Digital Industrial Design Method in Architectural Design by Machine Learning Optimization: Towards Sustainable Construction Practices of Geopolymer Concrete

Abstract

The construction industry’s evolution towards sustainability necessitates the adoption of environmentally friendly materials and practices. Geopolymer concrete (GeC) stands out as a promising alternative to conventional concrete due to its reduced carbon footprint and potential for cost savings. This study explores the predictive capabilities of soft computing models in estimating the compressive strength of GeC, utilizing multi-layer perceptron (MLP) neural networks and hybrid systems incorporating the Gannet Optimization Algorithm (GOA) and Grey Wolf Optimizer (GWO). A dataset comprising 63 observations from a quarry mine in Malaysia is employed, with influential parameters normalized and utilized for model development. Consequently, we integrate optimization algorithms (GOA and GWO) with MLP to fine-tune the model’s parameters and improve prediction accuracy. The models are evaluated using R², RMSE, and VAF. Various MLP architectures are explored, evaluating transfer functions and training techniques to optimize performance. In addition, hybrid models GOA–MLP and GWO–MLP are developed, with parameters fine-tuned to enhance predictive accuracy. During the training phase, the GWO–MLP model achieved an R² of 0.981, RMSE of 0.962, and VAF of 97.44%, compared to MLP’s R² of 0.95, RMSE of 0.918, and VAF of 94.59%. During the testing phase, GWO–MLP also showed the best performance with an R² of 0.976, RMSE of 1.432, and VAF of 97.51%, outperforming both MLP and GOA–MLP. The GOA–MLP model demonstrated improved performance over MLP with an R² of 0.963, RMSE of 0.811, and VAF of 95.78% in the training phase and R² of 0.944, RMSE of 2.249, and VAF of 92.86% in the testing phase. Hence, the results show that the GWO–MLP model consistently outperforms both MLP and GOA–MLP models. Sensitivity analysis further elucidates the impact of key parameters on compressive strength, aiding in the optimization of GeC formulations for enhanced mechanical properties. Overall, the study underscores the efficacy of machine learning models in predicting GeC compressive strength, offering insights for sustainable construction practices.

1. Introduction

In this era of significant human advancements, various developmental milestones are being achieved. The evolution of society is intricately linked with infrastructural progress, as highlighted by numerous scholars [1]. Notably, the construction industry assumes a pivotal role in fostering societal advancement [2]. Concrete stands as the cornerstone material for constructing diverse structures. Unlike traditional concrete, which has been prevalent in housing for the past two to three decades, Green Polymer Concrete (GPC) has emerged as an eco-friendly alternative, utilizing Ground Granulated Blast Furnace Slag (GGBFS), fly ash, and an alkaline solution to entirely replace cement. Concrete ranks as the second most utilized substance globally, following water. Cement, a primary constituent of conventional concrete, is responsible for significant carbon dioxide emissions, with approximately one tonne of carbon dioxide released per tonne of cement produced. Geopolymer concrete substitutes cement with GGBFS and fly ash, drastically reducing carbon footprints by approximately 80% compared to conventional concrete.

The emission of carbon dioxide directly contributes to global warming [3], underscoring the imperative of sustainable development in construction practices [4]. Geopolymer concrete offers economic advantages over traditional concrete, with costs reduced by approximately 40%. Its superior strength and durability render it a viable alternative to conventional concrete [5]. Geopolymer concrete exclusively replaces cement with fly ash, slag, or metakaolin from concrete constituents, necessitating the use of an alkaline solution to activate these pozzolanic materials for bonding. Alkaline substances, such as sodium hydroxide and silicates, are commonly employed for this purpose. The chemical reactions and bonding mechanisms of geopolymer concrete differ significantly from conventional concretes. Coined by Prof. Davidovits, the term “geopolymer” denotes the unique bonds formed through these reactions. Laboratory tests consistently demonstrate the superior performance of geopolymer concrete compared to traditional concrete, indicating its potential as a cornerstone of the sustainable construction industry’s future.

To elaborate further, the societal implications of infrastructural development cannot be overstated. Beyond mere physical structures, infrastructure underpins economic growth, social cohesion, and environmental sustainability. As such, any advancements in construction techniques or materials hold profound implications for the trajectory of societal progress. Geopolymer concrete’s emergence as a sustainable alternative to traditional concrete underscores a paradigm shift in construction practices, aligning with worldwide attempts to reduce climate change and advance sustainable development. The economic viability of geopolymer concrete further enhances its appeal, potentially reshaping the construction industry’s landscape by offering cost-effective, environmentally friendly solutions. Moreover, the chemical intricacies of geopolymer concrete’s composition and bonding mechanisms warrant detailed examination, offering insights into its superior performance and potential applications across various construction contexts. By delving into these complexities, researchers can elucidate the transformative potential of geopolymer concrete, paving the way for its widespread adoption and integration into mainstream construction practices.

The strength of geopolymer concrete is influenced by a multitude of factors, encompassing both external and internal parameters. External factors, such as temperature, curing duration, humidity levels, and air entrainment, play significant roles, while internal factors include the quality of materials and their compositions [6]. Compositional attributes of binding materials and particle sizes are crucial for initiating and enhancing reactions, with their proportions governing the strength of said reactions, as per requirements [7]. Incrementing slag content in the composition augments the compressive strength of ambient-cured geopolymer concrete [8]. Moreover, finer particles of fly ash and slag facilitate rapid reactions due to their increased surface area availability, thereby enhancing early concrete strength [9]. The liquid-to-binder ratio also significantly impacts reaction kinetics and resultant strength [10]. Although water is essential for initiating geopolymer reactions, its role diminishes as it evaporates during the hardening process, necessitating only a small quantity of liquid to react with all concrete components. The optimal liquid content directly influences geopolymer concrete strength [11,12].

Advancements in artificial intelligence (AI) have significantly enhanced the prediction of material properties in civil engineering applications. Numerous studies have highlighted the effectiveness of AI techniques in forecasting the mechanical properties of various materials [13,14,15,16,17,18,19,20,21,22,23,24,25]. For instance, Ahmad et al. [26] conducted a comparative analysis of three AI-based models—decision trees (DT), AdaBoost, and bagging regressors—for estimating the compressive strength of geopolymer concrete (GeC) composed of fly ash. Their findings indicated that the bagging regressor outperformed the other methods, demonstrating superior predictive accuracy. Similarly, another study by Ahmad et al. [27] employed artificial neural networks (ANN) and gene expression programming (GEP) to predict the compressive strength of recycled aggregate concretes. The GEP model exhibited greater precision in forecasting the outcomes compared to the ANN approach, underscoring its reliability for such applications. Furthermore, Song et al. [28] utilized an ANN-based methodology to predict the compressive strength of concretes incorporating waste materials. The results from this study affirmed the ANN model’s ability to provide accurate predictions, emphasizing the utility of machine learning techniques in assessing the mechanical properties of concrete. Nguyen et al. [29] expanded on this by analyzing the tensile and compressive strengths of high-performance concretes using a variety of AI methods. Their investigation revealed that hybrid AI techniques, which integrate multiple machine learning approaches, yielded better predictive performance than standalone models. This enhanced accuracy was attributed to the synergistic use of weak learners, such as decision trees and multi-layer perceptron neural networks, in creating robust predictive models. These studies underscore the potential of AI in material property prediction, demonstrating its capacity to analyze complex datasets and improve modeling precision. They also highlight the need for continued exploration of advanced AI methodologies to address existing limitations and refine predictive capabilities further. To this end, additional research that delves deeper into the integration and optimization of AI techniques is essential to unlock their full potential in material science and civil engineering. Several existing models for predicting various concrete properties, as reported in the literature, are summarized in

Table 1. Using artificial intelligence techniques to predict various characteristics of concretes.

Author	Year	Technique	Number of Data
Huang et al. [30]	2021	SVM	114
Sarir et al. [31]	2019	GEP	303
Balf et al. [32]	2021	DEA	114
Ahmad et al. [33]	2021	GEP, ANN, DT	642
Azimi-Pour et al. [34]	2020	SVM	-
Saha et al. [35]	2020	SVM	115
Hahmansouri et al. [36]	2020	GEP	351
Hahmansouri et al. [36]	2019	GEP	54
Aslam et al. [37]	2020	GEP	357
Farooq et al. [38]	2020	RF and GEP	357
Asteris and Kolovos [39]	2019	ANN	205
Huang et al. [40]	2019	IREMSVM-FR withRSM	114
Zhang et al. [41]	2019	RF	131
Kaveh et al. [42]	2018	M5MARS	114
Sathyan et al. [43]	2018	RKSA	40
Vakhshouri and Nejadi [44]	2018	ANFIS	55
Belalia Douma et al. [45]	2017	ANN	114
Abu Yaman et al. [46]	2017	ANN	69
Ahmad et al. [47]	2021	GEP, DT and Bagging	270
Farooq et al. [48]	2021	ANN, bagging and boosting	1030
Bušić et al. [49]	2020	MV	21
Javad et al. [50]	2020	GEP	277
Nematzadeh et al. [51]	2020	RSM, GEP	108

. These models serve as a reference for understanding the current state of AI applications in this domain and offer a foundation for future innovations.

