Machine Learning-Based Prediction of Optimum Design Parameters for Axially Symmetric Cylindrical Reinforced Concrete Walls

Abstract

This study presents a hybrid approach integrating metaheuristic optimization and machine learning methods to quickly and reliably estimate the optimum design parameters of dome-shaped axially symmetric cylindrical reinforced concrete (RC) walls. A comprehensive dataset was created using the Jaya algorithm to minimize total material cost for hinged and fixed support conditions. For each optimized design case, total wall height (H), dome height (Hd), dome thickness (hd), and fluid unit weight (γ) were considered as input parameters; optimum wall thickness (hw) and total cost were determined as output parameters. Using the obtained dataset, a total of thirteen different regression-based machine learning algorithms, including linear regression-based models, tree-based ensemble methods, and neural network models, were trained and tested. Hyperparameter adjustments for all models were performed using the Optuna framework, and model performances were evaluated using a ten-fold cross-validation method and holdout dataset results. The results showed that machine learning models can learn the optimum design space obtained from metaheuristic optimization outputs with high accuracy. In optimum wall thickness estimation, Gradient Boosting-based models provided the highest accuracy under both hinged and fixed support conditions. In total cost estimation, the Gradient Boosting model stood out under hinged support conditions, while the XGBoost model yielded the most successful results for fixed support conditions. The findings clearly show that no single machine learning model exhibits the best performance for all output parameters and support conditions. The proposed approach offers significantly higher computational efficiency compared to traditional iterative optimization processes and allows for rapid estimation of optimum design parameters without the need for any iterations. In this respect, this study provides an effective decision support tool that can be used especially in the preliminary design phases and contributes to sustainable, cost-effective reinforced concrete structure design.

1. Introduction

In the domain of engineering design processes, a variety of essential conditions must be met, encompassing economic, safety, and sustainability factors. The utilization of optimization techniques facilitates the swift identification of solutions that best align with these specific criteria. Consequently, a range of optimization methods have been applied to address various challenges within civil engineering, including optimizing the design of reinforced concrete cylindrical walls. Over the years, researchers conducted studies in this context, with an emphasis on sustainability factors via minimizing cost by using less material to provide ecological and sustainable design.

Melchers and Rozvany introduced optimal lower-bound solutions for circular cylindrical tanks that can be either open at the top or fixed/hinged at the base. Their research findings suggested potential savings in reinforcement of 20% to 25% [1]. Adidam and Subramanyam utilized the Davidon Fletcher Powell (DFP) algorithm to optimize the cost of water tanks, which included both the wall and floor components [2]. Saxena et al. presented a computational approach for nonlinear optimization to determine the minimum-cost design parameters for Intze-type water tanks of various capacities [3]. Thevendran and Thambiratnam explored the minimum weight design of circular conical concrete water tanks, with a focus on varying wall thickness as the design variable, using finite element methods for analysis [4]. Tan et al. aimed to minimize the material cost of reinforced concrete cylindrical water tanks through direct search methods, employing the beam on elastic foundation (BEF) analogy for tank analysis [5].

In recent decades, metaheuristic algorithms have gained popularity as a frequently used method for optimization. Consequently, metaheuristic methods have been applied in numerous studies to optimize the design of cylindrical walls.

Barakat and Altoubat employed three advanced optimization techniques (Shuffled Complex Evolution, Simulated Annealing, and genetic algorithm) to find the optimal combination of thicknesses, water tank depth, and wall inclination for conical reinforced concrete water tanks, with the objective of minimizing the total material cost, including concrete, reinforcement, and formwork [6]. Ansary et al. used genetic algorithms to compare retrofitting an existing tank and optimizing the design of new liquid-filled steel conical tanks [7]. Ansari et al. proposed a numerical tool consisting of a nonlinear finite element model coupled with an evolutionary-based genetic algorithm to analyze stiffened liquid-filled steel conical tanks while considering global and local buckling constraints [8].

Various metaheuristic algorithms and their hybrid forms have also been applied to optimize axially symmetric cylindrical reinforced concrete walls, both with and without post-tensioning. In a 2014 study, an optimum strategy based on the Harmony Search (HS) algorithm was used to investigate the correlation between wall thickness and height [9]. In another study, the Harmony Search algorithm was employed to optimize post-tensioned cylindrical walls and minimize the total material cost [10]. Hybrid forms of optimization algorithms, such as Harmony Search (HS), Teaching Learning-Based Optimization (TLBO), and the Flower Pollination Algorithm (FPA), were investigated to enhance optimization performance [11]. In the subsequent year, the impact of various optimization methods on the minimum-cost design of post-tensioned walls was studied, including TLBO, FPA, the Jaya algorithm, and three hybrid forms combining Jaya with TLBO and FPA [12]. While the objective function is typically defined as the total cost in these studies, Bekdaş and Nigdeli explored the limitation of internal forces on the wall as an alternative objective [13].

