Introduction

Many methods evaluate the pile settlement (PS) from empirical methods through experiments in an efficient way to generate the perspectives before operating projects, especially the bridged ones. Paying attention to assessing sensitive constructions dealing with people’s lives and considering intelligent solutions can help us find their actual conditions and calculate with low energy, time, and cost. Using computer-aid methods that are underlaid by the experimental data can give us a freer domain in terms of cost and time to realize the functioning of structures over the utilization period [1]. Several researchers have studied pile motion using the popular finite difference and finite element methods [2,3,4,5,6]. The key role of estimating the PS is played by the inputs of a model that wants to find the logical relationships between initial entering variables [7, 8].

However, testing on large models or full-scale experiments is required to adequately cover all aspects of pile-moving behaviour. Because these large-scale experimental researches are difficult to perform and expensive, many studies in the literature contain various simplified and numerical methods for predicting pile behaviour. Many experts have suggested a simplified method for predicting the initial pile motion responding to load settlement, given the elastic behaviour of the soil. In some simplified methods, piles are considered a means of reducing settlement [9], where the compacting ground soil layers support the pile structure’s bearing capacity, which is why the piles contribute to controlling the settlement rates. In another simplified method, the load is distributed between the piles and foundation regarding interactions between soil layers and pile [10]. Most simplified solutions do not adequately account for soil–structural interactions and cannot accurately predict the pile response to load settling.

In this regard, one article suggested a way to analyse the displacement of piles and presented the theoretical function for the examinations on earth pressure [11, 12]. Most experts assessed the PS, but all used the model that lacks the reflection of ground. However, artificial neural networks (ANNs) and techniques of machine learning have been operated in many types of research Lee and Lee [13], Hanna et al. [14], Liu et al. [15], Che et al. [16], and Shanbeh et al. [16]. Calibrating data for some research were used, and PS tests were done to generate the framework predicting the final capacity of the pile’s bearing. Samples from the in situ piles are selected to calibrate artificial intelligent (AI) developed models. The research used the technique of ANN to appraise the PS rates from some useful traits of piles socketed in rocks [17].

Several studies have broadly used methods related to functioning with regressions [18,19,20,21,22,23,24,25,26] regarding using regression methods such as stochastic regression machine of mini-max, adaptive regression of multivariate spline, and Gaussian trend regression [18, 23, 27, 28]. Also, many researchers use the powerful solution of support vector machines to compute the experimental parameters [29, 30]. The ability of this machine learning technique to estimate pile movements can be found in many studies with acceptable correlation results of more than 90 per cent [31, 32]. Further, one article evaluated the ultimate bearing capacity of piles using this method with desirable results [33, 34]. With this respect, the input dataset has properties of soil observed in a real field of projects, as the pile specimens should be provided.

Therefore, this study aims at estimating the PS factor as the key dependent variable in relation to many parameters of pile physical properties and the ground characteristics for the rocky region. For this reason, the support vector regression (SVR) as the means of model tries to indicate a perspective of pile movement over the operation period. As the novelty of the research and, secondly, estimating PS accurately, SVR is linked with the optimization algorithms to develop the SVR in the form of optimized frameworks with high accuracy besides the low-complex calculation net to compute PS in desirable conditions based on in situ measurements. SVR was coupled with marine predator algorithm (MPA) and the grasshopper optimization algorithm (GOA) to reach the goals. This study’s pile profiles and soil features data were collected from the Klang Valley Mass Rapid Transit (KVMRT) network in Kuala Lumpur, Malaysia.

Modifying the SVR-MPA and SVR-GOA models feeding data of KVMRT piles are used by the ratio of 70 for calibration and 30 per cent for validation, respectively. In this regard, the ratio of pile length to diameter, the rock hardness level showing with UCS, load masses effect over the pile, the standard penetration test (NSPT) results for rocks, the pile length beneath the soil, and the length of the pile in the rock was fed to models to estimate the PS for the megaproject of KVMRT [35]. Also, the indices of R, OBJ, VAF, MAE, and RMSE were opted to evaluate the impact of each proposed optimizer in fining pile settlement rates.

