ANN-based swarm intelligence for predicting expansive soil swell pressure and compression strength

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.

Firstly, the larger stresses in the form of swelling pressures (P s ), ASTM D4546, are generated when the volume change is blocked.The swelling pressure of ES (P s -ES) is a fundamental parameter in estimating the behaviour of soft clays as well as an imperative characteristic of designing geotechnical structures 7,9,10 .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 P s -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 P s 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 models 14 .Secondly, the unconfined compressive strength of ES (UCS-ES) is a desideratum for various parameters used in road design, primarily for highway construction [15][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 stabilization 18 .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 enhanced 15,19,20 .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. 21reported 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 P s UCS-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 ES 22 .Similarly, maximum dry density (MDD) is another major indicator of the compressibility of the ES, and its high value depicts larger UCS and lesser P s , whereas the OMC behaves vice-versa 23 .Additionally, the natural water content (w n ) also substantially impacts the swell-strength characteristics of various ES.At high values of the w n , more water enters the clay minerals which increases the swelling thereby leading to higher P s and lesser values of the UCS-ES 24,25 .
Various machine learning (ML) algorithms approaches have been widely considered in the recent past that are capable of accurately predicting many real-world problems [26][27][28][29] .The recently developed AI techniques include artificial neural networks (ANNs) 30 , genetic-based programming 31 , 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 others 35,36 .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 equations 38,39 .The ML models are deployed to compute the P s UCS-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 globe 31 .The main advantage of the ANN approach to calculate P s UCS-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 problems 40 .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 applications [40][41][42][43][44] .
ANNs are computer programs which are used to estimate and categorize issues related to the information handling of the data 45,46 .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 yielded 47,48 .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 well 49 .On the contrary, it was found by Das et al. 50that the SVM model outclassed the developed ANN models.In yet another study on P s -ES and UCS-ES (known as P s UCS-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 P s UCS-ES 31,51 .Therefore this study incorporates five advanced optimization methods, such as PSO 52 , GWO 53 , SMA 54 , MPA 43 alongside the ANN modelling to enhance the predictive capability.Ikizler et al. 55 formulated an ANN model to estimate the horizontal and vertical P s -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 behaviour 57 .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

ANN
These are simple yet dependable algorithmic models 40,41 .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 ES 31 .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 method 65 .The input value of the preceding layer (x i ) over every node is multiplied with the help of varying connection weight w ji .The addition process of weighted input signals took place on every node alongside the addition of a threshold value φ j , too.After that, a non-linear transfer function f ((. ) is used over the joint input I j for generating node output y j .It is important to state that the transfer functions commonly employed are linear and/or sigmoidal 66 .
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, 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 literature 22,41,47,64 . (1)

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 Eberhart 67 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 neighbourhood 68 .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):   www.nature.com/scientificreports/where, V t+1 i and V t i represent the particle i velocities in the case of iterations t + 1 as well as t, respectively.Similarly, Y t+1 i and Y t i denote the i th positions in the case of iterations t + 1 and t, respectively.The parameters w, indicates the cognitive social effects, m 1 andm 2 denote the inertial parameters, and n 1 andn 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 control 71 , geotechnical investigation 72 and, rainfall-runoff modelling 73 .

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 wolves 75 .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 fulfilled 76 .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.

− → A and
− → C are the coefficient vectors, t represents existing iterations, and the prey position vector is − → X prey , and the grey wolf position vector is − → X wolf .The calculation of vectors − → A and − → C is according to Eqs. ( 12) and ( 13); where r 1 and r 2 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 behaviour 77 , 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 metaheuristics 78 .This metaheuristic algorithm is also known for its simplicity, scalability, and special capability to keep the appropriate  www.nature.com/scientificreports/balance between diversification and intensification.In recent years, GWO has been employed for numerous engineering implications [79][80][81][82] .