Parallel to these developments, machine learning (ML), a sub-branch of artificial intelligence, has gained currency in structural and civil engineering. ML techniques are used to train a machine with an available dataset. The aim of this process is to find the best predictive model that represents the dataset. This allows a prediction to be made about the outcome that will correspond to new data. The development of this ML model is carried out by various ML algorithms. For concrete and steel members, several algorithms, such as linear regression, Ridge, Lasso, Elastic Net, Decision Tree, Random Forest, Extra Trees, k-Nearest Neighbors (KNNs), AdaBoost, Gradient Boosting, XGBoost, Light Gradient Boosting Machine (LightGBM), and Multilayer Perceptron (MLP), were run for the prediction of design outputs, including structural capacities and mechanical behaviors [14,15,16,17,18].

Studies on the prediction of structural performance and building energy requirements show that Gradient Boosting achieves high accuracy (R² ≈ 0.99) [19,20,21]. In the study conducted by Abdeyazdan et al., Random Forest also performed closely (R² ≈ 0.99) to Gradient Boosting [21]. Arbi et al. demonstrated the superior performance of XGBoost in the ML model developed for pile-bearing capacity estimation [22]. Similarly, XGBoost outperformed by achieving R² = 0.959 in predicting building performance [23]. Aldawish and Kulasegaram focused on the prediction of concrete compressive strength. In the study, methodologies including Artificial Neural Networks (ANNs), Support Vector Machines (SVMs), Decision Tree Regressions (DTRs), and other machine learning models were assessed, and Artificial Neural Networks (ANNs) were found to be highly adaptable in capturing nonlinear datasets depending on extensive and high-quality datasets [24]. Kondratieva et al. investigated several ML algorithms, including the multilayer feed-forward neural network (MLNN), to predict the damage behavior of RC frame buildings under single or multiple seismic excitations. Considering the results of their study, the authors proposed a user-friendly web application [25].

In maximizing model performance, algorithm selection is not the only factor. Hyperparameter tuning also has a significant role in this process. The Optuna framework has recently become crucial as an optimization tool in hyperparameter tuning. Kondratieva et al. used the Optuna framework, grid search, and random search in the hyperparameter tuning of a model developed to predict the strength of reinforced concrete beams. In their study, the Optuna framework achieved greater accuracy and convergence than traditional grid or random search approaches [26]. Khan et al. conducted a comprehensive comparative analysis of Automated Machine Learning (AutoML) frameworks for hyperparameter optimization. For this purpose, random search (RS), grid search (GS), Hyperopt, TPOT, Optuna, GP Minimize, Forest Minimize, GBRT Minimize, and Dummy Minimize were investigated, and it was concluded that Optuna outperformed in terms of predictive accuracy and computational training time [27].

Besides model efficiency, sustainability is also of increasing importance in AI model development. Data-driven modeling not only improves prediction accuracy but also contributes to environmental responsibility. For this purpose, Jegadeeswari and Rathipriya developed an Eco-Friendly Multi-Objective Framework (EFMOF) that optimizes hyperparameters focusing on energy efficiency and carbon reduction [28]. Another study proposed digital models for the sustainable construction of timber structures in order to enhance ductility and energy dissipation [29].

Recently, AI-driven automation has provided significant improvements in terms of design efficiency, precision, and sustainability. In order to predict the ultimate moment capacity of Ultra-High-Performance Concrete (UHPC) beams and U-shaped girders, RF, which is the base model, was integrated with different AutoML frameworks by KHAN et al. [27]. In this way, algorithm selection and hyperparameter adjustment are enabled without requiring human intervention. Mohanty et al. investigated the integration of AI and ML technologies for structural engineering. It was demonstrated that AI/ML-driven autonomous design achieved superior structural performance and minimized resource utilization [30]. Singh et al. integrated robust optimization and ML techniques with Building Information Modeling (BIM) to improve the safety of large-diameter tunnel construction [31].

Additionally, ML techniques are used for the optimization of energy performance and load-bearing behavior in buildings. For example, Iram et al. developed an ML model that improves thermal comfort and optimizes energy consumption in residential buildings, offering a practical approach to more sustainable and comfortable living environments [32]. Abbood et al. demonstrated superior predictive performance of Explainable ensemble ML models using Bayesian optimization in the prediction of the shear strength of RC deep beams [33]. Explainable Artificial Intelligence (XAI) methods, such as SHapley Additive exPlanations (SHAP) and Local Interpretable Model-Agnostic Explanations (LIME), have become increasingly important due to improved interpretability of ML models [34,35,36,37,38].

In parallel, metaheuristic methods have been integrated into deep learning. In biomedical and image processing fields, metaheuristic optimization algorithms, such as the genetic algorithm (GA), Particle Swarm Optimization (PSO), and Ant Colony Optimization (ACO), were used to tune deep neural networks, which provided significant improvement in model robustness and convergence [39,40,41,42,43]. These studies also inspire engineering optimization frameworks involving complex nonlinear relationships.