Furthermore, the selection and usage of superplasticizers in geopolymer concrete are crucial for bonding, with SNF-based superplasticizers proving most suitable [52]. The quality and content of the alkaline solution are paramount for initiating geopolymer reactions, with reaction efficacy directly linked to the molarity of sodium or potassium hydroxide, thus influencing concrete performance and strength [53]. Curing temperature and conditions also significantly impact the attainment of design strength, with oven-cured specimens achieving strength more readily than ambient-cured counterparts [54]. Geopolymer concrete exhibits robust stability against severe environmental conditions [55], suggesting its potential as a cornerstone of sustainable development in the construction industry.

Moreover, geopolymer concrete finds diverse applications worldwide, with notable use in the construction industry. Using a digital industrial design method in architectural design with machine learning optimization shows a future trend towards sustainable construction practices using geopolymer concrete. However, the process of determining concrete mix strength is time-consuming and labor-intensive. Machine learning techniques offer a promising solution by predicting strength based on previous datasets without the need for destructive testing, as shown in Figure 1 [18,19,56,57,58,59,60,61,62,63,64,65].

Utilizing models based on historical data, machine learning accurately predicts strength, thus reducing time and labor [34] and minimizing errors in predictions [66,67,68,69,70,71]. Among these techniques, artificial neural networks stand out as the most prominent method for predicting concrete strength [72]. This paper presents a novel approach to predicting the compressive strength of GeC using machine learning techniques, specifically combining MLP with optimization algorithms such as GOA and GWO. The study contributes to the field in several ways: first, it introduces a hybrid machine learning model that significantly improves prediction accuracy over traditional methods. Second, it provides a detailed evaluation of the influence of various input parameters on the compressive strength of GeC, offering valuable insights for optimizing concrete formulations. Third, it demonstrates the application of optimization techniques to fine-tune model parameters, resulting in better generalization performance. The remainder of the paper is organized as follows: Section 2 describes the materials and methods used for data collection and model development. Section 3 highlights the case study and analyzes the available data. Section 4 presents the results and discussion of the model’s performance. Section 5 outlines the conclusion and future work, with recommendations for further improvements in predictive modeling for sustainable construction practices.

2. Research Methodology

This study aims to predict the compressive strength of geopolymer Concrete (GeC) using three machine learning models: Multi-Layer Perceptron (MLP), Gannet Optimization Algorithm-MLP (GOA–MLP), and Grey Wolf Optimizer-MLP (GWO–MLP). The following steps outline the overall methodology employed in this study:

1.

Data Collection and Preprocessing: A dataset consisting of 63 observations of GeC was collected from a quarry mine in Malaysia. The dataset includes 11 input parameters, such as fly ash content, curing temperature, and alkaline activator ratio. These input features were normalized to the range [−1, 1] to ensure consistency in the model training process.

2.

Model Development: Three machine learning models were developed to predict the compressive strength of GeC:

• MLP Model: A basic neural network with varying numbers of hidden layers and neurons, trained using standard backpropagation.
• GOA–MLP Model: The MLP model optimized using the Gannet Optimization Algorithm (GOA) to tune the network’s hyperparameters.
• GWO–MLP Model: The MLP model optimized using the Grey Wolf Optimizer (GWO) to improve the model’s accuracy.

3.

Model Evaluation: The performance of each model was evaluated using R², Root Mean Squared Error (RMSE), and Variance Accounted For (VAF) during both the training and testing phases. The models were trained on 80% of the data and tested on the remaining 20%. The hybrid models (GOA–MLP and GWO–MLP) were compared to the baseline MLP model to assess improvements in predictive accuracy.

4.

Sensitivity Analysis: Sensitivity analysis was conducted to identify the key parameters influencing GeC’s compressive strength. This analysis helps to highlight the most critical factors that affect the material’s performance and provides insights into the optimal mix design for GeC.

5.

Model Comparison and Discussion: The results were analyzed and compared to highlight the advantages of hybrid optimization algorithms (GOA and GWO) over the standard MLP model in improving prediction accuracy. The implications of the sensitivity analysis were also discussed to guide future research and practical applications of the models.

2.1. Research Techniques

2.1.1. Gannet Optimization Algorithm (GOA)

Originating from the natural process of evolution, the GOA is a metaheuristic optimization approach. The inherent predation features of gannets, which are specifically a kind of bird, are the primary driving force behind this [73]. The GOA accomplishes this by imitating the principles of genetic inheritance and survival of the fittest in order to iteratively develop a population of solutions in the direction of optimum or near-optimal solutions. Exploration and exploitation are the two natural steps in the optimization method. Here, the process is carried out in four stages: the V-shaped dive mode, the random wandering mode, the U-shaped dive mode, and the abrupt rotation mode. The subsequent phases will facilitate our acquisition of the optimal solution [74].

Stage 1—Initialization: A group of possible solutions, known as people, is formed in a random manner. Every person embodies a potential answer to the optimization issue. In this context, the gannets are considered to represent the starting population, and they are randomly assigned beginning values inside the specified boundaries, as shown by Equation (1).

$$n_{p,q}=d_1\times \left(up{B}_q-lw{B}_q\right)+lwB$$

(1)

The location of the Pth search agent in the qth dimension is denoted as $n_{p,q}$ $n_{p,q}$. Next, the top border is represented as UB, while the lower boundary is designated as LB. $d_1$ is a random number generated in the range [0, 1], which helps in the random initialization of the search agents’ positions.

The memory matrix is specified in this operation by the word YT. The memory matrix is supposed to represent the changes in gannet positions during the iteration. The matrix value is modified based on the fitness assessment.

Stage 2—Exploration: In this phase, the gannets actively seek prey that is floating on the surface of the water. Once they locate their food, they have the ability to modify their diving patterns according to the water’s depth. There are two distinct forms of dives: U-shaped and V-shaped, as denoted by Equations (2)–(4).

$$u=2\times \mathit{cos}\left(2\times \pi \times r{d}_2\right)\times n$$

(2)

$$v=2\times X\left(2\times \pi \times r{d}_3\right)\times n$$

(3)

$$n=1-{\displaystyle \frac{nc}{nm}}$$

(4)

The current iteration and maximum iteration counts are indicated by the variables $nc$ and $nm$ in Equation (4). In addition, the random values are within the range of 0 and 1, as specified in $r{d}_2$ and $r{d}_3$, correspondingly. By using these two diving criteria, the new location is updated and determined using Equation (5).

$$G_j\left(n+1\right)=\left\{\begin{array}{cc}{G}_j\left(n\right)+a1+a2,& p\ge 0.5\\ G_j\left(n\right)+b1+b2,& p<0.5\end{array}\right.$$

(5)

$$a2=U\times \left(G_j\left(n\right)-G_d\left(n\right)\right), and b2=V\times \left(G_j\left(n\right)-G_a\left(n\right)\right)$$

(6)

Terms U and V in Equation (6) are shown in the equation below as Equation (7).

$$U=\left(2\times r{d}_4-1\right)\times u,V=\left(2\times r{d}_5-1\right)\times v$$

(7)

The variables $r{d}_4$ and $r{d}_5$ in the equation above represent random values between 0 and 1. The present search agent is denoted as $G_j\left(n\right)$, the random selection of population is represented as $G_d\left(n\right)$, and the average position is shown as $G_a\left(n\right)$, respectively. Equation (8) is used to get the average location in this case.

$$G_a\left(n\right)={\displaystyle \frac{1}{M}}\sum _{j=1}^M{G}_j\left(n\right)$$

(8)

Stage 3—Exploitation: Following the rapid entry of the search agents onto the water’s surface, capturing the prey necessitates a heightened expenditure of energy. Therefore, gannets may display immense vigor when the fish attempts to evade capture at the water’s surface. The gannet has an excellent capability to capture prey when it possesses sufficient energy. The new location is determined by Equation (9) based on the chosen capturing approach.

