ANN was initialized with 10 input neurons, 10 neurons in the hidden layer, and one output neuron for modelling the Ps-ES. On the contrary, for UCS-ES modelling, 11 neurons were used in the hidden layer to constitute 121 and 133 weights and biases for PsUCS-ES models, respectively. The optimum hyperparameters for PSO were set equal to 0.30, 1, and 2 as inertial weight (w), social coefficient (c1), and acceleration coefficient (c2), respectively.
In the case of ANN-GWO hybrid models, the wolf group was kept equal to 50 individuals. The number of inputs, hidden, and output neurons were adopted such that 97 and 121 weights and biases were obtained in the case of PsUCS-ES models. Based on the hidden neurons, the number of optimized weights as well as biases in the case of ANN-SMA and ANN-MPA are 145 and 157 for Ps-ES models whereas, 145 and 169 for UCS-ES models, respectively. The deterministic parameter “z” for the ANN-SMA was adopted as 0.20, whereas for ANN-MPA, Fish Aggregating Device (FAD) and P were set as 0.20 and 0.50, respectively, as listed in Table 3. Note that, the process for training the metaheuristic model is identical; however, the values of weights as well as biases in the case of the developed model are not the same in each case.
The convergence of the algorithm in searching local optima may be trapped; therefore, it is essential to investigate the merging behaviour of the optimization algorithm in assessing the robustness of the developed model. Furthermore, Fig. 12 as well as Fig. 13 display the convergence curves in the case of developed hybrid models (Ps-ES and UCS-ES, respectively). It is evident that ANN-PSO and ANN-GWO converge faster (almost equivalent) as compared to the other models, however, ANN-MPA surpasses other models in achieving higher accuracy. It is because the percent difference between ANN-PSO as well as ANN-GWO models is merely 1.5%, in contrast to the 15.11% and 64.58% difference in the case of ANN-SMA as well as ANN-MPA hybrid models, respectively. Moreover, the computational cost for the developed models using MATLAB was observed as 192.74 s, 189.87 s, 224.24 s, and 376.59 s in the case of ANN-PSO, ANN-GWO, ANN-SMA, as well as ANN-MPA, respectively, for 500 iterations of Ps-ES models. Similarly, for UCS-ES models, these values were recorded as 192.71 s, 194.18 s, 211.66 s, and 383.78 s, respectively. It is also stated that the number of iterations were finalized for the sake of comparison and this is why the local results were only derived. The curves show that further iterations may not significantly alter the accuracy of formulated models.
Convergence curves of hybrid ANN models in estimating Ps-ES.
Convergence curves of hybrid ANN models in estimating UCS-ES.
Performance evaluation of the formulated models
This portion evaluates the accuracy analysis of the formulated models by the statistical evaluation equations (Table 4 and Table 5)104. The performance evaluation of TrD is presented. The performance level for the developed models of Ps-ES was recorded in the range of 79.54% (R2 = 0.7954) to 85.4% (R2 = 0.854) in terms of coefficient of determination. Similarly, the UCS-ES models yielded an accuracy of 80.07% (R2 = 0.8007) to 86.22% (R2 = 0.8622). The TrD of both the developed models manifested a correlation (R greater than 0.8 which reflects a strong fit to the observed data points105,106. The results of the ANN-MPA and ANN-GWO (for Ps-ES), and ANN-MPA (for UCS-ES) were found to have R2 exceeding 0.80, and therefore they are considered to be yielding the best performance, i.e., low error indices. On the contrary, the ANN-PSO and ANN-SMA were observed to yield comparatively lower values while computing the swell-strength characteristics of the ES. The best R2 values in the case of the ANN and ANN-MPA modelling can be summarized as: (R2train of ANN = 0.864 and 0.9409, R2train of ANN-MPA = 0.8541 and 0.8624, and R2test of ANN = 0.7832 and 0.7921, R2test of ANN-MPA = 0.8796 and 0.8799). Furthermore, overfitting can be observed in ANN modelling of the PsUCS-ES where the testing R2 in both the cases is below 0.8. However, this issue is refined and the results have having higher degree of accuracy in ANN-MPA modelling where the training and testing R2 are almost equivalent.