Materials and Methodology

Case Study: Klang Valley Rapid Transit System (KVMRT)

Kuala Lumpur (Malaysia) has confronted many congestion issues as a populated city. Several projects have been done to solve these and are on the table to check the feasibility of executions. The Klang Valley Mass Rapid Transit (KVMRT) project was designed and constructed to reduce the traffic masses. The project of KVMRT was built to pass Kuala Lumpur’s Federal Territory and Selangor State, especially joining areas within the region of Klang Valley. The 51 km of KVMRT lines contain 35 stations in which the underground and surface constructions are involved. The total length of the underground tunnel is about 9.5 km. This mega project of KVMRT in Kuala Lumpur to decrease the traffic jam issues, including many piles to supporting bridges, has been chosen as the case study. Figure 1 shows the KVMRT’s position in Kuala Lumpur.

Fig. 1
figure 1

The location of the study area KVMRT

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Initial Data Set of Case Study

The 96 piles’ information in the KVMRT project were gathered into rocks like limestone, phyllite, sandstone, and granite. The San-trias class granite rocks were used for the KVMRT project piles. Materials and subsoil datasets were collected to reach the information on geological profiles. Based on the assessments done, the ground comprises residual rock materials. According to the dataset collected, the bedrock is placed between the depth of 70 cm underground and more than 1400 m.

The methodology for collecting field data sets for materials and subsoil databases in infrastructure projects typically involves a combination of geotechnical investigation techniques, such as drilling, sampling, and testing. The collected data includes information on the type and properties of soil, rock, and other subsurface materials, as well as their depth and location. However, the following lines provide information on the field sampling and bore log data collected:

  • The observed rock masses range from moderately to largely weathered.

  • The highest and lowest values for UCS, as per ISRM, are 68 and 25 MPa, respectively [36].

  • Bore log data taken to a maximum depth of 16.5 MPa shows the presence of highly weathered soil. The primary soil type is a hard sandy mud with a minimum and maximum \({N}_{{\text{SPT}}}\) of 4 and 167 blows per 300 mm, respectively.

  • Subsoil materials with \({N}_{{\text{SPT}}}\) values greater than 50 blows per 300 mm are observed in most areas at ground depth levels between 7.5 and 27.0 m.

Based on the gathered data, the recorded rocks’ masses are weathered as moderate to extreme. However, the UCS parameter for the lowest and highest values of rocks was registered, respectively, from 25 to 68 MPa based on the parameter of ISRM [37]. Further, the parameter of \({N}_{{\text{SPT}}}\) was reported for at least 4 to a high value of 167, respectively, per 30 cm. That the rock types vary, going from 750 to 2700 cm. Moreover, according to the registered data of bores under the depth of 16.5 m, the soil type is extremely weathered, as well as the soil type in, which is highly muddy with sand.

Making the leading dataset with effective impacts was the primary step to creating the predictive frameworks. It is fundamental to show the foremost vital variables processing the initial data of the model. The mentioned tests were performed utilizing analysis settings by Pile Dynamic, Inc. For this respect, several variables opted to investigate the outcomes of pile geometry: (1) Pile length to the diameter fraction (Lp/D); (2) pile length in the soil to the pile length in the rock layer ratio (Ls/Lr); (3) NSPT of rocks; (4) Qu, that is ultimate potential bearing (directly affects the pile motion rate); (5) UCS of rock. These parameters feeding models were used to simulate the PS values in this regard. Table 1 indicates the summary of inputs used for the developed models.

Table 1 The feeding dataset for developed models
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To estimate the settlement rate of a pile, the parameters that influence it are the length and diameter of the pile. Two specific parameters, the ratio of the pile length in the soil layer to the pile length in the rock layer (\({L}_{{\text{s}}}\)/\({L}_{{\text{r}}}\)) and the ratio of the total pile length to the pile diameter (\({L}_{{\text{p}}}/D\)), are analysed to determine the impact of pile geometry. Additionally, the model input for predicting pile subsidence is the UCS, which has a significant effect. The value of \({N}_{SPT}\) is also considered as input to indicate the condition of the soil layer. The pile load is another important factor that directly affects the settlement, so the final pile-bearing capacity (\({Q}_{u}\)) is included as input as well. In summary, five variables are selected as inputs to evaluate the pile settlement (SP).

Also, Fig. 2 indicates measured inputs and target values (PSs) with a diagram in which each string shows one pile sample that, based on relevant PS, has a specific colour.