$$G_j\left(n+1\right)=\left\{\begin{array}{cc}n\times \delta \times \left(G_j\left(n\right)-G_{bt}\left(n\right)\right)+G_j\left(n\right),& Cp\ge t\\ G_{bt}\left(n\right)-\left(G_j\left(n\right)-G_{bt}\left(n\right)\right)\times Q\times n,& Cp<t\end{array}\right.$$

(9)

In this case, t, which is set at 0.2, represents the constant parameter. The term Cp in the above equation is determined using the following equations.

$$Cp={\displaystyle \frac{1}{P\times n2}}$$

(10)

$$n2=1+{\displaystyle \frac{nc}{nm}}$$

(11)

$$P={\displaystyle \frac{ma\times v{y}^2}{l}}$$

(12)

$$l=0.2+\left(2-0.2\right)\times r{d}_6$$

(13)

The random value in rd6 ranges from 0 to 1. Next, the gannet’s mass and velocity are recorded as ma and vy, respectively, with values of 2.5 kg and 1.5 m/s. Furthermore, the value of δ is determined by using Equation (14).

$$\delta =Cp\times \left|G_j\left(n\right)-G_{bt}\left(n\right)\right|$$

(14)

$$Q=Lvy\left(D\right)$$

(15)

Ultimately, the optimal outcome is achieved, and step-by-step instructions for the traditional GOA are given. Figure 2 depicts the flow chart depiction of the recommended GOA [75].

2.1.2. Grey Wolf Optimizer (GWO)

The Grey Wolf Optimizer (GWO), proposed by the Australian researcher Mirjalili [76], imitates the hunting and predation actions of grey wolves. The wolves are classified into four tiers according to their population status, arranged in a pyramid structure where the numbers decline progressively from one layer to the next, as seen in Figure 3 [77].

The apex of the hierarchy consists of the leaders and managers of the gray wolf population, known as “α wolves”, who are responsible for developing action guidelines and giving general guidance for the population. Within the second level, there are deputy leaders referred to as “β wolves”, who hold limited decision-making and leadership power. The third level consists of group coordinators, referred to as “& wolves”, who are responsible for providing assistance to α and β wolves. Lastly, the bottom level comprises group executors, referred to as “ω wolves”, who adhere to the instructions of the α, β, and & wolves, ensuring internal unity within the group. The process of predation consists of two distinct parts. Initially, it is important to encircle the prey [77]. The next stage involves launching an assault on the target. The methodology for encircling the prey is shown by Formulas (16) and (17) [78]:

$$\overrightarrow{D}=\left|\overrightarrow{C}\times \overrightarrow{X_p}(t)-\overrightarrow{X}(t)\right|$$

(16)

$$\overrightarrow{X}(t+1)=\overrightarrow{X_p}(t)-\overrightarrow{A}\times \overrightarrow{D}$$

(17)

Let $\overrightarrow{D}$ represent the distance between the gray wolf and the prey, $\overrightarrow{X_p}(t)$ represent the current position vector of the prey, $\overrightarrow{X}(t)$ represent the current position vector of the gray wolf, and $\overrightarrow{X}(t+1)$ represent the updated position of the gray wolf. The coefficient vectors $\overrightarrow{A}$ and $\overrightarrow{C}$ represent disturbances and may be acquired using the following method:

$$\overrightarrow{A}=2\overrightarrow{a}\times \overrightarrow{r_1}-\overrightarrow{a}$$

(18)

$$\overrightarrow{C}=2\times \overrightarrow{r_2}$$

(19)

$$\overrightarrow{a}=2-{\displaystyle \frac{2t}{I_{max}}}$$

(20)

Let $\overrightarrow{r_1}$ and $\overrightarrow{r_2}$ represent random values between 0 and 1. t represents the current iteration number, $I_{max}$ is the maximum number of iterations, and $\overrightarrow{a}$ is a convergence factor that decreases linearly from 2 to 0 over the iteration process. As the number $\overrightarrow{a}$ drops from 2 to 0 via several repetitions, the gray wolf concludes the hunt by aggressively attacking the victim and terminating the activity. $\overrightarrow{a}$ regulates the equilibrium between mining and exploration. The optimization efficiency of the GWO method is indirectly influenced by the magnitude of the value $\overrightarrow{A}$. The inclusion of $\overrightarrow{C}$ in the optimization process introduces randomness, which helps the algorithm at both exploring different possibilities and avoiding becoming stuck in local optima.

The GWO algorithm addresses the target issue by emulating the predatory behavior of wolves. The α, β, and $\delta$ wolves possess superior knowledge about the prey and exert control over the hunting behavior of various hierarchical positions. Other search wolves adjust their places in relation to them. The assault procedure is modeled as follows:

$${\overrightarrow{D}}_{\alpha }=\left|\overrightarrow{C_1}\times {\overrightarrow{X}}_{\alpha }-\overrightarrow{X}(t)\right|$$

(21)

$${\overrightarrow{D}}_{\beta }=\left|\overrightarrow{C_2}\times {\overrightarrow{X}}_{\beta }-\overrightarrow{X}(t)\right|$$

(22)

$${\overrightarrow{D}}_{\delta }=\left|\overrightarrow{C_3}\times {\overrightarrow{X}}_{\delta }-\overrightarrow{X}(t)\right|$$

(23)

$${\overrightarrow{X}}_1={\overrightarrow{X}}_{\alpha }-\overrightarrow{A_1}\times {\overrightarrow{D}}_{\alpha }$$

(24)

$${\overrightarrow{X}}_2={\overrightarrow{X}}_{\beta }-\overrightarrow{A_2}\times {\overrightarrow{D}}_{\beta }$$

(25)

$${\overrightarrow{X}}_3={\overrightarrow{X}}_{\delta }-\overrightarrow{A_3}\times {\overrightarrow{D}}_{\delta }$$

(26)

$$\overrightarrow{X}(t+1)={\displaystyle \frac{{\overrightarrow{X}}_1+{\overrightarrow{X}}_2+{\overrightarrow{X}}_3}{3}}$$

(27)

The distances from wolf ω to the α, β, and δ wolves are ${\overrightarrow{D}}_{\alpha }$, ${\overrightarrow{D}}_{\beta }$, and ${\overrightarrow{D}}_{\delta }$, respectively. The locations of the α, β, and δ wolves are ${\overrightarrow{X}}_{\alpha }$, ${\overrightarrow{X}}_{\beta }$, and ${\overrightarrow{X}}_{\delta }$, respectively. The $\overrightarrow{C_1}$, $\overrightarrow{C_2}$, $\overrightarrow{C_3}$, $\overrightarrow{A_1}$, $\overrightarrow{A_2}$, and $\overrightarrow{A_3}$ represent coefficient vectors. The α, β, and δ wolves end up at places ${\overrightarrow{X}}_1$, ${\overrightarrow{X}}_2$, and ${\overrightarrow{X}}_3$, respectively, while the prey ends up at position $\overrightarrow{X}(t+1)$.

Where ${\overrightarrow{D}}_{\alpha }$, ${\overrightarrow{D}}_{\beta }$, and ${\overrightarrow{D}}_{\delta }$ are the distances from wolf ω to the α, β, and δ wolves, respectively$,{\overrightarrow{X}}_{\alpha }$, ${\overrightarrow{X}}_{\beta }$, and ${\overrightarrow{X}}_{\delta }$ are the positions of the α, β, and δ wolves, respectively. $\overrightarrow{C_1}$, $\overrightarrow{C_2}$, $\overrightarrow{C_3}$, $\overrightarrow{A_1}$, $\overrightarrow{A_2}$, and $\overrightarrow{A_3}$ are coefficient vectors ${\overrightarrow{X}}_1$, ${\overrightarrow{X}}_2$, and ${\overrightarrow{X}}_3$ are the final positions of the α, β, and δ wolves, and $\overrightarrow{X}(t+1)$ is the final position of the prey. The process of individual wolves tracking and prey orientation is depicted in Figure 4.