Metaheuristic optimization and machine learning methods are also used for structural design optimization. Nguyen generated a robust hybrid model integrating metaheuristic methods with an ML model to predict the mechanical properties of RC structures [18]. A hybrid method was proposed by Alkhawaldeh to improve the design of flat slabs. In this method, the Locust Swarm Algorithm (LSA) was used to optimize the model parameters and configurations [44]. In the success of prediction, data are a crucial parameter. In this context, optimum data will be an advantage for a machine in providing the best solutions. The study conducted by Yücel et al. aimed to determine the minimum carbon dioxide (CO₂) emissions and optimum cost estimation of reinforced concrete beams. For this purpose, an optimization process was carried out to obtain a dataset containing optimum data. These data were used to train Artificial Neural Networks (ANNs) [45]. Similarly, Bekdaş et al. generated optimum design parameters for a prestressed concrete cylindrical wall via the Harmony Search algorithm and then used them in an ML process [46].

In this study, a machine learning process was conducted to predict the optimum wall thickness of axially symmetric cylindrical reinforced concrete walls. For this purpose, two machine learning models were developed using two datasets, including optimum wall thicknesses. These data were generated via the Jaya algorithm (JA) under different support situations, dome thicknesses, and liquid unit volume weights [47]. Using the obtained optimum datasets, prediction models were developed with different basic and ensemble machine learning algorithms.

This study presents a comprehensive hybrid framework combining metaheuristic optimization and machine learning for the optimum design of axially symmetric cylindrical reinforced concrete walls with dome roofs. Unlike existing studies in the literature, in this research, machine learning models were trained not on randomly generated datasets but on a large-scale dataset consisting of engineering-consistent optimum design solutions obtained under physical constraints using the Jaya algorithm. Based on this dataset, surrogate machine learning models were developed to estimate the optimum wall thickness and total cost under hinged and fixed support conditions. The performance of thirteen different regression-based machine learning algorithms, whose hyperparameters were optimized using the Optuna framework, was evaluated comparatively. Furthermore, differences in model performance depending on support conditions and estimated output parameters were interpreted from an engineering perspective, and it was shown that no single machine learning model universally provides the best result for all design scenarios. The proposed approach provides an effective decision support tool, particularly for preliminary design phases, by enabling rapid and reliable estimation of optimum design parameters without the need for iterative optimization analyses.

The optimization process with the JA aimed to ensure minimum material costs, including concrete and steel bars (Section 2.1). The analyses of the wall were carried out by combining the theory of axially symmetrical cylindrical shells (Section 2.1) and flexibility theory (Section 2.1). This process was completed in the Matlab R2022b program [48]. In the training of the dataset with optimum data, several foundational and ensemble algorithms were implemented (Section 2.2). The k-fold cross-validation technique was used to separate the dataset for the training and testing phases of the ML algorithms (Section 2.3). The performance success of the developed models was evaluated with basic performance metrics (Section 2.3). This machine learning process was performed in the Python 3.10 program [49].

2. Materials and Methods

2.1. Optimization-Based Dataset Generation

The Jaya algorithm proposed by Rao [50] was used to obtain the optimum design data for axially symmetric cylindrical reinforced concrete (RC) walls with dome roofs. The main objective of the optimization was to minimize the total cost composed of concrete and reinforcement. The design variables included wall thickness, reinforcement diameter, and spacing, while the design constraints satisfied flexural, shear, and reinforcement requirements. The detailed explanations and mathematical formulations of the algorithm are available in Rao [50] and Kayabekir [47]. The Jaya algorithm was preferred because of its simple structure and parameter-free mechanism, which ensures stable and reliable convergence in optimization problems. Since the optimum design results used in this study were obtained from the original optimization study carried out by Kayabekir [47], the optimization process and validation were already completed in the aforementioned source.

Each optimization analysis produced one design case defined by the input parameters and the corresponding optimum wall thickness and total cost. The dataset obtained in this way consists not of randomly generated samples but of optimal design solutions obtained under physical constraints. This allows machine learning models to learn an optimal design space that is meaningful and consistent from an engineering perspective. The obtained optimum values were later used as data for the machine learning models.

The structure considered in this study is an axially symmetric reinforced concrete cylindrical wall with a dome roof, shown in Figure 1, typically used in liquid-containing structures. In Figure 1, R denotes the wall radius, H_w the wall height, and Hd the dome height, while h_w and h_d represent the wall and dome thicknesses, respectively. The structural behavior of the cylindrical wall was examined according to the classical theory of axially symmetric shells. In the analysis, axisymmetric loading was considered, and circumferential shear deformations were neglected. To simplify the governing relations, the wall was assumed infinitely long and subjected to hydrostatic pressure varying along its height. The relations between stress and deformation were obtained using the flexibility approach described in [47].