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 tendencies 44,54,83 .
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 location 54,84 .The comprehensive mathematical explanation of every phase is examined in this section and is given in Fig. 4.
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 ≥ 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 concen- tration 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 smellindex 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.
( 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 positioning 54 .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 SMA 85 .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 swarminspired 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 response 86 .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 interactions 43 .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 algorithms 43 .
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 × 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.( 31) www.nature.com/scientificreports/ 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 vari- able 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; (34) www.nature.com/scientificreports/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 behaviour 87,88 , as can be seen in Eqs.(47)  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 process 43 .The ultimate focus of this new function is to eliminate local points and recall the previous finest position to assist agents in increasing uniform convergence 89,90 .
As previously explained, MPA is referred to as solely a velocity-driven method, therefore introducing a binary multi-objective alternative can be a major enhancement 43,91 .Finally, Fig. 6 illustrates the construction steps of hybrid ANNs deployed in the current research to evaluate the P s UCS-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 ( 45) Flowchart for the mechanism of hybridization process of the metaheuristics used in the current study (With slight modifications after 92 ).www.nature.com/scientificreports/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 ranking 32 .As a result, this leads to a sustainable construction approach by enhancing the understanding of the swell-strength nature of the problematic soils.

Data preprocessing
To formulate the prognostic models, 168 and 145 observations of P s 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 P s -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 P s and 45 records for UCS).Therefore, 168 observation points of P s -ES and 145 points of UCS-ES were finally deployed to formulate the hybrid models.The important factors affecting the P s UCS-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 G s was considered for some of the datasets 31 .
Additionally, the contribution of G s (between 2.3 and 2.8) on the P s UCS-ES was negligible owing to its small range, however, it was considered by Akan and Keskin 93 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 P s UCS-ES.It is shown in Table 2, that the P s UCS-ES range between 12.5 and 521 Kpa, and 6.4 and 1060 kPa, respectively.Additionally, w n and sand content values for the P s have not been included because their impact is lesser for the given range of data.Note that the w n of the ES is different at different temperatures and drying times.However, the motive for selecting the w n as an input parameter is due to its close association with the plastic limit and a variety of environmental factors.According to Patel 94 , 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 w n near the PL or less may witness higher expansion.Also, the ES are problematic owing to their mechanical behaviour which is largely hydrophilic 95 .Also, the Pearson correlation coefficient (r) calculated for the w n 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 P s UCS-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 points 47 .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 (P s and UCS), respectively The distribution of input data for P s -ES is shown www.nature.com/scientificreports/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 P s -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.The Spearman rank correlation coefficient for P s 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 r 31,96 .In the current research, nine parameters were selected to model P s UCS-ES to avoid further complexity of the developed models.Note that, the P s -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    www.nature.com/scientificreports/(r = 0.47) and OMC (r = − 0.39), respectively.By and large, a high correlation prevails in the P s UCS-ES in the case of all input factors here.

AI-based analysis
The collected databases (168 instances for P s , 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 P s of clayey soils by partitioning the dataset into ratios of 70:30.Several other studies in the same field follow the same partitioning ratio 31,98,99 .
To evaluate the performance of the formulated models, commonly employed performance indices such as MAE, NSE efficiency, P i , R 2 , RMSE, RSR, VAF, WI, and WMAPE were determined [100][101][102] .The formulae of these indices can be expressed as Eqs.( 51) to (59), respectively:    where y i and 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 P s UCS-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 w n , sand and silt acted to be the input parameters, as mentioned earlier.As a result, 168 experimental results for P s -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 procedure 103 .The optimum number of neurons achieved from trials for both P s -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 (n s ), 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 P s -ES.On the contrary, for UCS-ES modelling, 11 neurons were used in the hidden layer to constitute 121 www.nature.com/scientificreports/and 133 weights and biases for P s UCS-ES models, respectively.The optimum hyperparameters for PSO were set equal to 0.30, 1, and 2 as inertial weight (w), social coefficient (c 1 ), and acceleration coefficient (c 2 ), 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 P s UCS-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 P s -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, Table 3. Parametric configuration of the developed hybrid ANN models.