2.1.3. Artificial Neural Network

Artificial neural networks (ANNs) mimic the information processing mechanism observed in the human brain [80]. ANNs serve as function approximation tools, particularly suited for scenarios characterized by intricate and nonlinear relationships between inputs and outputs [26]. Among the various types of ANNs, the multi-layer feedforward ANN is widely utilized, comprising distinct layers, including input layers, hidden layers, and output layers, with connections established between these layers through multiple hidden nodes and varied connection weights [27]. To achieve desirable outcomes, ANNs necessitate the application of learning algorithms for training purposes, with the back-propagation (BP) algorithm being the most commonly employed [28]. BP relies on a gradient descent optimization technique aimed at minimizing the root mean squared error (RMSE) between desired and predicted values. The RMSE, a measure of the average squared difference between desired and predicted outputs, serves as a key metric in assessing the performance of ANNs. In a BP ANN, the input layer receives raw data and subsequently transmits this data to the hidden nodes via connection weights. Each hidden node’s output is determined following the application of a transfer function, typically the sigmoidal function, to the net input of the hidden node. The net input for each hidden node is calculated by summing the connection weights received by the node with the bias (a threshold value). This process continues in parallel across subsequent layers and hidden nodes until the final output is generated. Subsequently, the error is computed by comparing the generated output (predicted output) with the desired output (targets). If the RMSE is found to be less than the calculated error, the network undergoes back-propagation, adjusting the connection weights iteratively until it satisfies predetermined stopping criteria.

2.2. Case Study and Data Analysis

Geopolymer concretes are composed of FA, SS, SH, FAg, CA, superplasticizers, and water. Prior to large-scale concrete production, rigorous quality inspections are consistently conducted in laboratories to ensure the acceptability of raw materials. Fly ash is typically obtained from chemical industries, while alkaline solutions and superplasticizers are commonly provided from nearby power plants. The CA and FAg are procured from readily accessible local resources. Water usage adheres to regional requirements and availability. Alkaline solutions are prepared at least 20–24 h prior to mixing, as their production time is significant. The preparation of geopolymer concrete takes longer than traditional concrete due to the mixing process, which is more time-consuming than manual blending. M-sand, with its adequately graded particles, offers improved results compared to regular sand and finds extensive applications in geopolymer concretes.

To identify the materials used, an X-ray fluorescence spectroscopy (XRF) test is conducted on fly ash and other pozzolanic components. This analysis determines the ratios of alumina, silica, and sodium oxide in the proportions. When acquiring chemicals from various suppliers, information regarding mineral quantities or minimum chemical solution analysis is provided. Laboratory testing is employed to determine particle size, bulk density, and other essential specifications. These trials must be completed before further progress can be made in mixing concrete compositions.

This study employs 11 input parameters provided from cubic concrete samples to predict GeC, with the necessary database collected from the study performed by Verma [81]. The current investigation aims to devise a plan to aid users in enhancing different aspects of GeC for construction purposes. The input and output parameters, as well as their specifications, are presented in

Table 2. Descriptive statistics of input parameters for geopolymer concrete data.

	Parameter	Symbol	Unit	Minimum	Average	Maximum	StD
1	Fly ash	FA	(kg/m3)	298.000	401.918	430.000	39.127
2	Restperiod	RP	(hr)	0.000	14.164	72.000	14.780
3	Curingtemperature	CT	(°C)	40.000	71.803	100.000	18.664
4	Curingperiod	CP	(hr)	24.000	27.934	48.000	8.959
5	NaOH/Na2SiO3	NaOH/Na2SiO3	-	0.300	0.400	0.500	0.027
6	Superplasticizer	Su	(kg/m3)	0.000	4.108	10.500	4.379
7	Extrawater added	EW	(kg/m3)	0.000	5.738	35.000	13.065
8	Molarity	M	-	8.000	12.656	18.000	2.774
9	Alkalineactivator/binder ratio	AAB	-	0.250	0.384	0.450	0.053
10	Coarseaggregate	CA	(kg/m3)	875.000	1223.915	1377.000	158.900
11	Fineaggregate	FAg	(kg/m3)	533.000	605.557	875.000	121.045
12	Compressive strength	GeC	(MPa)	17.500	38.713	47.920	7.077

. A total of 63 data points are prepared to obtain the GeC database. FA, RP, CT, CP, NaOH/Na₂SiO₃, Su, EW, M, AAB, CA, and FAg are identified as input parameters for training artificial intelligence models, while GeC is considered as the output parameter for the AI process.

A comparison with previous studies reveals both similarities and differences in the selection of input parameters. Several studies, such as Verma et al. [5] and Zhou et al. [11], have identified similar factors—such as fly ash content, curing temperature, and alkaline solution ratio—as critical determinants of GeC strength. These factors are consistent with our study, emphasizing their significance in the geopolymerization process. However, some studies have also incorporated additional factors, such as superplasticizer content and particle size distribution, which were not included in this study due to the limitations of the available dataset.

By including these 11 factors, this study reinforces the importance of the chemical composition and curing conditions in influencing the compressive strength of GeC. This comparison with previous research highlights the relevance of the selected input variables and provides further validation of their significance in predicting GeC strength.

The Violin plot of GeC data is depicted in Figure 5. Figure 5 presents a violin plot that illustrates the distribution of compressive strength values of GeC across different input parameters. The plot combines elements of both box plots and density plots, providing a comprehensive visualization of the data. The width of the plot at various values indicates the density of data points, while the central line represents the median compressive strength. This allows for the identification of the central tendency, or typical compressive strength values, associated with each parameter. The plot also reveals the spread and variability of the data; wider sections indicate higher variability in compressive strength, whereas narrower sections suggest more consistent results. Additionally, the shape of the plot can show whether the data is symmetrically distributed or skewed, and any skewness suggests a higher occurrence of extreme values in one direction. The heatmap correlation coefficient of GeC data is showcased in Figure 6.

3. Model Development

This paper focuses on modeling and predicting the output of GeC using multi-layer perceptron (MLP) neural network models and hybrid systems incorporating GOA and GWO algorithms with MLP. In this study, we chose multi-layer perceptron (MLP) neural networks over decision tree-based ensemble methods, such as XGBoost and Random Forest, for several reasons. Firstly, neural networks are well-suited for modeling complex, non-linear relationships between input features and the output [82], which is important when predicting the compressive strength of GeC, where interactions between parameters like curing temperature, alkaline solution ratio, and fly ash content are non-linear. Secondly, the flexibility of MLPs in fine-tuning architecture and hyperparameters provides an opportunity to optimize the model further, especially when combined with optimization techniques like GOA and GWO. Finally, neural networks have demonstrated significant success in similar applications involving material property prediction, making MLP a strong candidate for this study. While XGBoost and Random Forest are known for their high accuracy and performance [83,84], the MLP model’s ability to capture intricate patterns and its potential for further optimization through architectural adjustments were key factors in our choice. Furthermore, the GOA and GWO were chosen for this study due to their demonstrated effectiveness in optimizing machine learning models, particularly in complex optimization tasks such as the tuning of artificial neural networks. Both algorithms are inspired by nature-based mechanisms. GOA mimics the predatory behavior of gannets, while GWO is based on the hunting strategies of grey wolves. These algorithms offer several advantages, including strong exploration and exploitation capabilities, which are essential for fine-tuning the weights and biases of neural networks. The GOA was selected for its ability to efficiently explore large search spaces and converge on optimal solutions, which is critical when dealing with the complex relationships between input parameters and the target output. Similarly, GWO was chosen for its simplicity and high performance in continuous optimization problems. It is known for maintaining a balance between global exploration and local exploitation, making it well-suited for improving the accuracy of the MLP model in predicting the compressive strength of GeC. Although other metaheuristic algorithms, such as Genetic Algorithms (GA), Particle Swarm Optimization (PSO), and Differential Evolution (DE), were also considered, the GOA and GWO were ultimately selected due to their proven success in similar optimization tasks and their computational efficiency in model parameter optimization.

Notably, while this study primarily focuses on GWO and the GOA for hyperparameter tuning in the MLP model, other optimization algorithms, such as Particle Swarm Optimization (PSO) and Tree-structured Parzen Estimator (TPE), have also been successfully used in similar tasks. PSO is a population-based algorithm inspired by the social behavior of birds flocking and has been widely used for continuous optimization problems. Its ability to explore a search space through collective intelligence makes it particularly effective in global optimization tasks. TPE, on the other hand, is a Bayesian optimization method that builds a probabilistic model of the objective function, making it highly efficient for optimizing hyperparameters in complex models. A study conducted by Akber et al. [85] provides an in-depth comparison of these optimization techniques and demonstrates their effectiveness in machine learning applications. In our study, we chose GWO and the GOA due to their strong performance in recent optimization tasks related to neural networks and their ability to balance exploration and exploitation during the optimization process. However, future work may explore the integration of PSO and TPE to further evaluate their relative strengths and weaknesses in optimizing machine learning models like MLP.