The optimization analysis was performed using flexibility theory, which allowed the interaction between the dome and the wall to be modeled accurately. The fixed-base cylindrical wall was idealized as an equivalent isostatic system, and redundant forces were found by compatibility equations to ensure equilibrium under hydrostatic pressure. This theoretical background provided the basis of the optimization analysis, from which the dataset for machine learning was generated.

After the optimization stage, the study continued with the development of machine learning models to estimate the optimum wall thickness and total cost for different geometric and material conditions. In this study, machine learning models were trained using only the geometric and loading parameters that are critical in the preliminary design phase, with the aim of focusing on these variables.

2.2. Description of the Optimization Dataset

The dataset used in this study was generated from optimization analyses that achieved the optimum design of axially symmetric cylindrical reinforced concrete walls. Each optimization analysis represents a single design case for the specified geometric and material parameters, and optimum wall thickness and total cost values were obtained from each analysis result. Wall height, dome geometry parameters, unit weight, and support conditions were used as input parameters in the optimization process (

Table 1. Input and output parameters of the dataset.

Type	Parameter	Description
Input	H	Wall height
Input	Hd	Dome height
Input	hd	Dome thickness
Input	γ	Unit weight
Input	Support condition	Fixed/pinned
Output	Hw	Optimum wall thickness
Output	C	Total cost

). The obtained optimum results were evaluated as output data in the development of machine learning models. The dataset thus created contained a total of 13,650 different design cases corresponding to different geometric and structural conditions.

The input parameters used in this study were selected as fundamental geometric and loading parameters that can be directly controlled by the designer during the preliminary design phase and have a dominant effect on optimum wall thickness and total cost. Limiting the input parameters both increases the stability of the developed machine learning models and strengthens their practical usability in engineering applications.

2.3. Machine Learning Methodology

The results obtained using the Jaya algorithm were used in the development of machine learning (ML)-based regression models to predict the optimum wall thickness and total cost of dome-shaped axially symmetric cylindrical reinforced concrete (RCC) walls. The dataset consists of optimization-based design cases representing different geometric configurations and material conditions, with each data record containing four input variables and two output parameters. The main objective of the machine learning approach is to establish reliable prediction relationships between input variables and optimum design outputs, enabling fast and efficient predictions without the need to repeat the optimization process. In this context, the developed machine learning models act as surrogate models replacing traditional optimization analyses, enabling the rapid estimation of optimum design parameters without the need for repetitive metaheuristic optimization processes.

The machine learning workflow consists of four main stages: data preparation, model training, hyperparameter optimization, and performance evaluation. Various regression algorithms were evaluated to determine the most suitable model for the dataset. Hyperparameter optimization was performed using the Optuna framework, aiming to reduce the risk of over-learning by selecting appropriate parameters. The performance of the models was evaluated using various performance metrics commonly used for regression problems (e.g., R² and RMSE). Details regarding the dataset and the algorithms used are presented in the following subsections.

The purpose of the machine learning models in this study is not to replace structural mechanics or classical optimization, but to serve as a computational surrogate for the optimization process. While the Jaya algorithm provides rigorous and physically consistent optimal solutions, it requires repetitive and time-consuming iterative analyses for each new design case. Once trained, the machine learning models can estimate optimum wall thickness and total cost in milliseconds, enabling rapid preliminary design, parametric studies, and real-time decision support. Therefore, the proposed approach bridges metaheuristic optimization and practical engineering design by providing a fast, data-driven approximation of the optimal design space.

2.3.1. Machine Learning Process

The general machine learning workflow followed in this study is shown in Figure 2. The optimum design data obtained from the optimization analyses were used as input data for the machine learning models. First, the dataset was prepared in a way suitable for machine learning, and preprocessing steps were applied. Then, different regression-based models were trained, and hyperparameter optimization was performed to improve model performance. The performance of the trained models was evaluated using validation and performance metrics, and in the final stage, the optimum wall thickness and total cost were estimated.

Regression Models

The regression models considered in this study were selected to represent commonly used machine learning approaches in engineering problems, exhibiting varying levels of complexity and learning capabilities. These models are frequently preferred in the literature due to their ability to capture linear and nonlinear relationships and their robust prediction performance. Comparative evaluation of multiple models allows for an objective examination of their predictive performance for the optimization dataset under consideration.

Linear Regression-Based Models

The linear regression method is based on determining a linear relationship between dependent and independent variables. In the linear regression model, a linear relationship is assumed between the dependent variable and the independent variable. This relationship can be expressed using Equation (1) [51].

$$y_i={\beta }_0+{\beta }_1{x}_i+{ϵ}_i\to \mathrm{i}=1, 2, \dots , n$$

(1)

In Equation (1), y_i represents the dependent variable, β₀ and β₁ represent the regression coefficients, while x_i represents the independent variable and ε_i represents the statistical error term; i is the sample index, and n denotes the total number of observations. Linear regression is widely used in many forecasting problems because the output coefficients are easy to apply and interpret [52].