Fig. 2
figure 2

The input and target value diagram: a Lp/D, b Ls/Lr, c NSPT, d UCS, and e Qu, f PS

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Support Vector Regression, SVR

The machine learning way of support vector regression (SVR) was designed to compute the regression boundaries [30]. SVR is operated for sorting regressions, in which the error range (ε) is considered to determine regressions. Categorizing classes of regressions can be operated for creating hyperplane boundaries. The SVR learning technique worked as the supervised and found the regressions explained in Eq. (1) [38].

$${\text{min}}_{w,b} = \frac{1}{2}w^{2} + C\mathop \sum \limits_{i = 1}^{m} \left( {\xi_{i} + \xi_{i}^{*} } \right)$$
(1)
$${\text{s}}.{\text{t}}. \left\{ {\begin{array}{*{20}c} {y_{i} - \left( {w^{T} x_{i} + b} \right) \le \varepsilon + \xi_{i} } \\ {\left( {w^{T} x_{i} + b} \right) - y_{i} \le \varepsilon + \xi_{i}^{*} } \\ {\xi_{i} , \xi_{i}^{*} \ge 0} \\ \end{array} } \right\}$$

where the variable of \(\xi\) shows the boundary violations; \(w\) represents the weight of inputs; \(C\) denotes modifying a variable in order; \(b\) is the bias rate and denotes the rate of deviation from the hyperplane boundaries is shown with \(\varepsilon\).

The fitness function has 2 terms that are explained as follows:

$$\frac{1}{2}\left\| w \right\| ^{2}$$
(2)
$$C \mathop \sum \limits_{i = 1}^{m} \left( {\xi_{i} + \xi_{i}^{*} } \right)$$
(3)

Equation (2) was proposed for enhancing the area among the samples and boundaries of the hyperplane, afterwards preserving the intervals between the inputs with the boundaries. Equation (3) works like the modifying tool. The weight and bias were generated over solving the objective function as the target of the hyperplane’s boundaries [24]. This article uses the quadratic objective function to do the dedicated tasks accurately. The crucial SVR duty has been to figure out the key components as optimal magnitudes. To achieve the key factors mentioned, such as sigma, C, and ε needs a smart algorithm, the optimizing strategies of MPA and GOA were used to join with the SVR to appraise the above-mentioned factors at optimal rates.

Grasshopper Optimization Algorithm (GOA)

The grasshopper optimization algorithm (GOA), known as the colony base algorithm, can simulate the grasshoppers’ behaviour as insects to locate the best solution [39]. These creatures in adulthood reveal their colony base habits in going a long distance having traits of longer ranges and sudden motions. Equation (4) reveals the mathematical form of the habits.

$$x_{i} = S_{i} + G_{i} + A_{i}$$
(4)

In Eq. (4), the location of the grasshopper \(i\) is defined by \(x_{i}\) and the social interaction of grasshoppers is shown by \(S_{i}\).

$$S_{i} = \mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1} \\ {i \ne j} \\ \end{array} }}^{N} s\left( {d_{ij} } \right)\hat{d}_{ij} { },\;d_{ij} = \left| {x_{i} - x_{j} } \right|{ },\;\hat{d}_{ij} = \frac{{x_{i} - x_{j} }}{{d_{ij} }}$$
(5)

In Eq. (5), \(d_{ij}\) represents the distance among grasshoppers of \(i\) and \(j\); also \(\hat{d}_{ij}\) reveals the vector as being a unit among \(i\) th and \(j\) th grasshoppers. Equation (6) considers the strength of the social force (\(s\)).

$$s\left( x \right) = fe^{{\frac{ - x}{l}}} - e^{ - x}$$
(6)

where the parameters of \(f\) and \(l\) refer to the intensity of attraction and length, alternatively. Notably, the Nymph types of grasshoppers do not have wings, which leads to the wind being an important factor in moving direction.

$$A_{i} = u\hat{e}_{{\text{w}}} { },\;G_{i} = - g\hat{e}_{{\text{g}}}$$
(7)

where Eq. (7), the parameters of gravity force and wind advection of grasshoppers are mentioned with \(G\) and \(A\). The variables of \(u\) and \(g\) are known as, respectively, the wind drifts and the gravity constants. Moreover, the unit vector specifying the wind advection directions and the gravity force directions are represented via \({\widehat{e}}_{{\text{w}}}\) and \({\widehat{e}}_{{\text{g}}}\). Consequently, by contributing Eqs. (5)–(7), grasshoppers’ behaviours can be formed with the below relation.