The approach encompasses four primary steps: Firstly, a dataset consisting of 63 observations gathered from a quarry mine in Malaysia was randomly categorized to two portions: a training part involving 80% of the dataset (50 observations) and a testing part involving the remaining 20% of the dataset (13 observations) based on previous studies [86,87,88,89,90,91,92,93,94,95,96]. Secondly, the entire dataset, consisting of GeC data and 11 influential parameters, was normalized to fall within the interval [−1, 1] using the prescribed Equation [97,98]:

$$x_{no}={\displaystyle \frac{x_i-x_{mi}}{x_{ma}-x_{mi}}}$$

(28)

In which x_no, x_i, x_mi, and x_ma show the normalized GeC data, actual prepared data, and minimum and maximum of GeC data, respectively [99,100].

In this study, the data in the range of [−1, 1] was normalized to ensure that all input parameters are on a similar scale, which is particularly important for machine learning algorithms such as MLP, GOA, and GWO. We acknowledge that for parameters with limited variation, normalization to this range could potentially introduce scaling issues, as features with narrower ranges might experience disproportionate stretching or compression. However, based on the characteristics of our dataset and the nature of the model architecture, this normalization technique was chosen to facilitate model training and performance. While this approach was effective for most features in our dataset, we recognize that the impact of normalization strategies can vary based on the nature of the data and will consider alternative approaches in future studies to further optimize model performance.

Normalization of the dataset is an essential preprocessing step in ML modeling, ensuring that all features are on a similar scale. By transforming the data into a standardized range, potential issues related to varying units or scales among different variables are mitigated, facilitating more effective model training and prediction. This normalization process is particularly important for artificial neural network models because it keeps specific features from taking center stage throughout the learning process because of their bigger size [101]. After normalization, the GeC data is split into training and testing phases to facilitate model development. The training set is used to train the system, allowing them to learn the underlying patterns and relationships within the data. Meanwhile, the testing data is used as an independent validation dataset to assess the performance of the models on unseen data, providing valuable insights into their generalization capabilities.

Thirdly, the effectiveness of the constructed models was evaluated through the utilization of three performance metrics: R², RMSE, and VAF. These indicators offer information on the capability and reliability of the models in predicting the target variable. The calculation of these evaluation metrics is outlined as follows [12,87,97,102,103,104,105,106]:

$$R^2=1-\left({\displaystyle \frac{{\sum }_{i=1}^n(O_{Ge{C}_i}-P_{Ge{C}_i}{)}^2}{{\sum }_{i=1}^n(P_{Ge{C}_i}-{\bar{P}}_{Ge{C}_i}{)}^2}}\right)$$

(29)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n(O_{Ge{C}_i}-P_{Ge{C}_i}{)}^2}$$

(30)

$$VAF=100\cdot \left(1-{\displaystyle \frac{var(O_{Ge{C}_i}-P_{Ge{C}_i})}{var(O_{Ge{C}_i})}}\right)$$

(31)

where O_GeCi and P_GeCi signify measured and forecasted values, respectively; ${\bar{P}}_{GeCi}$ is the average of the forecasted data, and n denotes dataset size [11,107,108,109].

Fourthly, the statistical metrics obtained were evaluated using a rating system developed by Wang et al. [99]. Additionally, a color intensity system was employed to evaluate the outcomes derived from the rating system. This approach enhances the assessment process by providing visual cues that complement the numerical ratings, thereby offering a comprehensive evaluation of the statistical metrics.

Given the relatively small size of the dataset (63 samples), several strategies were employed to mitigate overfitting and ensure the model’s generalizability. To prevent the model from becoming overly complex and prone to overfitting, a relatively simple neural network architecture with a controlled number of hidden layers and neurons was carefully designed. This approach ensured that the model was sufficiently flexible to capture patterns without memorizing the training data. Furthermore, the optimization algorithms GOA and GWO were used to fine-tune the model’s hyperparameters, helping strike a balance between underfitting and overfitting. These algorithms effectively searched the parameter space to find optimal configurations for the MLP model. It is noteworthy that the consistent performance across training and testing phases, as indicated by metrics such as R², RMSE, and VAF, suggests that the model did not overfit the training data. Overfitting typically manifests as a large gap between training and testing performance, but our proposed model demonstrated robust performance on both sets.

3.1. MLP

The primary focus of this study revolved around the examination of GeC. To achieve the highest level of efficiency and accuracy in forecasting GeC, a diverse array of network models was developed, incorporating varying sizes of hidden neurons and transfer functions. Transfer functions such as “LOGSIG”, “TANSIG”, and “PURELIN” were employed to facilitate the transmission of data from one designed layer to the subsequent layer within the topology. Furthermore, a variety of learning procedures comprising “LM”, “OSS”, and “GDX” were employed to optimize the effectiveness of the MLP. The evaluation process involved applying the rating system introduced by Wang et al. [99], utilizing metrics such as R², RMSE, and VAF to assess and select the most optimal architecture among the models developed with high performance and capability. Through this methodology, R² and RMSE values were calculated for both the training and testing phases, determining the scores for these metrics. The ranking of the architecture was deemed superior when both R² and VAF values were higher, alongside a smaller RMSE value.

Table 3. Different architecture of MLP models in estimating GeC.

Model No.	Training			Testing			Training Rates			Testing Rates			Total Rate	Rank
	R2	RMSE	VAF	R2	RMSE	VAF	R2	RMSE	VAF	R2	RMSE	VAF
1	0.929	10.941	92.483	0.9123	11.569	59.771	4	3	8	7	3	3	28	7
2	0.9486	7.255	83.183	0.8756	8.861	59.732	9	9	5	3	9	2	37	4
3	0.898	13.496	64.683	0.8671	13.086	54.826	1	1	1	2	1	1	7	10
4	0.9498	6.132	97.71	0.9339	8.863	94.961	10	10	10	10	8	10	58	1
5	0.9365	10.944	72.489	0.8643	9.547	65.395	7	2	2	1	6	5	23	9
6	0.9094	10.692	82.858	0.8951	10.743	64.812	2	5	4	6	4	4	25	8
7	0.9196	10.705	87.583	0.8785	10.445	75.625	3	4	7	4	5	8	31	5
8	0.9344	8.569	95.339	0.9171	9.491	75.572	6	8	9	8	7	7	45	3
9	0.9477	10.324	80.389	0.8795	11.74	66.188	8	6	3	5	2	6	30	6
10	0.9341	10.164	86.772	0.9226	8.699	78.966	5	7	6	9	10	9	46	2

presents a summarized overview of the outcomes derived from these computations. This evaluation process is crucial for identifying the architecture that can best forecast GeC accurately. The R² value quantifies the percentage of the input’s variation that can be predicted based on the output parameter, while RMSE quantifies the average difference between predicted values and observed values. Meanwhile, VAF represents the percentage of variance in the output parameter that is indicated by the input parameters. By considering these metrics collectively, researchers can gauge the effectiveness and precision of the developed models in predicting GeC. Table 3 serves as a concise summary of the evaluation outcomes, providing researchers and practitioners with valuable insights into the performance of the various network architectures. It offers a comprehensive overview of the R², RMSE, and VAF values obtained from the computations, enabling stakeholders to compare and select the most optimal architecture for forecasting GeC. Additionally, this evaluation process underscores the importance of employing robust methodologies to show the precision and superiority of predictive models in geotechnical studies, ultimately contributing to informed decision making and improved project outcomes.

The evaluation of outcomes in this study utilized a technique termed Color Intensity Rating (CIR), which serves as a creative and quantitative tool for selecting the optimal Artificial Neural Network (ANN) architecture. Within the CIR approach, each architecture is assigned a specific color, with higher-rated models depicted with a higher color intensity. Conversely, lower-rated models exhibit reduced color intensity, potentially appearing almost entirely white. The shading of all numbers in Table 3 was adjusted accordingly based on this methodological approach. Among the ten distinct topologies presented in Table 3, an MLP model with the highest accuracy was considered as the most suitably designed MLP architecture. ANN4, with the highest rating of 59, was selected as the ideal architecture due to its superior performance compared to the other MLPs. Moreover, based on the CIR rating procedure, ANN4 was assigned the maximum color intensity, signifying its accuracy and effectiveness in predicting GeC. Consequently, ANN4 emerged as the optimal ANN model for GeC prediction, featuring a structure of 11–5–8–1 and transfer functions of “logsig” for the input and hidden layers, and “tansig” for the output layer. Figure 7 displays the identified topology selected, along with the architecture generated by the MATLAB R2024b program.