The values of MAE were calculated in the range of 5.63% to 6.71% and 5.11% to 6.49% for the TrD of Ps-ES and UCS-ES models, respectively. RMSE values were recorded in the acceptable range of 7.10% to 8.42% and 6.66% to 8.1% for Ps-ES and UCS-ES models, respectively. The results reveal that ANN-MPA outperforms other models from the viewpoint of correlation as well as accuracy. The maximum values of R2 were obtained for the ANN-MPA as 0.854 and 0.8624 for PsUCS-ES models, respectively. Moreover, the lowest MAE (5.63% and 5.11%) and RMSE (7.10% and 6.66%) were also obtained for PsUCS-ES, respectively, in the case of ANN-MPA models. Apart from correlation and mentioned errors, the models were also evaluated using the Nash–Sutcliffe (NS) performance index. The values for NS (in ANN-MPA models) were recorded in the range of 0.79 to 0.8622, with the maximum value of 0.854 and 0.8622 for PsUCS-ES, respectively. The values of NS > 0.75 are found to yield excellent performance. Hence, the currently developed models also manifest strong goodness of fit.
The accuracy of the formulated models was also evaluated with the help of an error histogram and slope of the regression line obtained using the plot of experimental to predicted results, as shown in Figs. 14, 15, 16, and 17 (Ps-ES) and Figs. 18, 19, 20, as well as Fig. 21 (UCS-ES), respectively. It is evident that the scatter of data points for all the developed models mainly lies within the slope of ± 20% deviation from the best-fit line, which also represents the close agreement of predicted and actual results31. The error histogram showed 78%, 88%, 82%, and 85% of the TrD of Ps-ES models within ± 10% relative error for ANN-PSO, ANN-GWO, ANN-SMA, and ANN-MPA, respectively. Similarly, UCS-ES models yielded 85%, 90%, 78%, and 89% of the predictions within ± 10% relative error for ANN-PSO, ANN-GWO, ANN-SMA, as well as ANN-MPA, respectively.
Illustration of performance through scatter plots and error histograms (ANN-PSO of Ps-ES prediction).
Illustration of performance through scatter plots and error histograms (ANN-GWO of Ps-ES prediction).
Illustration of performance through scatter plots and error histograms (ANN-SMA of Ps-ES prediction).
Illustration of performance through scatter plots and error histograms (ANN-MPA of Ps-ES prediction).
Illustration of performance through scatter plots and error histograms (ANN-PSO of UCS-ES prediction).
Illustration of performance through scatter plots and error histograms (ANN-GWO of UCS-ES prediction).
Illustration of performance through scatter plots and error histograms (ANN-SMA of UCS-ES prediction).
Illustration of performance through scatter plots and error histograms (ANN-MPA of UCS-ES prediction).
Furthermore, a few other visual representations, such as Taylor diagrams as well as the Accuracy matrix, are also given to assess the performance of the formulated ANN-based models. The former refers to the mathematical 2-D representation of the comparative evaluation of the model from the standpoint of root mean squared error (RMSE), R (between predicted and experimental values), and the ratio of their standard deviation. Each model is identified within the diagram by a marker, character, or point, which quantifies its evaluation on a linear and radial scale. The position of the marker depicts the model performance; the closer the marker is to the reference point, the higher the accuracy of the developed model. Figure 22 manifests Ps-ES models with R values > 0.8, representing a strong agreement among observed as well as predicted values. The correlation values for UCS-ES models are also ≥ 0.78, depicting a good fit to experimental results (Fig. 23). The marker points of almost all the models are in proximity to reference points, however, the ANN-MPA being the closest one, represents a relatively more robust model.
Taylor diagrams: (a–c) for Ps-ES modelling and (d–f) for UCS-ES modelling.
Illustration of WCB values: (a) for Ps-ES modelling and (b) for UCS-ES modelling.
For evaluating the accuracy of the formulated models, the accuracy matrix is also presented in Figs. 24 and 25. The percentage accuracy of the model is expressed in terms of ρ relative to their ideal values.
Accuracy matrix for the hybrid ANN models in predicting Ps-ES.
Accuracy matrix for the hybrid ANN models in predicting UCS-ES.
For instance, ideal values for mean absolute error (MAE), RMSE, and R2 are 0, 0 and 1, respectively. Table 4 shows MAE, RMSE, and R2 for the ANN-MPA Ps-ES model observed as 0.0563, 0.0710, and 0.8541, respectively. Hence, the accuracy of the ANN-MPA is 94.37% (100–5.63), 92.51% (100–7.10) and 85.41% from the viewpoint of MAE, RMSE, and R2, respectively. Correspondingly, the accuracy of the ANN-MPA model in the case of UCS-ES approaches 94.89%, 93.34%, and 86.24% in terms of MAE, RMSE, and R2, respectively.