$$X_{i} = \mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1} \\ {i \ne j} \\ \end{array} }}^{N} s\left( {\left| {x_{j} - x_{i} } \right|} \right){ }\frac{{x_{j} - x_{i} }}{{d_{i,j} }}{ } - {\text{g}}\hat{e}_{{\text{d}}} + {\text{u}}\hat{e}_{{\text{w}}} { }$$
(8)

Nonetheless, utilizing the mathematical model to tackle optimization problems is unfeasible as the grasshoppers promptly attain their comfort zone, and the swarm fails to converge towards a predetermined point. To address optimization issues, a revised version of Eq. (8) is suggested as follows according to [39]:

$$X_{j}^{d} = c\left[ {\mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1} \\ {i \ne j} \\ \end{array} }}^{N} c\frac{{ub_{{\text{d}}} - lb_{{\text{d}}} }}{2}s\left( {\left| {x_{i}^{d} - x_{j}^{d} } \right|} \right)\frac{{x_{i} - x_{j} }}{{d_{ij} }}} \right] + \hat{D}_{{\text{d}}} { }$$
(9)

where the parameters of the upper and lower boundary are indicated via \(ub_{{\text{d}}}\) and \(lb_{{\text{d}}}\); the \(d\) th dimensions values are represented by \(\hat{D}_{{\text{d}}}\); the population number is indicated via \(N\); \(c\) is a reducing coefficient that improves the exploitation as the iteration number goes up, and increasing the iteration leads to a balance of exploration and exploitation.

$$c = c_{{{\text{max}}}} - {\text{Iter}} \times \frac{{c_{{{\text{max}}}} - c_{{{\text{min}}}} }}{{M.{\text{Iter}}}}$$
(10)

where for the current article, the values of \({c}_{{\text{max}}}\) and \({c}_{{\text{min}}}\), respectively, are determined to be 1 and 0.0001.

Marine Predator Algorithm (MPA)

The novel metaheuristic algorithm of the natural marine predators algorithm (MPA) was developed by Faramarzi et al. [40]. In the marine predators’ interactions with prey, predators employ a wide range of foraging strategies called MPA-inspired Brown and Levy accidental migration. In hunting grounds, if the focus of the prey is higher, the predators will use the Brown way, and if the prey is little, the predator will use the Levy method. Levi’s movements include jumps and fast motions that enhance the exploration speed. Brown motion involves fixed steps in the same task to optimize the work process. On the other hand, environmental issues such as fish aggregating devices (FADs) and eddy formations are among the factors changing the behaviour of predators. Figure 3 shows the process of optimization by MPA [41].

Fig. 3
figure 3

The flowchart of the marine predator algorithm (MPA)

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The MPA main stages are explained as follows [6]:

When prey moves with the Brown movement, upgrades the matrices of prey with the following relations:

$$\overrightarrow {{{\text{step}}_{j} }} = \overrightarrow {{R_{L} }} \otimes \left[ {\overrightarrow {{{\text{elite}}_{j} }} - \left( {\overrightarrow {{R_{L} }} \otimes \overrightarrow {{{\text{prey}}_{j} }} } \right)} \right]$$
(11)
$$\overrightarrow {{{\text{prey}}_{j} }} = \overrightarrow {{{\text{prey}}_{j} }} + \left( {P \cdot \vec{R} \otimes \overrightarrow {{{\text{step}}_{j} }} } \right)$$
(12)

The variable of RL denotes the vectors containing changing numbers based on the movement of levy type. Another population society can be upgraded as:

$$\overrightarrow {{{\text{step}}_{j} }} = \overrightarrow {{R_{B} }} \otimes \left( {\overrightarrow {{R_{B} }} \otimes \overrightarrow {{{\text{elite}}_{j} }} } \right) - \overrightarrow {{{\text{prey}}_{j} }}$$
(13)
$$\overrightarrow {{{\text{prey}}_{j} }} = \overrightarrow {{{\text{elite}}_{j} }} + \left( {P \cdot cf \otimes \overrightarrow {{{\text{step}}_{j} }} } \right)$$
(14)

Afterwards, the elite matrix is equal to multiplying by \(cf\). That the \(cf\) is defined as follows:

$$cf = \left[ {1 - \left( {{\text{iter}}/{\text{max}} \times {\text{iter}}} \right)} \right]^{{\left( {2 \cdot {\text{iter}}/{\text{max}} \times {\text{iter}}} \right)}}$$
(15)