Multiple linear regression is used to determine the extent to which a given set of independent variables explains the dependent variable [53].

Least Absolute Shrinkage and Selection Operator (LASSO) Regression: Least Absolute Shrinkage and Selection Operator (LASSO) regression is a method that performs variable selection by shrinking the insignificant coefficients in the model towards zero, thus ensuring that only significant variables are included in the model. Lasso reduces some coefficients to zero by applying the penalty term on the variables, leaving only significant variables in the model [54]. Lasso regression improves variable selection while simultaneously reducing overfitting [55].

Ridge Regression: Ridge Regression was developed as an alternative to the least squares method in cases where multicollinearity problems are present. The goal of the Ridge is to obtain more stable parameter estimates by minimizing the influence of explanatory variables. The Ridge estimator was developed to reduce the variance of the estimators by solving the multicollinearity problem. This method reduces large coefficients but does not reduce them to zero [56].

Elastic Net Regression: One of the common biased predictors is Elastic Net, which narrows the coefficients and performs variable selection to increase the accuracy and interpretability of the regression model. In order to overcome the disadvantages of the Lasso estimator, the Elastic Net estimator is proposed. ElasticNet has a penalty function that combines the advantages of both LASSO and Ridge Regression. This method offers an alternative approach that preserves the strengths of Ridge Regression while improving LASSO’s variable selection capability [57].

Tree-Based Regression Models

Decision Tree: Decision trees are hierarchical structures that perform prediction or classification by dividing a dataset into branches based on specific criteria. Using independent variables, they optimize data splits to increase the probability that the dependent variable belongs to a specific value or class [58]. The primary goal of decision trees is to partition the data space into a specific number of leaf nodes and make predictions at each leaf. As this partitioning process progresses in regression trees, the prediction is generated by averaging the resulting values of the observations at each leaf [59].

Random Forest: Random Forest is a method based on the collaboration of multiple decision trees. In this algorithm, n decision trees are created using the given data, and the outputs of these trees are combined by taking the average value in the regression. The most significant advantage of the Random Forest method is that it reduces the risk of overfitting thanks to the contribution of multiple different trees. Random Forest provides separation by using the best randomly selected variables at each node [60].

Extra Trees: Extra Trees is an algorithm based on decision trees. The Extra Trees algorithm generates a large number of unpruned decision trees using random splits from the training dataset. In the Extra Trees algorithm, a random variable and a random cut point are preferred for splitting. This model consists of many decision trees, similar to the Random Forest algorithm. However, the level of randomness used in creating the trees is higher [61].

Ensemble and Boosting Models

Gradient Boosting: Gradient Boosting is an ensemble method that builds a robust prediction model by repeatedly training weak learners. The basic approach is based on learning each new tree to reduce the errors made by previous models. Thus, the model improves itself by addressing the remaining error at each step, resulting in a structure that provides higher overall accuracy [62]. The goal is to gradually reduce the prediction error by utilizing the derivative of the loss function [63].

eXtreme Gradient Boosting (XGBoost): eXtreme Gradient Boosting (XGBoost) is an optimized version of the Gradient Boosting algorithm. It incorporates several efficient features designed to address the overfitting problems and missing data challenges inherent in traditional Gradient Boosting during the training process. XGBoost’s principle is that each new tree is trained to reduce the prediction errors of previous trees [64].

Light Gradient Boosting Machine (LightGBM): Light Gradient Boosting Machine (LightGBM) was developed to overcome the efficiency and scalability issues of the Gradient Boost algorithm for high-dimensional features or large datasets. LightGBM utilizes various optimization techniques, primarily histogram-based methods, to both increase computational speed and improve prediction accuracy. By processing data using histogram-based splitting, the algorithm can efficiently learn complex relationships and patterns. LightGBM’s success is evaluated using various performance metrics, including model accuracy, computational speed, consistency, and ease of use.

Adaptive Boosting (AdaBoost): Adaptive Boosting (AdaBoost) regression is one of the ensemble methods used in machine learning, and its main goal is to build a strong model by sequentially combining a series of weak learners. The algorithm prioritizes the training data and, with each iteration, prioritizes incorrectly predicted examples, allowing the next learner to focus on these difficult examples. This minimizes errors and increases the overall accuracy of the model [65]. The final prediction is achieved by combining the weighted contributions of all trees [66].

Distance-Based and Neural Network Models

K-Nearest Neighbor (KNN): The working principle of the KNN algorithm is to determine the k training examples closest to the example to be predicted and to perform the prediction using the information of these k neighbors. The performance of the KNN algorithm depends on the distance/similarity measure used [67]. Functions such as Euclidean or Manhattan are used for distance measurement [68].

