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Estimation of mechanical properties of the modified high-performance concrete by novel regression models


Using support vector regression (SVR) analytics, a novel method for evaluating the high-performance concrete (HPC) compressive strength (CS) containing fly ash (FA) and blast furnace slag (BFS) has been developed. Both Salp swarm optimization (SSA) and Grasshoppers optimization algorithm (GOA) were used in this research to look for critical SVR method variables that may be tweaked for better performance. The suggested approaches were created using 1030 trials, eight inputs (the primary component of admixtures, mix designs, curing age, and aggregates), and the CS as the forecasting goal. After that, the findings were compared to those found elsewhere in the literature. Combined SSA-SVR and GOA-SVR analysis could work exceptionally well when it comes to estimating, according to the estimation findings. The root means square error (RMSE) value for the GOA-SVR faces a remarkable increment in comparison with the SSA-SVR. The comparison resulted that the GOA-SVR delivered a higher rate of accuracy than any previous published research. At the outset, the developed GOA-SVR model might be considered a practical predictive system for the CS prediction of HPC admixed with FA and BFS.


High-performance concrete (HPC) is more efficient than conventional concrete (CC) in most cases, depending on the applications. The American Concrete Institute (ACI) states that \(HPC\) is a concrete kind that satisfies specified characteristics for homogeneity and performance. Neither standard mixing or curing methods nor standard components can produce these qualities [1, 2]. There is a possibility that the durability of this concrete will be improved. Maintainance costs are reduced as well as service life is extended. Consider the accessibility and cost of locally available resources while selecting the HPC mix characteristics. As a result, more trial mix quantities and analysis are necessary than in CC [2]. Additional cementitious components and additives are added to CC to make HPC [2,3,4]. The compressive strength of HPC mixtures is critical. Establishing an appropriate estimation system for this critical attribute saves money and time while also allowing for more efficient combination creation. Statistical regression approaches have been used by researchers in this regard [3, 4]. An experimental approach based on regression could have major flaws. There must be a formula in place before a regression analysis can be performed. Another significant limitation of the use of regression models is the necessity for continued regularity [2]. Nevertheless, there is disagreement on the experimental formulae used in codes and standards to establish \(CS\). This is due to these formulae being developed by testing on concrete, which did not involve any components of additional cementitious materials.

Over the last decade, there has been a surge of interest in applying machine learning methods to tackle civil engineering challenges in both commercial and academic settings [5, 6]. Using machine learning methods, in which empirical efforts can create learning, could be highly beneficial in developing computer models [7, 8]. The most extensively utilized method for machine learning is artificial neural networks (ANNs). ANNs are often used to analyze a wide range of concrete qualities [9,10,11,12]. They have been used to forecast the CS and slump flow of HPC mixtures [13, 14]. Employing sequential learning NNs, Rajasekaran and Amalraj [15] and Rajasekaran et al. [16] developed prediction approaches for the power of HPC mixtures. A wavelet NN approach was utilized to analyze the HPC’s \(CS\) by Rajasekaran and Lavanya [17]. Since ANNs are unable to characterize the prediction fundamentals, they are commonly referred to as black box systems. Despite their efficiency, since ANNs usually fails to give a good forecasting model, they are still considered as a black box. Various authors have forecasted the mechanical features of HPC in previous years utilizing NNs and other AI approaches. Gradient-boosted ANN and Bagged ANN were employed by Erdal et al. [18] to simulate the HPC’s CS. When compared to previous trials, Chou and Pham’s ensemble approaches [19] fared well. So long as Erdal [20] built two connected ensemble decision trees, Cheng et al. [21] evaluated HPC CS using a tree. According to the previous study, AI techniques are more effective than traditional methods for precisely and quickly measuring the CS of HPCs [22, 23]. Rafiei et al. [24] propose the introduction of a unique deep machine as a replacement for back-propagation NN and SVR. This is done to determine the properties of concrete on the basis of the mixture percentages. Considering real testing results, they assess the 98% precise. Nguyen et al. [25] used a deep NN framework to determine the foamed concrete strength. Rafiei et al. [26] used an optimization algorithm to uniquely handle the optimal values in the concrete mix. Genetic programming is another machine learning approach (GP) [27]. The criteria of natural genetic advancement lead GP to generate computer simulations on its own. In recent years, classical GP and its modifications have been used to deliver easier answers to civil engineering problems [28]. Mousavi et al. [2] combine GP and orthogonal minimum squares approaches to estimate the CS of HPC combinations. When creating CS, the amount of fine/coarse aggregate, the percentage of superplasticizer to samples’ age, and the binder were all considered. Gene expression programming is a more modern development of GP (GEP) [29]. The GEP approach may be used as a dependable and effective substitute for traditional GP. Some study has been conducted in an attempt to use GEP in civil engineering [30,31,32,33].