Validation of the developed models
The attainment of higher accuracy of the VdD indicates a more robust and accurate model. Therefore, in this study, the developed models were validated with the help of two levels of validation. Firstly, 30% of the unused data separated from the main dataset was divided equally among TsD and VdD. In the second level of validation, a simulated dataset was used for parametric analysis, which is presented to see the effect of variable change and its impact on the PsUCS-ES.
First level validation
A portion of the primary dataset was used in K-fold cross-validation having K = 5 to validate the ANN-based formulated models. The statistical evaluation of all the proposed models is furnished in Tables 4 and 5 for PsUCS-ES models, respectively. The results reveal that ANN-MPA manifest a more robust model, yielding R2 = 0.8826, RMSE = 0.0701, and MAE = 0.0568 and R2 = 0.8766, RMSE = 0.066, as well as MAE = 0.0548 for TsD and VdD respectively, for predicting Ps-ES. Similarly, ANN-MPA manifested R2 = 0.8608, RMSE = 0.0695, as well as MAE = 0.0567 for test data and R2 = 0.8990, RMSE = 0.0511, and MAE = 0.0361 for VdD, in the case of UCS-ES model. It is pertinent to mention that the magnitudes of correlation are greater, whereas the magnitude of errors for the test and VdD lies below the TrD, which represents no overfitting during the training stage of the ANN-MPA. Figure 14b,c also depicts that most of the prediction of the ANN-MPA lies in between ± 20% of the deviation of the best-fit line. In the case of Ps-ES models, the performance of other models is depicted in Figs. 14, 15, 16, and 17, which reflects that the accuracy of the formulated models is equivalent to ANN-MPA. ANN-GWO furnished second robust results in forecasting the Ps-ES, whereas, the UCS-ES prediction exhibited overfitting in the training process (Figs. 18, 19, 20, and 21, respectively).
Uncertainty and statistical testing
The credibility evaluation of a typical AI model is necessary in the case of a prediction model to estimate the target variable for the new dataset. The current study employed UA to evaluate the quantifiable assessment of errors of the developed models to predict PsUCS-ES. This analysis was performed on 1st level of validation, i.e., on the TrD, TsD, and VdD, including 118, 25 and 25 for Ps-ES and 101, 22, and 22 experimental results for UCS-ES, as listed in Tables 6 and 7, respectively. To perform the UA, an absolute error was initially calculated between the predicted and experimental values for all three datasets. Subsequently, the mean of error (MOE) and standard deviation (SD) were computed for the said data. Furthermore, the margin of error (ME) was determined at a 95% confidence interval to yield the width of confidence bound (WCB). Upper bound (UB), lower bound (LB), as well as standard error (SE) were also determined to compute WCB. The results of WCB for the formulated models have been provided in Tables 8 and 9 for Ps-ES and UCS-ES, respectively.
Second level validation
The value of WCB for a good model shall be as small as possible; hence, the model with minimum WCB reflects a robust model.
For both cases, Ps-ES and UCS-ES, the ANN-MPA manifested minimum WCB, therefore, it ranked first in robustness for TrD, TsD, and VdD data, which is also depicted in Fig. 23.
Second-level validation
Owing to the overfitting problem while formulation of AI models, the models generated in this study were validated on different sets. For this purpose, simulated datasets were created as shown in Table 10. Moreover, as depicted in Fig. 26 the effect of changing parameters has been studied by keeping remaining variables constant. The details of the parametric and sensitivity analysis are given below.
Illustration of level-2 validation phase.
Parametric analysis
Table 10 illustrates the details of the simulated datasets produced alongside the fluctuating range of the considered input parameters107. It is pertinent to mention that, the summation of all the input parameters had been 100% the same everywhere to simulate the real-world scenario. Moreover, the LL, PI, Gs, MDD, OMC, SP, wn, sand, and silt were designated at their minimum, maximum, and mean entities.