The predator moves using the movement type of Levy, and then the matrix of Preys can be upgraded via the relations (16) and (17):

$$\overrightarrow {{{\text{step}}_{j} }} = \overrightarrow {{R_{L} }} \otimes \left( {\overrightarrow {{R_{L} }} \otimes \overrightarrow {{{\text{elite}}_{j} }} } \right) - \overrightarrow {{{\text{prey}}_{j} }}$$
(16)
$$\overrightarrow {{{\text{prey}}_{j} }} = \overrightarrow {{{\text{elite}}_{j} }} + \left( {P \cdot cf \otimes \overrightarrow {{{\text{step}}_{j} }} } \right)$$
(17)

After either of the iterations, the elite matrix would be upgraded, accompanied by the best answers, and the ultimate answers are presented after the last iteration.

Evaluation of Models: SVR-MPA and SVR-GOA

Five indicators were used to evaluate the accuracy of models SVR-GOA and SVR-MPA to estimate the pile settlement (PS) ranges in the calibration and validation phases, as revealed in Table 2.

Table 2 Evaluating indices for analysing the proposed frameworks
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In relations (18)–(22), the predicted subsidence of piles is indicated via \({p}_{N}\); target value of measurements is indicated by \({t}_{n}\); \(\overline{t }\) is showing the averaged pile settlement that is observed; the calculated PSs are indicated using \(\overline{p }\) as the average value. Afterwards, alternatively, the number of samples gathered for the train and test phase is indicated by the ntrain and ntest variables.

Results and Discussion

SVR-MPA and SVR-GOA are hybrid algorithms combining SVR with a population-based metaheuristic optimization algorithm. While SVR-MPA mimics the hunting behaviour of marine predators, SVR-GOA simulates the swarming behaviour of grasshoppers. These algorithms have shown promising results in improving the performance of SVR in estimating pile settlement.

The results of the PS model were generated via the SVR-MPA and SVR-GOA frameworks, which have used the machine learning technique to estimate the rates of pile settlements. Thus, we considered the modelling process to have the cost and complexity at the lowest level accompanied by the highest accuracy of PS estimation; this event is practical when using the optimization algorithms to modify SVR in solving kernels close to target values. Mathematics operations and models were performed in the MATLAB environment. Measured pile settlement rates for the project KVMRT as the case study are indicated using Fig. 4, in which 70% of data are considered for the training phase and 30% for the testing phase. As shown in Fig. 4, the placement of the PS rate for both testing and training phases ranges in various scales of values.

Fig. 4
figure 4

Pile settlement rates measured in project KVMRT

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Table 3 shows the key variables of SVR optimized with MPA and GOA.

Table 3 The key factors of variables magnitudes of each optimizer
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The error rates entered in the modelling process are shown in Fig. 5 for both SVR-MPA (a) and SVR-GOA (b). In this regard, the SVR-MPA (a) modelled pile settlement rates with lower errors than SVR-GOA (b). The former model has an error range of − 4 to + 7% of the suitable condition, while on the other hand, SVR-GOA (b) has an error range of − 7 to + 15%. Extending the error range of 75% and more than 100% for mentioned lowest and highest bounds can signify the MPA optimization algorithm’s capability to remove the error rates rather than the GOA method. However, the overestimated and underestimated PS values can be found in both the validation and calibration stages. For SVR-GOA (b), the 80th pile is calculated with the highest error rate and for another model.

Fig. 5
figure 5

The errors and PS rates modelled by: a SVR-MPA and b SVR-GOA

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Focusing on the error, matters are discussed, and showing the target values with the modelling PS values can help us understand the models’ performance to appraise the PS magnitudes. For this respect, Fig. 6 indicates the modelled PS magnitudes with the measured ones as the target values. Regarding the objective rates of PS based on Fig. 6, SVR-MPA (a) shows a better coincidence for a continuous blue solid line with a red dashed line representing the measured PSs. However, this status is conducted for two phases of training and testing. On the other side, SVR-GOA (b) could not do modelling operations as well as the others. According to Fig. 6 (b), overlapping the two lines mentioned above is not fitted as happened for SVR-MPA. Actually, in the calibrating phase, the GOA optimizer made the highest mistake computing the PS rates, especially for the piles of 78–82, with approximately 12% error rates. In comparison, this rate for the model with MPA optimizer was 5%. Therefore, the difference of 140% between the two models shows the goodness of the MPA algorithm.