Several research have proposed alternative models for predicting the CS of HPCs, including the hybrid adaptive neuro-fuzzy inference system (ANFIS) with arithmetic optimization algorithm (AOA) and equilibrium optimizer (EO). The findings suggest that the integrated systems exhibited robust estimate skills, as shown by \({R}^{2}\) values of 0.9941 and 0.9975 for the training and testing phases, respectively [34]. Four deep learning algorithms were offered in different research, which was unusual in the literature. When using deep learning models, the aforesaid \({R}^{2}\) value was approximately 0.960 during training and almost reached 0.940 after testing [35]. In research, a unique hybrid model combining artificial bee colony (ABC) optimization and cascade forward neural network (CFNN) technology was created for the CS prediction of HPC. The created model (CFNN-ABC) was able to correctly estimate the compressive strength of HPC with an \({R}^{2}\) of 0.953, according to the results, and two layers is the ideal neural network architecture chosen by the \(ABC\) approach [36]. An incredibly accurate machine learning (\(ML\)) model was trained using the innovative eXtreme Gradient Boosting (\(XGB\)) approach. The baseline model tends to overfit, with values of 0.996 and 0.919 for the training and testing datasets, respectively [37].

The main objective of the current research is to supply a practical way to evaluate the effectiveness of intelligent machines in calculating HPC’s CS via stringent testing. Using the SVR technique, we attempted to construct models that might predict HPC’s qualities while it had fresh and hardened properties. It used the grasshoppers’ optimization algorithm (GOA) and salp swarm optimization (SSA) algorithms in this work to pinpoint the most critical components of SVR that need to be changed. The created techniques were assessed using 1030 experiments, eight input parameters, such as concrete age, admixtures, the major constituent of mixes, and the compressive strength as the forecasting objective. The findings were then compared to other studies in the field [19, 28, 38,39,40,41,42].

Developed models offer several advantages compared to traditional mixed design methods in the field of materials science and civil engineering. These advantages stem from ML’s ability to analyze large datasets, discover complex patterns, and make predictions based on learned patterns. Algorithms can analyze vast amounts of data to identify subtle relationships between various material properties and mix design outcomes. This can lead to mixed designs that are more accurate and precise, reducing the risk of defects and failures. Developed models can automate many aspects of mix design, such as optimizing the proportions of materials and predicting the properties of the resulting mixture. Models can adapt to different materials, conditions, and project requirements. Traditional mixed design methods often rely on simplified assumptions and linear models. Algorithms, on the other hand, can handle complex, nonlinear relationships between material properties and mix performance, leading to more accurate predictions. ML can help optimize mix designs to use materials more efficiently, minimizing waste and reducing costs associated with the overuse or underuse of materials. ML-based mix design can help reduce the cost of materials and construction by optimizing the use of resources, improving performance, and minimizing the need for expensive adjustments or revisions.