It is depicted in Fig. 26 that, as anticipated, all the trends are shown by smooth curves. Figure 26a–e,g depict the expected increase in Ps-ES with rising CF, LL, PI, Gs and MDD, respectively, while, Fig. 26f, i–j displays the reverse decreasing trend in Ps-ES with increasing OMC, wn, sand and silt, respectively. These results are consistent with the R-value reflected by the given matrix in Fig. 11, as well as they are in good agreement with the findings of Jalal et al.31. However, the decrease in the Ps-ES at higher water content is associated with larger values of Ps-ES with LL (Fig. 26b), which is reflected by the Δw = 0.6(PI/LL)108. On the contrary, the forecasted UCS-ES elevated with increasing PI, MDD, OMC, and SP, as shown in Fig. 26m–q, while, it lowered down in the case of Gs and silt content, Fig. 26n,t, respectively, which are are in good agreement with the GEP parametric study results of Jalal et al.31. But, for OMC and SP, the Ps-ES was observed to follow an increasing trend after some time since the OMC and MDD are significantly influenced by the particle size being fine. It is stated that the greater impact is by the content of CF109. Figure 26k–l shows that the UCS-ES lowered down with the increase in CF and LL of the original ES110. Figure 26r showed that with the increase in soil water content, the UCS-ES was observed to decrease111. Also, the trends between swell-strength characteristics and the nine aforementioned input parameters attained in the level-2 validation stage are in good agreement with the behaviour of the actual dataset (as shown in Figs. 7 and 8, respectively), which verifies the robustness of the proposed model.
Sensitivity analysis
Sensitivity analysis (SA) evaluates the impact on the output of a formulated model with changing input parameters. It gives an idea about the most significant input parameters, and as a result, by eliminating the relatively trivial parameters, the number of inputs could be lessened, thereby lowering the perplexity of the model alongside the time required for training a specific model. To conduct the SA for the current study on PsUCS-ES, the generally employed cosine amplitude technique (referred to as, CAM) was incorporated wherein the data pairs assist in the construction of data array, = [x1,x2,x3,…, xi,…,xn], such that the variable xi in the array, X, refers to the length vector of m in the form of:
The association among Aij (strength of the relation) versus the datasets of xi as well as xj is determined with the help of Eq. (61):
The Aij values for PsUCS-ES versus the input parameters are depicted in Fig. 27. In the TrD of Ps-ES, the CF and MDD are the governing parameters whose effect exceeds 0.90 whereas the wn, sand and silt have the lowest impact on the Ps-ES. The results of ANN-SMA and ANN-PSO are higher than those of ANN-GWO for all studied input parameters except PI and SP. On the contrary, in the TrD, TsD, and VdD of UCS-ES, the MDD and sand appear to largely govern the strength of the ES. In the TrD of UCS-ES, the effect of PI, OMC, and wn is recorded to be the least, respectively. Furthermore, the efficiency of results with various algorithms in the TrD of UCS-ES case follows the order: of ANN-SMA > ANN-GWO > ANN-PSO. Similarly, in the case of TsD and VdD of Ps-ES, CF and PI are the most significant input parameters whereas wn, sand and silt are the least significant parameters. Interestingly, the efficiency of results is higher for ANN-SMA and ANN-PSO in the case of TsD (PsUCS-ES). However, ANN-PSO and ANN-SMA yield the lowest results for PsUCS-ES in the case of VdD. Hence, ANN-SMA exhibits the most reliable results for Ps-ES while ANN-SMA and ANN-PSO are equally efficient algorithms in the case of UCS-ES.
Sensitivity analysis of PsUCS-ES.
Summary and conclusions
In various civil engineering projects, the swell-strength properties of expansive soils (ES) are crucial for evaluating the design of structures resting on the ES. Usually, laboratory tests are conducted for computing the swell pressure as well as the unconfined compression strength of the ES (referred to as ‘PsUCS-ES’) which are not only time-consuming but also expensive. Thus, this study aims to find a robust and efficacious alternative to conduct the actual laboratory tests with efficient AI-based models. This would help to estimate the PsUCS-ES based on available experimental databases from the past literature. This study concentrates on the formulation of metaheuristics by deploying PSO, GWO, SMA, and MPA for the evaluation of the PsUCS-ES obtained from ANN modelling. A database of 168 Ps and 145 UCS observations was considered by consulting 61 and 99 internationally published papers, respectively, after a detailed literature search. 70% of the dataset was selected randomly as the TrD, whereas the rest of the unused dataset was deployed to test and validate the developed models. Based on the aforementioned modelling, the following conclusions are drawn:1. All the models were trained using the best hyperparameters of the ANN model resulting from PSO, GWO, SMA, and MPA. In the case of Ps-ES modelling, the fixation of several neurons in the hidden layer is purely a trial-and-error method. Furthermore, the ANN models of PsUCS-ES using PSO were uniformly optimized with inertial weights equalling 0.3, social coefficient of unity, and acceleration coefficient of 2. The ANN-GWO metaheuristic (189.87 s) exhibited superior performance from the standpoint of computational cost, whereas PSO (192.71 s) surpassed in the case of the UCS-ES models.