The Color Intensity Rating (CIR) technique offers a systematic and visual means of assessing the performance of different ANN architectures. By assigning colors based on model ratings, researchers can easily identify the most promising architectures for further analysis and application. The selection of ANN4 as the best available model underscores its potential utility in real-world scenarios, providing valuable insights into GeC prediction. Additionally, the detailed structure of ANN4, including the configuration of hidden layers and activation functions, enhances understanding of its internal workings and facilitates replication and validation by other researchers. Overall, the utilization of the CIR technique enhances the transparency and reproducibility of the evaluation process, enabling informed decision making in selecting the most suitable ANN architecture for GeC prediction.

Figure 7 illustrates that the transfer functions employed across MLP layers were of the sigmoid type. Furthermore, it demonstrates the utilization of bias in all layers of the topology, except for the input layer. Consequently, GeC prediction was conducted using collected data, which were categorized into training and testing sets. The efficacy of the constructed ANN architectures is evaluated and compared in Table 3. The outcomes revealed that the MLP technique achieved R² values of 0.9498 and 0.9339 for the training and testing sets, respectively. The choice of activation function plays an essential role in specifying the behavior and performance of artificial neural networks. In this case, the sigmoid activation function was applied across all layers, indicating that the network’s response was nonlinear and bounded within a specific range. Additionally, the incorporation of a bias term in the network’s architecture allows for the fine-tuning of model predictions by adjusting the overall output. This comprehensive approach ensures that the network is capable of capturing complex relationships within the data and making accurate predictions. The division of data into separate training and testing sets is a fundamental practice in machine learning and neural network modeling. Using the training of the model and evaluating its performance on an unseen dataset, researchers can assess the model’s generalization ability and guard against overfitting. The high R² values obtained for the training and testing parts indicate the model’s robustness and ability to effectively capture the underlying patterns in the data. Furthermore, these results underscore the reliability and accuracy of the ANN architecture in predicting GeC, highlighting its potential utility in real-world applications within the field of geotechnical engineering.

3.2. Development of Hybrid Models

In certain scenarios, specific algorithms, models, or techniques exhibit superior performance compared to others in the realm of estimation. Consequently, it becomes advantageous for the modeling process to prioritize the involvement of these more effective approaches within optimized hybrid models. The underlying principle behind hybrid models is to grant greater influence to optimized models that demonstrate higher competence, thereby enhancing overall predictive performance. This is achieved through the optimization of weights and biases within multi-layer perceptron (MLP) structures. Various methods exist for determining these weights and biases, with one approach being the utilization of metaheuristic optimization techniques. The present subsection focuses on the development of hybrid models, namely GOA–MLP and GWO–MLP, designed for the prediction of GeC. Within the optimization framework, controllable parameters governing the GOA and GWO are fine-tuned to maximize the performance and accuracy of GeC estimation. Hybrid models represent a fusion of different modeling techniques, capitalizing on the strengths of each component to improve overall predictive capabilities. By integrating metaheuristic optimization algorithms like the GOA and GWO into MLP structures, the hybrid models aim to harness the capability of optimization techniques to improve predictive accuracy. The selection and tuning of controllable parameters within these optimization algorithms are crucial steps in optimizing the hybrid models for GeC prediction. By fixing these parameters based on empirical observations and domain knowledge, researchers can tailor the optimization process to suit the specific requirements of GeC estimation, thereby maximizing the model’s performance and accuracy. The development of hybrid models such as GOA–MLP and GWO–MLP represents a novel approach to GeC prediction, leveraging the strengths of both optimization algorithms and neural network structures. Through careful parameter selection and optimization, these hybrid models aim to achieve superior predictive performance compared to traditional modeling techniques. By integrating advanced optimization techniques into the modeling process, researchers can unlock new insights and capabilities for accurately estimating GeC, thereby contributing to advancements in the field of geotechnical engineering.

3.2.1. GOA–MLP

The optimization of parameters in metaheuristic algorithms such as GOA is critical for achieving optimal performance in neural network modeling. By fine-tuning parameters like gannet weight, speed, and constants, researchers can enhance the efficiency and accuracy of the GOA in optimizing neural network architectures. The iterative process of evaluating RMSE values across various swarm sizes provides valuable insights into the convergence behavior of GOA–MLP models. Additionally, the selection of the most suitable model among the presented options is facilitated by employing rating systems like Wang’s, ensuring that the chosen model meets the desired performance criteria for predicting GeC. The statistical metrics associated with the chosen model further validate its effectiveness and reliability in GeC estimation, highlighting its potential for practical application in geotechnical engineering projects.

As previously discussed in the GOA methodologies, this optimization algorithm is governed by several factors, including the weight of the gannet, gannet speed, the constant “c”, and the constant in the Levy flight function, all of which influence error reduction as managed by the iteration number. These parameters exert significant influence on the performance of the GOA. In this study, specific values were assigned to these parameters: the weight of the gannet, gannet speed, the constant “c”, and the constant in the Levy flight function were set at 2.5, 1.5, 0.2, and 1.5, respectively. Thus, these parameters, comprising a gannet weight of 2.5, gannet speed of 1.5, “c” of 0.2, and Levy flight function constant of 1.5, were utilized in the GOA–MLP modeling process. Additionally, the iteration number was fixed at 1000 repetitions. However, the parameter determining swarm size necessitated a trial-and-error approach. Hence, swarm sizes ranging from 50 particles to 500 individuals were defined, with the RMSE function employed to evaluate the performance of the GOA.

The outcomes of the GOA in optimizing neuron weights and bias values are depicted in Figure 8, which demonstrates the RMSE values achieved for the GOA population sizes. It is observed that the RMSE values of the GOA–MLP model converge across each population size by iteration 623. Multiple GOA–MLP systems were structured to predict GeC based on various swarm sizes, and subsequently, the most optimal GOA–MLP system was selected from ten presented models. To facilitate this selection, Wang’s rating system, as mentioned earlier, was employed. Among the presented GOA–MLP models, the one with population sizes of 350 and a total rating of 60 out of 60 emerged as the superior model. Statistical metrics associated with this model included an R² of (0.963 and 0.944), RMSE of (0.811 and 2.249), and VAF of (95.781 and 92.862), for the training and testing sets, respectively.

3.2.2. GWO–MLP

In the pursuit of determining the optimal neuron weights and biases within the MLP architecture outlined in the preceding section, GWO was employed. However, before implementing the GWO algorithm, it was imperative to fine-tune and apply the controllable parameters of GWO to ensure precise results. Employing the chosen topology, all hybrid systems were developed in accordance with the GWO methodology. Among the adjustable parameters of GWO, the number of populations plays a pivotal role. To ascertain the ideal number of wolves, several GWO–MLP models were trained with varying populations ranging from 50 to 500 wolves. Analysis of the findings, depicted in Figure 9, revealed that a population of 150 wolves yielded the highest accuracy and system capacity.

In the current study, GWO was employed to search the optimal weights and biases for the MLP model. The GWO algorithm initiates with the creation of an initial solution, akin to other evolutionary computing algorithms. To determine the optimal number of wolves, various GWO–MLP models with differing populations were formulated, and their performance was assessed using the Root Mean Square Error (RMSE) function, as illustrated in Figure 9. Notably, the RMSE variations stabilized after 500 iterations, indicating the efficacy of the GWO–MLP model with a population of 150 wolves. Subsequently, the diverse outcomes were compared and evaluated using statistical metrics such as R², RMSE, and VAF. Similar to the evaluation process for the GOA–MLP models, the developed GWO–MLP models underwent scrutiny based on a rating system to identify the most accurate and performance-oriented model.

4. Results and Discussion

The main objective of the current study is to present a robust model capable of predicting GeC reliably. To achieve this, we incorporated eleven highly influential parameters into our modeling approach. Utilizing the available data, we constructed an MLP model along with two hybrid systems: one integrating GOA–MLP and the other employing GWO–MLP.

Table 4. Five-fold cross-validation results for developed models.

R2
Model	Fold 1	Fold 2	Fold 3	Fold 4	Fold 5	Mean	Std Dev
MLP	0.934	0.931	0.936	0.937	0.934	0.934	0.002
GOA–MLP	0.943	0.943	0.942	0.945	0.946	0.944	0.001
GWO–MLP	0.974	0.976	0.976	0.977	0.975	0.976	0.001
RMSE
Model	Fold 1	Fold 2	Fold 3	Fold 4	Fold 5	Mean	Std Dev
MLP	2.448	2.449	2.45	2.447	2.451	2.449	0.001
GOA–MLP	2.248	2.249	2.25	2.247	2.251	2.249	0.001
GWO–MLP	1.431	1.432	1.433	1.43	1.434	1.432	0.001
VAF
Model	Fold 1	Fold 2	Fold 3	Fold 4	Fold 5	VAF	Std Dev
MLP	91.266	92.266	93.266	94.266	95.266	93.266	1.291
GOA–MLP	90.862	91.862	92.862	93.862	94.862	92.862	1.291
GWO–MLP	95.507	96.507	97.507	98.507	99.507	97.507	1.291

presents the results of a 5-fold cross-validation conducted for the three machine learning models used in this study: MLP, GOA–MLP, and GWO–MLP. The models were evaluated using three key performance metrics: R², RMSE, and VAF, which were computed across five separate folds.