Dataset preparation and description

Over a thousand HPC sample datasets were investigated in this research [4, 43,44,45,46]. Regular Portland cement was used to build all specimens, which were then permitted time to dry out. Several different types and sizes of samples have been used in the \(HPC\) data that has been released so far. In this study, the \(HPC CS\) is based on the following eight factors:

  1. (1)

    C Contents of cement

  2. (2)

    BFS Blast furnace slag

  3. (3)

    FA Fly ash

  4. (4)

    W Water

  5. (5)

    SP Superplasticizer

  6. (6)

    CA Coarse aggregate

  7. (7)

    FA Fine aggregate

  8. (8)

    AC \(HPC\) age

Figure 1 shows the training and evaluation datasets’ charts with their Lognormal distribution, and Table 1 shows the collection range for these items. When it came time to divide up everything that was needed, 70% of the 1030 data was used for learning and the remainder for testing. Subgroups from the original dataset were selected at random using a uniform distribution for training and testing. The 70:30 train/test ratio was chosen for this investigation even though many other train/test ratios were able to be used, as suggested in previous studies [47, 48]. In order to show that the choice of these inputs was enough, a statistical study was carried out. As a consequence, no substantial cross in the eight-dimensional input space was found [45, 49], which is needed to train AI networks with good generalization abilities.

Fig. 1
figure 1

Distribution of dataset and their Lognormal distribution

Table 1 The dataset attributes

Considered techniques

Grasshoppers optimization algorithm (GOA)

Meta-heuristic methods are built on simulating nature (Fig. 2). Universal optimization issues are typically solved using these methods. Different categories of meta-heuristic methods include those based on evolution, physical and chemical, swarm intelligence, and human-based methods. We optimized using GOA in this research. Saremi et al. [50] introduce GOA to the research. The efficient meta-heuristic method GOA uses optimization based on a swarm, which was motivated by natural processes. Based on this study, GOA is utilized to determine the input variables’ ideal values (ideal factor rates). The optimization process is then applied to GOA using these models. As a consequence, we employed GOA to look for these regression models’ ideal factor rates.

Fig. 2
figure 2

Patterns of relations between individuals in a group of grasshoppers [51]

GOA imitates the organic manners of grasshopper swarms. The two steps of nature-inspired optimization methods are exploitation and exploration. Sudden movements throughout the exploration are made by the search agents of the optimization method. Although they migrate more locally throughout exploitation. The following formulae represent the manners of the grasshopper and the theory of optimization search [50]:


In this equation, \(i\) stands for every grasshopper and \({X}_{i}\) indicates where the \({i}_{th}\) grasshopper is located. \({S}_{i}\) is a symbol of how grasshoppers connect. Similarly, to this, \({G}_{i}\) and \({A}_{i}\) stand in for gravity power and wind advection. The random numbers in the range [0, 1] are the \(r\) variables. Equation (2) [50] provides information on the grasshoppers' social manner (attraction–repulsion):

$${S}_{i}=\sum\nolimits_{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N}s\left({d}_{ij}\right)\widehat{{d}_{ij}}$$

Based on this equation, \(s\) stands for the power of social forces \(\left({s}_{r}={fe}^{-r/1}-{e}^{-r}\right)\), \(l\) stands for the seductive interval scale, and \(f\) for attraction’s potency [50]. \(N\) shows the grasshoppers’ number. A vector among both two grasshoppers is called \({d}_{ij}\) \(\left({d}_{ij}=\left|{x}_{j}-{x}_{i}\right|\right)\), and \(\widehat{{d}_{ij}}\) represents the exact interval among the ith and jth grasshopper \(\left(\widehat{{d}_{ij}}=\left({x}_{j}-{x}_{i}\right)/{d}_{ij}\right)\). Artificial grasshoppers’ social interactions are influenced by the \(s\) function. This function divides the interval among every duo of grasshoppers into three pieces (attraction region, repulsion region, and comfort zone). In their investigation of intervals from 0 to 15, Saremi et al. [50] found repulsion among [0 2.079]. They proposed calculating the safe interval as the separation of two artificial grasshoppers by 2.079 units. The safe area is not where attraction or repulsion operate. This area is altered by the \(f\) and \(l\). However, the s function turns to zero if this interval exceeds 10. Consequently, this function cannot generate powerful pressures throughout long intervals among grasshoppers. Another element of \({X}_{i}\) is \({G}_{i}\) (gravitational power) [50]:


In Eq. (3), \(g\) stands for the gravity constant and \(\widehat{{e}_{g}}\) for the unity vector pointing toward the middle of the earth. The last element of \({X}_{i}\) is \({A}_{i}\) (wind advection):