ANN-PSO ANN-GWO ANN-SMA ANN-MPA ANN-PSO ANN-GWO ANN-SMA ANN-MPA
n h 10 8 k 500 500 500 500 500 500 500 500  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 (P s -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 P s -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.

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 P s -ES was recorded in the range of 79.54% (R 2 = 0.7954) to 85.4% (R 2 = 0.854) in terms of coefficient of determination.Similarly, the UCS-ES models yielded an accuracy of 80.07% (R 2 = 0.8007) to 86.22% (R 2 = 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 points 105,106 .The results of the ANN-MPA and ANN-GWO (for P s -ES), and ANN-MPA (for UCS-ES) were found to have R 2 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 R 2 values in the case of the ANN and ANN-MPA modelling can be summarized as: (R 2 train of ANN = 0.864 and 0.9409,         www.nature.com/scientificreports/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    www.nature.com/scientificreports/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 P s -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, P s -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.www.nature.com/scientificreports/trained using the best hyperparameters of the ANN model resulting from PSO, GWO, SMA, and MPA.In the case of P s -ES modelling, the fixation of several neurons in the hidden layer is purely a trial-and-error method.Furthermore, the ANN models of P s UCS-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 (192.71s) surpassed in the case of the UCS-ES models.UCS2.Validation of the ANN-based P s UCS-ES models using wide statistical indices (such as MAE, NS, ρ, R 2 , RMSE, RSR, VAF, WI, and WMAPE) was performed.It was recorded that all the developed models for P s -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 P s -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 P s UCS-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.

Figure 1 .
Figure 1.Architecture of developed ANN Model for estimation of P s UCS-ES in this study.

Figure 5 .
Figure 5.A pseudo-code-driven comprehensive flowchart of the marine predator algorithm (MPA).

Figure 9 .
Figure 9. Box plots of input parameters and the output value (i.e., Swell Pressure P s ).

Figure 11 .
Figure 11.Correlation coefficient matrix results plotted for swell pressure as well as UCS of the expansive soils (P s UCS-ES).

Figure 12 .Figure 13 .
Figure 12.Convergence curves of hybrid ANN models in estimating P s -ES.

Figure 14 .
Figure 14.Illustration of performance through scatter plots and error histograms (ANN-PSO of P s -ES prediction).

Figure 15 .Figure 16 .
Figure 15.Illustration of performance through scatter plots and error histograms (ANN-GWO of P s -ES prediction).

Figure 17 .Figure 18 .
Figure 17.Illustration of performance through scatter plots and error histograms (ANN-MPA of P s -ES prediction).

Figure 19 .
Figure 19.Illustration of performance through scatter plots and error histograms (ANN-GWO of UCS-ES prediction).

Figure 20 .
Figure 20.Illustration of performance through scatter plots and error histograms (ANN-SMA of UCS-ES prediction).

Figure 21 .
Figure 21.Illustration of performance through scatter plots and error histograms (ANN-MPA of UCS-ES prediction).
Figure 22 manifests P s -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.

Figure 22 .
Figure 22.Taylor diagrams: (a-c) for P s -ES modelling and (d-f) for UCS-ES modelling.

Figure 25 .
Figure 25.Accuracy matrix for the hybrid ANN models in predicting UCS-ES. 0

Table 1 .
Researches ID and research references of the two expansive soil databases collected in this study.

Table 4 .
Details of performance indices for P s -ES during ANN-based modelling.

Table 5 .
Details of performance indices for UCS-ES during ANN-based modelling.

Table 6 .
Results of Uncertainty analysis (UA) for P s -ES during ANN-based modelling.

Table 7 .
Results of Uncertainty analysis (UA) for UCS-ES during ANN-based modelling.

Table 8 .
Results of one-tailed t-test for P s -ES during ANN-based modelling.

Table 10 .
Details of simulated datasets for P s UCS-ES for validation purposes.