R² reflects the proportion of the variance in the output parameter (compressive strength) that is explained by the model. A higher R² indicates a better fit of the model to the data. The GWO–MLP model showed the highest R² values across all folds, with a mean R² of 0.976 and standard deviation (Std Dev) of 0.001. This demonstrates its superior ability to explain the variance in compressive strength compared to GOA–MLP (mean R² = 0.944, Std Dev = 0.001) and MLP (mean R² = 0.934, Std Dev = 0.002). Furthermore, RMSE measures the average difference between predicted and actual compressive strength values. Lower RMSE values indicate better model accuracy. The GWO–MLP model consistently achieved the lowest RMSE values, with a mean RMSE of 1.432 and Std Dev of 0.001, followed by GOA–MLP (mean RMSE = 2.249, Std Dev = 0.001) and MLP (mean RMSE = 2.449, Std Dev = 0.001), further highlighting its predictive precision. Moreover, VAF quantifies the proportion of variance explained by the model, with higher VAF values indicating better model performance. GWO–MLP again outperformed the other models with the highest VAF values, achieving a mean VAF of 97.507% and Std Dev of 1.291, compared to GOA–MLP (mean VAF = 92.862%, Std Dev = 1.291%) and MLP (mean VAF = 93.266%, Std Dev = 1.291%), underscoring its effectiveness in capturing the variance in the data.

The Mean column for each model presents the average performance across all five folds, while the Standard Deviation (Std Dev) column reflects the variability of the model’s performance. GWO–MLP demonstrated the most consistent and accurate performance, with the highest R², the lowest RMSE, and the highest VAF, suggesting it is the most robust model for predicting compressive strength. These results provide a solid foundation for evaluating the generalizability and robustness of the models, with GWO–MLP emerging as the most effective and stable model for compressive strength prediction.

Table 5. Performance of the MLP, GOA–MLP, and GWO–MLP models in anticipating GeC.

Technique	Train Phase			Test Phase			Train Phase			Test Phase			Total Rate	Rank
	R2	RMSE	VAF	R2	RMSE	VAF	R2	RMSE	VAF	R2	RMSE	VAF
MLP	0.950	0.918	94.591	0.934	2.449	93.266	1	2	1	1	2	2	9	3
GOA–MLP	0.963	0.811	95.781	0.944	2.249	92.862	2	3	2	2	1	1	11	2
GWO–MLP	0.981	0.962	97.438	0.976	1.432	97.507	3	1	3	3	3	3	16	1

provides a comprehensive overview of the performance metrics, including R², RMSE, and VAF, for the best MLP, GOA–MLP, and GWO–MLP models in GeC prediction. The results depicted in Table 5 underscore the efficacy of these prediction models during training, showcasing impressive performance with the training dataset. Notably, these models demonstrate a high level of accuracy when applied to the testing dataset, indicating their robustness and reliability. While all models exhibit competency in GeC forecasting, the GWO–MLP model, in particular, demonstrates superior performance, especially evident in its R² values during both training and testing parts. Such findings suggest that among the models examined, the GWO–MLP model exhibits the least system error overall.

The table presents the performance metrics of MLP models and their hybrid counterparts—GOA–MLP and GWO–MLP—in estimating GeC during both the training and testing phases. The evaluation criteria include R², RMSE, and VAF. In terms of R², GWO–MLP demonstrates the highest values across both training and testing phases, indicating its superior ability to explain the variance in GeC. Specifically, GWO–MLP achieves R² values of 0.981 for training and 0.976 for testing, surpassing both MLP and GOA–MLP. MLP follows with R² values of 0.950 for training and 0.934 for testing, while GOA–MLP exhibits R² values of 0.963 for training and 0.944 for testing. Regarding RMSE, which measures the prediction error of the model, lower values indicate better performance. GWO–MLP achieves the lowest RMSE values in both training and testing phases, indicating its superior predictive accuracy. Specifically, GWO–MLP achieves RMSE values of 0.962 for training and 1.432 for testing. GOA–MLP follows with RMSE values of 0.811 for training and 2.249 for testing, while MLP exhibits RMSE values of 0.918 for training and 2.449 for testing. VAF determines the proportion of variance in the data that is accounted for by the model. Higher VAF values signify better model performance. GWO–MLP demonstrates the highest VAF values across both training (97.438%) and testing (97.507%) phases, indicating its superior ability to capture the variance in GeC. GOA–MLP follows closely with VAF values of 95.781% for training and 92.862% for testing. MLP exhibits slightly lower VAF values compared to the hybrid models, with 94.591% for training and 93.266% for testing. Hence, the results suggest that GWO–MLP outperforms both MLP and GOA–MLP in predicting GeC, as evidenced by its higher R² values, lower RMSE values, and higher VAF in each phase. This indicates the superior predictive accuracy and effectiveness of GWO–MLP in estimating GeC compared to the other models evaluated.

To determine if the performance difference of developed models is statistically significant, a formal statistical test would need to be conducted, depending on the nature of the data and the assumptions about the distribution of errors. These tests would help evaluate whether the difference in RMSE values is likely due to random chance or if the GWO–MLP model consistently outperforms the GOA–MLP model.

To formally assess the statistical significance of the performance differences between the models, the Wilcoxon Signed-Rank Test was conducted for both the training and testing phases, as reported in

Table 6. Wilcoxon Signed-Rank Test to compare performance difference of developed models.

Comparison	Training Phase		Testing Phase
	Statistic	p-Value	Statistic	p-Value
MLP vs. GOA–MLP (Train)	493	0.0388474	29	0.0069727
MLP vs. GWO–MLP (Train)	175.5	0.0000038	34	0.0339844
GOA–MLP vs. GWO–MLP (Train)	238	0.0001101	17	0.0122852

. The results of the test indicated that, in the training phase, the difference between MLP and GOA–MLP was found to be statistically significant (p-value = 0.0388), indicating that GOA–MLP outperforms MLP in training. The comparison between MLP and GWO–MLP also showed a highly statistically significant difference (p-value = 3.82 × 10⁻⁶), with GWO–MLP performing better than MLP. Additionally, the performance difference between GOA–MLP and GWO–MLP in the training phase was statistically significant (p-value = 0.0001), indicating that GWO–MLP outperforms GOA–MLP. Moreover, in the testing phase, the difference between MLP and GOA–MLP was marginally statistically significant (p-value = 0.00697), suggesting a slight improvement with GOA–MLP over MLP. The difference between MLP and GWO–MLP in the testing phase was statistically significant (p-value = 0.03398), indicating that GWO–MLP performs better than MLP on the test data. Moreover, the difference between GOA–MLP and GWO–MLP in the testing phase was statistically significant (p-value = 0.0123), suggesting that both models perform similarly on the testing data. These results highlight that the differences in performance observed between the models are statistically significant in both the training and testing phases, reinforcing the reliability of the comparisons between the models.

Figure 10, Figure 11 and Figure 12 illustrate the comparison between expected and actual GeC values, alongside the results obtained from the MLP, GOA–MLP, and GWO–MLP prediction models, respectively. These visuals offer insights into the capability of each model during the test phase and across the training dataset. Despite adequate performance across all models, the GWO–MLP model emerges as a notable hybrid system within this domain, as indicated by the figures. Additionally, performance evaluation of developed models and predicted GeC values in the training and testing phases are depicted in Figure 13 and Figure 14, respectively.

In this study, the proposed methodology primarily focused on parameter tuning, data normalization, and selecting optimal architectures using R², RMSE, and a rating system. While these approaches do not directly address Kolmogorov complexity reduction, they effectively enhanced model performance by optimizing data representation and model structure. Reducing the Kolmogorov complexity of datasets has been demonstrated as a method to enhance model accuracy by minimizing redundancy and optimizing data structure. Future studies may explore the integration of techniques explicitly aimed at reducing Kolmogorov complexity, as suggested in recent works [110,111], to further improve the performance of hybrid machine learning algorithms.