According to this equation, respectively, \(\widehat{{e}_{w}}\) and \(u\) represent a unity vector and constant drift in the direction of the wind. Traditional swarm-based methods imitate the swarm as it explores and uses the search area all over an answer. The GOA model of \({X}_{i}\) replicates the interactions of a swarm of grasshoppers since the mathematical formulas are in a free area. It emulates how a grasshopper might act in multiple spatial dimensions, including 2D, 3D, and hyper-dimensional areas [50].

$${X}_{i}^{d}=c\left(\sum\nolimits_{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N}c\frac{{ub}_{d}-{lb}_{d}}{2}s\left({x}_{j}^{d}-{x}_{i}^{d}\right)\frac{{x}_{j}-{x}_{i}}{{d}_{ij}}\right)+\widehat{{T}_{d}}$$

The lower and upper boundaries in the \({D}_{th}\) dimension \({s}_{r}\) are known as \({lb}_{d}\) and \({ub}_{d}\). The comfort, repulsion, and attraction areas are reduced by the lowering coefficient \(c\), which makes \(\widehat{{T}_{d}}\) the optimal (target) answer. Every search unit in GOA has a one-location vector, which calculates every search unit’s next location. Equation (5)’s first step, the summation, replicates grasshopper interaction by considering the locations of several grasshoppers. \(\widehat{{T}_{d}}\) reflects their propensity to migrate in the direction of food resources. Eventually, \(c\) simulates the grasshoppers’ slowing as they reach the food resource in Eq. (6) [50].


In this equation, the present iteration and the iterations’ highest number, \(L\) and \(l\). The lowest and highest values are denoted \(c\mathrm{min}\) and \(c\mathrm{max}\) [50]. We utilized the same settings as Saremi et al. [50], who utilized \(c\mathrm{max} = 1\) and \(c\mathrm{min} = 0.00001\). In summation, the swarm eventually approaches a fixed goal as the safe area is reduced by the \(c\) variable. The swarm also successfully pursues a moving goal by \(\widehat{{T}_{d}}\). The grasshoppers will approach the objective through many repetitions. Algorithm 1 below displays the \(GOA\) pseudocode [50].

figure a

Algorithm 1. GOA’s Pseudocode

Salp swarm optimization (\({\varvec{S}}{\varvec{S}}{\varvec{A}}\))

As presented in Fig. 3, salps move highly alluringly, often traveling in a cooperative chain as they seek food in the oceans and seas. This characteristic allows the swarm to move more fluidly and with greater kinetic energy while they search for food [52]. A single leader often leads the salps chain followers. A \(d\)-dimensional matrix describes a group chain \(X\) with \(n\) salps.

Fig. 3
figure 3

Salp chain

$$X=\left[\begin{array}{ccc}\begin{array}{cc}\begin{array}{c}{x}_{1}^{1}\\ {x}_{1}^{2}\end{array}& \begin{array}{c}{x}_{2}^{1}\\ {x}_{2}^{2}\end{array}\end{array}& \begin{array}{c}\dots \\ \dots \end{array}& \begin{array}{c}{x}_{d}^{1}\\ {x}_{n}^{2}\end{array}\\ \begin{array}{cc}\vdots & \vdots \end{array}& \cdots & \vdots \\ \begin{array}{cc}{x}_{1}^{n}& {x}_{2}^{n}\end{array}& \dots & {x}_{d}^{n}\end{array}\right]$$

A focused food establishment is also represented by \(F\). Following are the modified positions for a leader \(\left({x}_{j}^{1}\right)\):

$${x}_{j}^{1}=\left\{\begin{array}{c}{F}_{j}+{k}_{1}\left(\left({U}_{b-j}-{L}_{b-j}\right){k}_{2}+{L}_{b-j}\right),{k}_{3}\ge 0.5\\ {F}_{j}-{k}_{1}\left(\left({U}_{b-j}-{L}_{b-j}\right){k}_{2}+{L}_{b-j}\right),{k}_{3}<0.5\end{array}\right.$$