Multilayer Perceptron (MLP): Multilayer Perceptron (MLP) is a type of artificial neural network method capable of modeling nonlinear relationships using multiple layers. The MLP model is a multilayer, fully connected neural network. Because neural networks can recognize and learn patterns in large datasets, this method is particularly effective in complex and high-dimensional datasets [69].

2.3.2. Hyperparameter Optimization Using Optuna

The hyperparameters of all machine learning models used in this study were optimized using the Optuna framework [70]. Optuna was chosen for the hyperparameter optimization process because, in addition to utilizing advanced search algorithms such as the Tree-structured Parzen Estimator (TPE), it efficiently performs automatic hyperparameter optimization. Optuna explores the predefined search space for each model to find the hyperparameter combination that minimizes the prediction error on the validation dataset. The search spaces used for the models in this process are listed in

Table 2. Hyperparameter search spaces for machine learning models.

Model	Parameter Type and Range
Random Forest	n_estimators (data type: int; range: 10–200) max_depth (data type: int; range: 2–32; log = True) min_samples_split (data type: float; range: 0.1–1.0)
Gradient Boosting	n_estimators (data type: int; range: 10–200); learning_rate (data type: float; range: 1× 10-3−–1; log); max_depth (data type: int; range: 1–7).
Decision Tree	max_depth (data type: int; range: 1–32); min_samples_split (data type: float; range: 0.1–1.0); min_samples_leaf (data type: float; range: 0.1–0.5).
KNN	n_neighbors (data type: int; range: 1–20); weights (categorical; uniform/distance); p (data type: int; range: 1–2)
Linear Regression	No tunable hyperparameters (defaults only)
Ridge	alpha (data type: float; range: 1× 10-5–1× 105; log)
Lasso	alpha (data type: float; range: 1× 10-5–1× 105; log)
Elastic Net	alpha (data type: float; range: 1× 10-5–1× 105; log); l1_ratio (data type: float; range: 0–1).
MLP	hidden_layer_sizes (categorical; (50,), (100,), (50, 50), (100, 50)); activation (categorical; tanh/relu); alpha (data type: float; range: 1× 10-5–1× 10-1; log); learning_rate (categorical; constant/adaptive).
AdaBoost	n_estimators (data type: int; range: 10–200); learning_rate (data type: float; range: 1× 10-3–1; log).
Extra Trees	n_estimators (data type: int; range: 10–200); max_depth (data type: int; range: 2–32; log); min_samples_split (data type: float; range: 0.1–1.0).
XGBoost	n_estimators (data type: int; range: 10–200); max_depth (data type: int; range: 1–10); learning_rate (data type: float; range: 1× 10-3–1; log); subsample (data type: float; range: 0.5–1.0); colsample_bytree (data type: float; range: 0.5–1.0).
LightGBM	n_estimators (data type: int; range: 10–200); max_depth (data type: int; range: 1–10); learning_rate (data type: float; range: 1× 10-3–1; log); subsample (data type: float; range: 0.5–1.0); colsample_bytree (data type: float; range: 0.5–1.0).

. Optimal hyperparameters were obtained by performing 200 trials for each model.

2.3.3. Model Training and Validation

In order to train and validate the machine learning models, the dataset was divided into training (80%) and holdout (20%) subsets to ensure a balanced assessment of the model’s learning capacity and its generalization performance on unseen data. A set of hyperparameters was later optimized for each algorithm with 5-fold cross-validation using the training. A k-fold cross-validation approach was applied to ensure the generalization ability of the models during the best model selection process and ensure that the results were independent of randomness. In this process, the training data were separated into different layers, and the performance of the models was evaluated through multiple iterations. As a result of this process, models with the best hyperparameters were selected. Following this, the selected models for each algorithm were evaluated using the training and holdout sets. The experiment results were provided as a rounded average of runs with 10 random seeds between (42–52) to ensure the stabilization of the provided results. The results for each algorithm in the following section present the metrics obtained from the holdout set.

The predictive ability of the models was evaluated using various performance metrics commonly used for regression problems. In this context, the coefficient of determination (R²) and root mean square error (RMSE) were considered as key performance metrics for model evaluation.

3. Results and Discussion

3.1. Prediction Performance for Optimum Wall Thickness

3.1.1. Pinned Support

Figure 3 presents the comparative performance of the machine learning models developed for estimating optimum wall thickness under hinged support conditions on a holdout dataset.

According to the results, Gradient Boosting-based models (Gradient Boosting, XGBoost, and LightGBM) provide significantly higher accuracy compared to the other models. In particular, the Gradient Boosting model stands out as the most successful model in predicting optimum wall thickness, with the highest R² value and the lowest error metrics. This indicates that wall thickness under hinged support conditions exhibits a nonlinear but regular relationship with the input parameters.

Figure 4 shows the predicted vs. true graph obtained on the holdout dataset of the Gradient Boosting model, which exhibited the best performance for hinged support conditions. The graph clearly shows that the predicted optimum wall thickness values have a high agreement with the actual values, and scattering is quite limited. This result confirms that the model has a strong generalization ability not only to training data but also to test data.