UCS2. Validation of the ANN-based PsUCS-ES models using wide statistical indices (such as MAE, NS, ρ, R2, RMSE, RSR, VAF, WI, and WMAPE) was performed. It was recorded that all the developed models for Ps-ES exhibited R significantly exceeding 0.8 for the TrD, TsD, and VdD. However, ANN-MPA excelled in yielding high R values and exhibited the lowest absolute error for all these three distinct.
3. The results of UCS-ES models performance revealed that R only exceeded 0.9 in the case of TrD, but, not for TsD and VdD. Also, the ANN-MPA model yielded higher R values (0.89, 0.93, and 0.94), and comparatively low MAE values (5.11%, 5.67, and 3.61%) in the case of PSO, GWO, and SMA, respectively. UCSUCS.
4. All the ANN-base models were also tested using the a-20 index. For all the formulated models, maximum points were recorded to lie within ± 20% error. In addition, the ANN-SMA interpreted higher accuracy in terms of the a-20 index, and its superiority was also supported by the results depicted in Taylor’s diagram and the WCB values.
5. The uncertainty analysis UA for Ps-ES models showed that the ANN-MPA is observed to be the most accurate model followed by ANN-GWO, ANN-SMA, and ANN-PSO for the TrD. This type of trend was also recorded for the TsD and VdD except that ANN-PSO outperformed ANN-SMA. On the other hand, in the case of UCS-ES models, the ANN-MPA exhibited the highest accuracy followed by ANN-GWO, ANN-PSO, and ANN-SMA, for TrD. The parameter and sensitivity analyses of ANN-based PsUCS-ES models also revealed coherent variation of the considered input parameters with the outputs.
This study is limited to the range of the parameters mentioned in the available dataset considered in this paper. Also, the inherent time and cost attributed to the initial creation of the aforementioned experimental database are still challenging. The models formulated here are based on specific soil characteristics and environmental conditions. In addition, the presence of biases or inaccuracies in this database could affect the robustness of the developed models. The validation of these models is also limited to the existing database. Moreover, trial and error in model optimization, overfitting issues, and computational costs are other noteworthy limitations while developing models. It is suggested to evaluate other optimization techniques including random forest and support vector machines in future research.
Data availability
The data used in the manuscript may be provided upon requesting Fazal E Jalal (jalal@szu.edu.cn).
Change history
06 November 2024
A Correction to this paper has been published: https://doi.org/10.1038/s41598-024-76107-4
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FE Jalal contributed to the research project by conceptualizing the study, designing the methodology, and data collection process, conducting data analysis, and experimental design and implementation. He also played a significant role in interpreting the results and drafting the manuscript. Mudassir Iqbal provided critical insights during the analysis, and interpretation of the findings, and provided valuable feedback for improvement. His expertise significantly enhanced the quality and rigour of the research. He offered guidance and advice throughout the study. Waseem Akhtar Khan made substantial contributions to the literature review, gathering relevant research articles, and synthesizing the information. Arshad Jamal played a crucial role in the literature review section by gathering relevant research articles and contributing to an overview of the optimization algorithms. Kennedy Onyelowe played a significant role in interpreting the results and drafting the manuscript. Lekhraj contributed in reviewing, editing and writing the manuscript.
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The original online version of this Article was revised: In the original version of this Article author Arshad Jamal was incorrectly affiliated with ‘Department of Civil and Environmental Engineering, King Fahd University of Petroleum and Minerals, KFUPM, Box 5055, 31261, Dhahran, Saudi Arabia’. The correct affiliation is ‘Department of Civil Engineering, College of Engineering, Qassim University, Buraydah 51452, Saudi Arabia’ and author Mudassir Iqbal was omitted as a corresponding author. Correspondence and requests for materials should also be addressed to mudassiriqbal@uetpeshawar.edu.pk.
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Jalal, F.E., Iqbal, M., Khan, W.A. et al. ANN-based swarm intelligence for predicting expansive soil swell pressure and compression strength. Sci Rep 14, 14597 (2024). https://doi.org/10.1038/s41598-024-65547-7
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DOI: https://doi.org/10.1038/s41598-024-65547-7
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