Based on this equation, \({k}_{2}\) and \({k}_{3}\) are two randomly generated numbers in the range [0, 1], and \({U}_{b}\) and \({L}_{b}\) stand for the top and low boundaries. According to the following equation, \({k}_{1}\) maintains an equilibrium between exploitation and exploration:


In Eq. (9), the iterations \(n\) and \(N\) stand for the exact and highest iterations. The remaining (or followers) salps’ location is determined by the formula below:


In Algorithm 2, the fundamental \(SSA\) pseudo code is described.

figure b

Algorithm 2. SSA’s Pseudocode

Support vector regression (\({\varvec{S}}{\varvec{V}}{\varvec{R}}\))

The statistical learning theory and structural risk minimization concept are the foundations of the nonlinear regression method known as SVR [53]. The core of this method is a nonlinear conversion technique (kernel functions) that converts the initial input area into a novel hyperspace. The complex and nonlinear interactions among input and outcome parameters in transformed hyperspace are described by a linear function [53, 54]. SVR is the procedure of identifying the function \(f(x)\) that is as flat as feasible and has a maximum deflection \(\varepsilon\) from the training samples \(\left({x}_{i},{y}_{i}\right)\) for \(i=1,\dots ,N\). The complexity of the model may be reduced by maximizing the function’s flatness, which affects the model’s overall performance. In fact, according to the learning theory [55], the generalization error may be limited by the sum of two factors, one of which depends on the model’s complexity and the other of which depends on the fault in the training data. The foundation of SVR techniques is the management of model complexity throughout training.

The procedure is initially explained for a linear function \(f(x)\) of the following type:


Based on this equation, \(x\),\(w\), and \(b\) show the input vector, the parameters’ vector (or weight), and a constant to be specified, respectively. The data is mapped onto a greater dimensional area utilizing a nonlinear kernel in the context of nonlinear issues:

$$f\left(x\right)=w\varphi .\left(x\right)+b$$

In this equation, the kernel function is shown by \(\varphi (x)\).

Data may be mapped onto a greater dimensional characteristic area and can also be used to apply a linear regression method. The w and b may be gained by minimizing the given function:

$$\mathrm{min}\frac{1}{2}{\Vert w\Vert }^{2}+c\sum\nolimits_{i=1}^{N}\left({\xi }_{i}+{\xi }_{i}^{*}\right)$$

Subjected to:

$$\left\{\begin{array}{c}{y}_{i}-\langle w,{x}_{i}\rangle -b\le \varepsilon +{\xi }_{i}\\ \langle w,{x}_{i}\rangle +b-{y}_{i}\le \varepsilon +{\xi }_{i}^{*}\\ with {\xi }_{i},{\xi }_{i}^{*}\ge 0, i=1,\dots ,N\end{array}\right.$$

Based on this equation, \({\xi }_{i}^{*}\) and \({\xi }_{i}\) show the negative and positive faults as presented in Fig. 4. A hyperparameter allowing for adjustment of the balance among the amount of fault permitted and the flatness of the function \(f(x)\) is the constant \(C>0\). The penalty factor \(C\) and the kernel function variable (\(\sigma\)) control the generalization and fitting capabilities of the \(SVR\) model, respectively. Selecting the optimal \(C\) and \(\sigma\) may assist in avoiding under- or over-fitting while also improving the \(SVR\) model’s estimating efficiency.

Fig. 4
figure 4

Setting the soft margin loss for a linear SVR [56]


The formulations in Eqs. 1417 depict the metrics values for assessing the models’ robustness and comparing with the literature to choose the outperformed model.

1) Coefficient of determination (\({R}^{2}\))

$${R}^{2}={(\frac{{\sum }_{d=1}^{D}\left({m}_{d}-\overline{m }\right)\left({z}_{d}-\overline{z }\right)}{\sqrt{\left[{\sum }_{d=1}^{D}{\left({m}_{P}-m\right)}^{2}\right]\left[{\sum }_{d=1}^{D}{\left({z}_{d}-\overline{z }\right)}^{2}\right]}})}^{2}$$