The findings reveal that tree-based ensemble methods can represent the complex relationship structure more effectively.

3.1.2. Fixed Support

Figure 5 shows the comparative performance of the machine learning models used for predicting optimum wall thickness under fixed support conditions on the holdout dataset.

Under fixed support conditions, it is observed that the internal force distribution becomes more complex with increasing support rigidity, creating more pronounced nonlinear effects on the optimum wall thickness. Therefore, models that can strongly represent nonlinear relationships, such as Gradient Boosting and XGBoost, were found to provide higher accuracy compared to the other models. According to the test results, the Gradient Boosting model emerged as the most successful model for fixed support conditions.

Figure 6 presents the prediction–actual value graph for the holdout dataset of the Gradient Boosting model, which showed the best performance under fixed support conditions. The concentration of the predicted and actual optimum wall thickness values around the diagonal line shows that the model successfully captures the complex behavior under fixed support conditions.

These results clearly demonstrate that support conditions have a significant impact on model performance in optimum wall thickness estimation and that nonlinear modeling approaches should be preferred in the case of fixed supports.

3.2. Prediction Performance for Total Cost

3.2.1. Pinned Support

Figure 7 presents the comparative performance of the machine learning models developed for total cost estimation under hinged support conditions on the holdout dataset.

The results show that under hinged support conditions, the total cost exhibits a more regular and approximately linear relationship; however, local nonlinear effects arising from the optimized wall thickness distribution still make ensemble-based models more advantageous. Therefore, it is seen that linear regression-based models, as well as powerful ensemble methods such as Gradient Boosting and XGBoost, achieve perfect accuracy levels. However, considering the error metrics, it is determined that Gradient Boosting-based models offer more stable and reliable predictions.

Figure 8 shows the prediction–actual value graph of the holdout dataset for the Gradient Boosting model, which exhibits the best performance for hinged support conditions. The graph shows that the predicted total cost values are quite close to the actual values, and large deviations are limited.

This indicates that total cost estimation under hinged support conditions can be achieved with high accuracy through appropriate model selection, and the proposed approach can be used as an effective cost forecasting tool in the preliminary design phases.

3.2.2. Fixed Support

Figure 9 shows the comparative performance of the machine learning models used for total cost estimation under fixed support conditions on the holdout dataset.

Under fixed support conditions, the increase in support rigidity and changes in internal force distribution create a more complex and fragmented effect on the total cost. This limits the performance of linear regression and neural network-based models, while tree-based ensemble methods allow for more effective capture of this complex relationship structure. According to the test results, the Gradient Boosting and XGBoost models yielded the most successful results in total cost estimation under fixed support conditions. Although the performance difference is quite small, the XGBoost model achieved slightly higher accuracy metrics compared to Gradient Boosting, demonstrating a very close and competitive performance between the two models.

Figure 10 presents the prediction–actual value graph of the holdout dataset for the XGBoost model, which showed the best performance for fixed support conditions. The graph shows that the model can produce predictions close to actual values, even in the high cost range.

The results obtained reveal that support conditions and output parameters are decisive in model selection for total cost estimation and that nonlinear, ensemble-based models should be preferred in the case of fixed supports.

3.3. Comparative Evaluation of Machine Learning Models

In this section, the performance of the developed machine learning models is compared using R², MSE, MAE, and MAPE metrics calculated on holdout datasets (

Table 3. Performance metrics of models for cost prediction under fixed support conditions.

Model	R2	MSE	MAE	MAPE (%)
XGBoost	0.999935	3.55 × 105	470.01	0.28
Gradient Boosting	0.999934	3.60 × 105	465.01	0.27
LightGBM	0.999831	9.19 × 105	734.20	0.40
Extra Trees	0.978303	1.18 × 108	7717.98	3.88
Elastic Net	0.977599	1.22 × 108	9004.50	5.63
Ridge	0.977599	1.22 × 108	9004.50	5.63
Lasso	0.977599	1.22 × 108	9004.50	5.63
Linear Regression	0.977599	1.22 × 108	9004.50	5.63
kNN	0.974086	1.41 × 108	9239.07	5.10
AdaBoost	0.970123	1.62 × 108	10,500.01	6.41
MLP	0.962503	2.04 × 108	11,352.99	6.95
Random Forest	0.953452	2.53 × 108	11,828.33	6.27
Decision Tree	0.871965	6.96 × 108	20,810.76	11.17

,

Table 4. Performance metrics of models for optimum thickness under fixed support conditions.