Fig. 6
figure 6

The PS target values and modelled by: a SVR-MPA and b SVR-GOA

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As seen in the presented figures, the modelling process is so important that in this part, Fig. 7 tries to show the error distribution based on the relevant frequency. The condition of error distribution for both models does not obey a typical rule to create a tall bar showing the concentration of errors around a certain number, especially zero. In this regard, both optimization algorithms have the most error accumulation, around − 4%, with about 27 counts as the frequency. This event has led to creating the short-height bars for either proposed framework.

Fig. 7
figure 7

The error distribution of models with the normal distribution curve

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For comparing the differences of models to estimate the pile settlement rates, Fig. 8 is attempted to show the discrepancy percentage in simulations. Regarding Fig. 8, most of the piles experienced high rates of differences in modelling PS that reached 7 per cent for the pile number of 34. However, in some cases in which the PS estimated by two models are close to each other, reducing the differences near the zero line, such as 10, 12, 16, and 22, totally the average amount of differences reaches 3 per cent.

Fig. 8
figure 8

The difference in PS modelled using SVR-MPA and SVR-GOA

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Figure 9 shows the best-fit line between the measured PS (horizontal axis) against the modelled one (vertical axis). In the light of the lower error rate for SVR-MPA (b), the best-fit line for the PS points is closer to the bisector line, rather than SVR-GOA with more error rates. Moreover, the correlation index of R2 shows that the MPA algorithm has done its job better than the GOA optimizer. This fact results from the error rates shown via MAE and RMSE that are high for SVR-GOA, respectively, with 102 and 108% differences compared to SVR-MPA.

Fig. 9
figure 9

Modelled PS with the measured values for a SVR-GOA and b SVR-MPA

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From the OBJ indicator viewpoint, which encompasses the error indexes of RMSE, MAE, and correlation factor of R2, the SVR-GOA framework (a) was rated for 0.581, about 108 per cent higher than the SVR-MPA with a value of 0.279.

Conclusion

Assessing the immunization of constructions such as bridges and buildings over the operation stage is paramount for experts and engineers. Considering essential factors such as the rate of piles’ settlement (PS) as a vital issue in projects should be investigated before operating them. Various items have to be used to examine the pile movement, and these measures can help us realize the perspective of projects over the operation of the project. In this regard, several intelligent solutions are designed to calculate the rates of pile motions. Thus, the current research used machine learning SVR to model the magnitude of PSs for the constructed project of KVMRT in Malaysia. Two optimization algorithms, the marine predator algorithm (MPA) and the grasshopper optimization algorithm (GOA), were utilized to accurately determine SVR’s main variables. The optimizers were combined with SVR to create SVR-MPA and SVR-GOA frameworks. To examine the capability of developed models, five metrics were considered apprising the PS magnitudes. In this regard, the R2 correlation index of SVR-MPA in the training phase was calculated at 0.996 at the desired level and 0.985 for SVR-GOA. These values for the testing phase were calculated with a 0.72% difference in favour of SVR-MPA, obtained at 0.998, and for SVR-GOA, 0.991. In the training stage, the RMSE index was calculated for SVR-MPA as 0.286 mm, while SVR-GOA was 0.590 mm with 106.38%. This index for validation for SVR-MPA was calculated at 0.281 mm and for SVR-GOA at 0.596 mm with a 112.5% difference. In addition, using the OBJ that includes the error criteria in both training and testing phases, the SVR-MPA was rated at 0.279 and the SVR-GOA 0.581. The condition of error distribution for both models did not obey a typical rule showing the concentration of errors around a certain number, especially zero. In this regard, both optimization algorithms had the most error accumulation, around the − 4%, with about 27 counts as the frequency. In summary, the SVR-MPA model could remove the most errors seen in SVR-GOA through a better simulation process. Existing the error range of − 7.34 to 14.48% entered by GOA compared to the MPA with the errors of − 4.14 to 6.68% implies the latter model’s accuracy and higher capability to reduce error rates.

Considering the project directly related to people’s lives earns high attention. However, covering all aspects of pile settlement behaviour on a large and full scale is difficult and requires a large cost range regarding energy, time, and finance. To cease these costs and reach an adequate estimation of pile settlement, introduced models in the present study proved an acceptable convergence rate of estimation which defines their workability and performance in estimating pile settlement behaviour without mentioned costs and difficulty. Therefore, it is suggestible that these models and specially SVR-MPA, in practical situations and projects.