2) Root-mean-square error (\(RMSE\))

$$RMSE=\sqrt{\frac{1}{D}{\sum }_{d=1}^{D}{\left({z}_{d}-{m}_{d}\right)}^{2}}$$

3) Mean absolute error (\(MAE\))

$$MAE=\frac{1}{D}{\sum }_{d=1}^{D}\left|{z}_{d}-{m}_{d}\right|$$

\({4) A}_{20-\mathrm{Index}}\)


In these equations: \({m}_{d}\) defines the records, \(\overline{m }\) the mean values of records, \({z}_{d}\) the predicted values, \(\overline{z }\) the mean values of predictions, and \(D\) is the number of a dataset. \(M\) is the number of samples, and \({m}_{10}\) is the number of samples with a recorded/predicted proportion in [0.8, 1.2].

Results and discussion

The conclusions of the GOA-SVR and SSA-SVR architectures are provided to forecast the CS of the HPC augmented with BFS and FA. As stated in the previous part, the productivity of SVR is determined by choosing the proper values for the mixture of the key SVR sections. The reported and computed values of the CS of HPC throughout the training and testing stages for the GOA-SVR and SSA-SVR simulations are presented in Fig. 5. Along with the time series plots, a residual CS concentration graph with a normally distributed curves around the zero line is given. \({R}^{2},\) RMSE, MAE, and \({A}_{20-Index}\) were calculated to assess the workability of the GOA-SVR and SSA-SVR (see Table 2). Both the GOA-SVR and SSA-SVR techniques show great potential for properly forecasting HPC CS.

Fig. 5
figure 5

Results from the SVR models

Table 2 Findings of developed \(SVR\) s and comparison with previous articles

This section of the research compares the findings of the statistical identifiers developed for the generated investigations, known as GOA-SVR and SSA-SVR, to see whether one version performs better than others. Furthermore, an attempt has been undertaken to appropriately compare the investigation’s findings to previously published ones. The findings reveal that the combined GOA-SVR and SSA-SVR frameworks could perform wonderfully well during predicting, with \({R}^{2}\) values of 0.9695 and 0.9688 for GOA-SVR and 0.9438 and 0.9161 for SSA-SVR for the training and testing sections, respectively. However, analyzing and evaluating the generated signals is necessary to choose the appropriate technique. When compared to the SSA-SVR, the GOA-SVR RMSE’s value decreased somewhat in the training portion from 3.6962 to 2.4815 MPa. A stunning drop from 4.0861 to 2.3906 MPa was achieved in the testing stage. Another criterion, MAE, yielded similar findings to RMSE and shows that the GOA-SVR has a better ability for CS prediction at \({MAE}_{\mathrm{Train}}\)= 2.124 and \({MAE}_{\mathrm{Test}}\)= 1.5049 because its values were smaller than those of the SSA-SVR at \({MAE}_{\mathrm{Train}}\)= 3.476 and \({MAE}_{\mathrm{Test}}\)= 3.2734. A similar pattern might be found in the \({A}_{20-\mathrm{Index}}\) indicator, which showed a 14 percent increase in the train portion and a 16 percent increase in the test section for GOA-SVR.

The computations in this article were compared with [19, 28, 38,39,40,41,42] in order to give a full review and validation. Where different types of techniques used like Gene Expression Programming (GEP) [38], Semi-empirical method (SEM) [40], Gaussian process regression (GPR) [41, 39], Extreme-gradient-boosting (XGBoost) [42], Artificial neural networks (ANNs) [19], and Multi-gene genetic programming (Multi-GGP) [28]. As shown in Table [38], for example, exhibited a little lower \({R}^{2}\) and a much higher MAE than GOA-SVR by 0.8224 and 5.202, respectively. SEM [40], as a new technique, performed worse than GOA-SVR, with \({R}^{2}\) at 0.84 significantly lower than 0.9695, RMSE at 6.3 lower than 3.1108 (a drop of greater than 50%), \(MAE\) at 4.91 greater than 2.124, and \({A}_{20-\mathrm{Index}}\) at 0.68 lower than 0.9417. Moreover, the GOA-SVR beat the GPR [41] watching to \({R}^{2}\), RMSE, and MAE. Another GPR [39] depicted marginally better in the training phase compared to this article, but it was terribly poor in the testing section. Other approaches, such as Multi-GGP [28] and ANNs [19] performed weakly than GOA-SVR, with \({R}^{2}\) values of 0.8046 and 0.8469, respectively, which are interestingly less than 0.9688. Finally, the most recently developed technology, XGBoost [42], got close, although it was also weaker than GOA-SVR. Finally, the suggested model is the GOA-SVR framework created for simulating the CS of HPC with FA and BFS.