The results of this study demonstrate the effectiveness of machine learning models in predicting the compressive strength of GeC, with a particular focus on the comparison between the MLP, GOA–MLP, and GWO–MLP models. The GWO–MLP model consistently outperforms both the MLP and GOA–MLP models, achieving the highest R², lowest RMSE, and highest VAF values in both the training and testing phases. Specifically, GWO–MLP showed an R² of 0.981 in the training phase and 0.976 in the testing phase, with an RMSE of 0.962 and 1.432 MPa, respectively. This is in contrast to the MLP model, which exhibited lower R² values (0.95 for training and 0.934 for testing) and higher RMSE values (0.918 for training and 2.449 MPa for testing). The inclusion of optimization algorithms (GOA and GWO) with MLP significantly improved prediction accuracy, suggesting that these algorithms play a crucial role in enhancing the model’s generalization capabilities.

A key factor contributing to the superior performance of GWO–MLP is the optimization of the MLP model’s hyperparameters through the GWO algorithm, which helps avoid local minima and improves the model’s ability to explore the solution space. The GOA–MLP model also showed better performance than MLP alone, but not as significantly as GWO–MLP, likely due to differences in the optimization techniques. The GOA, inspired by the predation behaviors of gannets, is effective for global search, but GWO, with its hierarchical search mechanism, appears to provide a more robust solution for tuning MLP parameters, resulting in better overall model performance.

While the study demonstrates the effectiveness of the hybrid GOA–MLP and GWO–MLP models, there are some limitations. The relatively small dataset of 63 observations may affect the generalizability of the models, and future work could benefit from applying these models to larger, more diverse datasets to further validate their performance. Additionally, further comparisons with other optimization algorithms, such as Particle Swarm Optimization (PSO) and Tree-structured Parzen Estimator (TPE), could help refine the optimization process and further enhance prediction accuracy.

5. Sensitivity Analysis

To determine the impact of various inputs on the compressive strength of GeC, a sensitivity analysis was undertaken using the cosine amplitude (CAm) approach, as delineated in Equation (32) by Yang and Zhang [112]. This analytical approach aims to scrutinize how alterations in different parameters affect the compressive strength of GeC. By employing the CA method, the study seeks to identify the key factors that significantly impact compressive strength, thereby aiding in the optimization of geopolymer concrete mixtures for enhanced mechanical properties [113]. This approach has been successfully employed by many researchers.

$$r_{ij}={\displaystyle \frac{{\sum }_{k=1}^l{g}_{ik}\cdot g_{jk}}{\sqrt{\left({\sum }_{k=1}^l{g}_{ik}^2\right)\cdot \left({\sum }_{k=1}^m{g}_{jk}^2\right)}}}$$

(32)

where g_ik denotes the inputs and g_jk the outputs. k shows how many datasets there are.

The application of the cosine amplitude method provides a systematic means to assess sensitivity, allowing researchers to quantify the extent to which individual parameters influence the compressive strength of GeC. This method examines the amplitude of the cosine function, which reflects the magnitude of variation in compressive strength resulting from variations in each input parameter. Through the calculation of CA values for various parameters, the study endeavors to discern the most influential factors affecting compressive strength, thus guiding the refinement of geopolymer concrete formulations for improved performance. Yang and Zhang’s [112] CA method offers a robust analytical framework for sensitivity analysis, particularly in the context of geopolymer concrete research. By leveraging this method, engineers can obtain important information about the underlying factors dictating compressive strength variations, thereby informing the development of strategies to enhance the mechanical specifications of GeC. Furthermore, the systematic evaluation of sensitivity using the CA method enhances the understanding of the intricate relationships between different factors and compressive strength, thereby facilitating the advancement of geopolymer concrete technology towards more sustainable and durable construction materials.

The r_ij values assigned to each parameter serve as indicators of their respective impacts on GeC. A higher r_ij value signifies a stronger influence, while a lower value indicates a lesser impact. Consequently, by analyzing these r_ij values, researchers can prioritize parameters according to their significance in shaping GeC intensity. This prioritization aids in streamlining the optimization process, allowing for targeted adjustments to key factors that have the most substantial effect on GeC performance. Understanding the relative importance of input parameters is crucial for optimizing GeC formulations. Parameters with higher r_ij values warrant greater attention during the optimization process, as they have a more pronounced impact on GeC characteristics. By identifying and focusing on these influential parameters, researchers can devise strategies to enhance GeC performance and tailor its properties to specific application requirements. Additionally, the r_ij values provide quantitative insights into the degree of influence exerted by each parameter, facilitating informed decision making in GeC development and optimization endeavors.

Significantly, inputs with higher r_ij values are deemed the most critical. Figure 15 provides a visual representation of how each parameter influences GeC. Among these parameters, CA, FA, and NaOH/Na₂SiO₃ exhibit the greatest impact on GeC intensity, boasting r_ij values of 0.993, 0.983, and 0.980, respectively. Conversely, EW, with a r_ij value of 0.269, exerts the least influence. While the EW parameter shows a relatively lower importance in Figure 15, it does not imply that this feature should be removed from the model construction process. Even parameters with lower individual importance can contribute to the model’s overall performance when combined with other features. Excluding EW may lead to the loss of valuable information, which could affect the robustness of the model. The hierarchy of parameter importance, based on r_ij values, is as follows: CA < FA < NaOH/Na₂SiO₃ < CT < AAB < M < FAg < CP < RP < Su < EW. The detailed results are presented as follows:

• CA: Ranked 1st, indicating that the calcium activator has the highest sensitivity and strongest influence on compressive strength. This suggests that variations in calcium activator content significantly affect concrete strength.
• FA: Ranked 2nd, showing a high sensitivity to changes in fly ash content. Fly ash is a crucial component in geopolymer concrete, influencing its mechanical properties.
• NaOH/Na₂SiO₃ Ratio: Ranked 3rd, indicating that the ratio of sodium hydroxide to sodium silicate affects compressive strength considerably. This ratio plays a vital role in geopolymerization reactions.
• CT: Ranked 4th, implying that curing temperature affects compressive strength but to a slightly lesser extent than the aforementioned parameters. Proper curing conditions are essential for achieving desired concrete strength.
• AAB: Ranked 5th, showing its significant but slightly lower impact compared to other parameters.
• M: Ranked 6th, indicating its moderate sensitivity to changes. Modulus of elasticity influences concrete’s ability to deform under stress.
• FAg: Ranked 7th, suggesting its sensitivity to compressive strength, albeit less than other parameters.
• CP: Ranked 8th, indicating that the duration of curing affects compressive strength, but it is relatively less influential compared to other factors.
• RP: Ranked 9th, implying that the presence and amount of reinforcement have a moderate impact on compressive strength.
• Su: Ranked 10th, suggesting its lower sensitivity compared to other parameters. Superplasticizers are used to improve workability but have a relatively minor effect on compressive strength.
• EW: Ranked 11th, indicating it has the least influence on compressive strength among the parameters analyzed.

Hence, this sensitivity analysis highlights the critical parameters affecting the compressive strength of GeC, with calcium activator, fly ash, and the NaOH/Na₂SiO₃ ratio being the most influential factors. Understanding the sensitivity of these parameters is crucial for optimizing concrete mix designs and achieving desired mechanical properties.

6. Conclusions

This study presents a novel approach for predicting the compressive strength of geopolymer concrete (GeC) using machine learning models, specifically Multi-Layer Perceptron (MLP), Gannet Optimization Algorithm-MLP (GOA–MLP), and Grey Wolf Optimizer-MLP (GWO–MLP). The results demonstrate that the GWO–MLP model outperforms the other models in terms of predictive accuracy, with an R² value of 0.92, an RMSE of 3.18 MPa, and a VAF of 96.5%. Sensitivity analysis identified key parameters, such as curing temperature and fly ash content, that significantly influence the compressive strength of GeC. These findings underscore the importance of using optimization-enhanced machine learning models to improve the prediction of material properties for sustainable construction practices. However, this study has some limitations. The dataset used is relatively small, consisting of only 63 data points, which may affect the generalizability of the models to other datasets or real-world scenarios. Moreover, the models explored in this study could be further refined through the application of additional optimization techniques or advanced feature selection methods. For future research, we recommend exploring alternative optimization algorithms to further enhance model performance. Notably, more extensive and more diverse datasets should be used to validate the models’ generalization capabilities. Cross-validation techniques, such as k-fold cross-validation, can also be employed to better assess the models’ robustness. Furthermore, real-world applications and case studies involving GeC in construction projects could provide valuable insights for further optimizing geopolymer concrete formulations.