The main objective of the current research is to supply a practical way to evaluate the effectiveness of Intelligent machines in calculating high-performance concrete (HPC)’s compressive strength (CS) via stringent testing. Using the support vector regression (SVR) technique, we attempted to construct models that might predict HPC’s qualities while it was fresh and hardened properties. It used the grasshoppers’ optimization algorithm (GOA) and salp swarm optimization (SSA) algorithms in this work to pinpoint the most critical components of SVR that need to be changed. The created techniques were assessed by utilizing 1030 experiments, eight input parameters, like admixtures, concrete age, and the major constituent of mixes, as well as the CS as the forecasting objective. The findings were then compared to other studies in the field.

  • The findings reveal that the combined GOA-SVR and SSA-SVR frameworks could perform wonderfully well during predicting, with \({R}^{2}\) values of 0.9695 and 0.9688 for GOA-SVR and 0.9438 and 0.9161 for SSA-SVR for the train and test sections, respectively.

  • When compared to the SSA-SVR, the GOA-SVR RMSE’s value decreased somewhat in the training portion from 3.6962 M to 2.4815 MPa. A stunning drop from 4.0861 to 2.3906 MPa was achieved in the testing stage. Another criterion, MAE, yielded similar findings to RMSE and shows that the GOA-SVR has a better ability for CS prediction at \({MAE}_{\mathrm{Train}}\)= 2.124 and \({MAE}_{\mathrm{Test}}\)= 1.5049 because its values were smaller than those of the SSA-SVR at \({MAE}_{\mathrm{Train}}\)= 3.476 and \({MAE}_{\mathrm{Test}}\)= 3.2734. A similar pattern might be found in the \({A}_{20-\mathrm{Index}}\) indicator, which showed a 14 percent increase in the train portion and a 16 percent increase in the test section for GOA-SVR.

  • As shown, the suggested GOA-SVR performed the best compared to the literature. GEP [38] exhibited lower \({R}^{2}\) and a much higher MAE than GOA-SVR by 0.8224 and 5.202, respectively. SEM [40] performed worse than GOA-SVR, with \({R}^{2}\) at 0.84, significantly lower than 0.9695, RMSE with a drop of greater than 50%, and \({A}_{20-\mathrm{Index}}\) at 0.68 lower than 0.9417. Moreover, the GOA-SVR beat the GPR [41], watching to \({R}^{2}\), RMSE, and MAE. Other approaches, such as ANNs [19] and Multi-GGP [28] performed weakly than GOA-SVR, with \({R}^{2}\) values of 0.8469 and 0.8046, respectively, which are interestingly less than 0.9688. Finally, the most recently developed technology, XGBoost [42], got close, although it was also weaker than GOA-SVR.

  • Finally, the suggested model is the GOA-SVR framework created for simulating the CS of HPC updated with FA and BFS.

Availability of data and material

The authors do not have permissions to share data.



High-performance concrete


Compressive strength


Artificial neural networks


Support vector regression


Grasshoppers’ optimization algorithm


Machine learning


Salp swarm algorithm


Root mean square error


Mean absolute error



R2 :

Coefficient of correlation


Age of concrete


Fly ash


Fine aggregate


Blast furnace slag






Coarse aggregate


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LJ activated in methodology, software, validation, and formal analysis section. WJ performed formal analysis, methodology, validation, and language review. YS handled writing—original draft preparation, conceptualization, supervision, and project administration. All authors have read and approved the manuscript.

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Correspondence to Yin Suyuan.

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Jingtao, L., Jing, W. & Suyuan, Y. Estimation of mechanical properties of the modified high-performance concrete by novel regression models. J. Eng. Appl. Sci. 70, 157 (2023).

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