Model	R2	MSE	MAE	MAPE (%)
Gradient Boosting	0.999485	0.000014	0.00291	0.43
XGBoost	0.999452	0.000014	0.00308	0.47
LightGBM	0.999283	0.000019	0.00349	0.53
MLP	0.990715	0.000245	0.01232	1.78
Extra Trees	0.976815	0.000612	0.01993	3.01
kNN	0.972513	0.000726	0.02094	3.04
Lasso	0.96913	0.000815	0.02228	3.29
Elastic Net	0.969129	0.000816	0.02228	3.29
Ridge	0.969126	0.000816	0.02228	3.30
Linear Regression	0.969126	0.000816	0.02228	3.30
AdaBoost	0.955607	0.001173	0.02903	4.66
Random Forest	0.930195	0.001844	0.03540	5.36
Decision Tree	0.761044	0.006312	0.06679	10.28

,

Table 5. Performance metrics of models for cost prediction under hinged support conditions.

Model	R2	MSE	MAE	MAPE (%)
Gradient Boosting	0.999993	4.91 × 104	129.84	0.07
XGBoost	0.999985	9.60 × 104	238.19	0.13
LightGBM	0.999772	1.49 × 106	635.27	0.31
kNN	0.981752	1.20 × 108	7934.09	4.37
Extra Trees	0.979838	1.32 × 108	8615.06	4.39
Lasso	0.979633	1.33 × 108	9069.15	5.84
Elastic Net	0.979622	1.34 × 108	9062.79	5.85
Ridge	0.979618	1.34 × 108	9062.92	5.85
Linear Regression	0.97961	1.34 × 108	9063.35	5.85
AdaBoost	0.969632	1.99 × 108	11,776.70	7.25
MLP	0.968032	2.10 × 108	11,712.76	7.33
Random Forest	0.925405	4.89 × 108	17,876.38	9.58
Decision Tree	0.800002	1.31 × 109	29,803.84	16.99

and

Table 6. Performance metrics of models for thickness prediction under hinged support conditions.

Model	R2	MSE	MAE	MAPE (%)
Gradient Boosting	0.999737	0.000008	0.00189	0.29
LightGBM	0.999696	0.000009	0.00237	0.36
XGBoost	0.999694	0.000009	0.00243	0.37
MLP	0.983188	0.000480	0.01761	2.65
kNN	0.982030	0.000513	0.01691	2.51
Extra Trees	0.978409	0.000617	0.01921	2.87
Elastic Net	0.970199	0.000851	0.02275	3.36
Lasso	0.970197	0.000851	0.02276	3.36
Ridge	0.970175	0.000852	0.02277	3.37
Linear Regression	0.970173	0.000852	0.02278	3.37
AdaBoost	0.955318	0.001276	0.02999	4.79
Random Forest	0.922300	0.002219	0.03906	5.87
Decision Tree	0.807304	0.005504	0.06099	9.21

). Evaluations are conducted separately for hinged and fixed support conditions, considering both optimum wall thickness and total cost outputs.

The test set results presented in Table 3, Table 4, Table 5 and Table 6 show that the performance of the machine learning models varies significantly depending on both the support condition and the predicted output parameter. Tree-based ensemble methods and Gradient Boosting-based models achieved the highest accuracy values in both optimum wall thickness and total cost estimation. In particular, the Gradient Boosting, XGBoost, and LightGBM models stood out with low error metrics and high R² values in all scenarios.

In contrast, linear regression-based models were relatively successful in total cost estimation, but their performance was limited in outputs where nonlinear relationships, such as optimum wall thickness, were dominant. While the neural network-based MLP model provided high accuracy in some cases, it performed worse in cost estimation compared to tree-based ensemble methods.

The results clearly demonstrate that no single machine learning model provides the best results for all output parameters in engineering problems. Therefore, it is concluded that model selection should be made depending on both the problem type and the predicted engineering output.

4. Discussion and Conclusions

This study proposes a hybrid approach combining metaheuristic optimization and machine learning methods to quickly and reliably estimate the optimum design parameters of dome-shaped axially symmetric cylindrical reinforced concrete (RC) walls. Optimum wall thickness and total cost outputs were estimated using machine learning models on a comprehensive dataset generated using the Jaya algorithm for hinged and fixed support conditions.

A single Jaya optimization run typically requires iterative structural analyses that may take several minutes per case, whereas the trained ML model produces predictions in milliseconds. Therefore, for large parametric studies involving hundreds of cases, the proposed approach can reduce total computation time from days to seconds.

In this study, the high performance values obtained by machine learning models (especially R² ≈ 0.99 and above) stem from the optimization-based and deterministic structure of the dataset. The dataset used is not composed of random or noisy experimental measurements, but rather of optimum design solutions obtained under physical constraints using the Jaya algorithm. Therefore, the predicted output parameters (optimum wall thickness and total cost) have strong and direct engineering relationships with the input variables. In this context, the fundamental role of machine learning models is not to reduce uncertainty or filter noise, but rather to enable the rapid and reliable approximation (surrogate modeling) of a defined optimum design space. The high accuracy values obtained are due not to data leakage, but to the deterministic character of the problem structure and the optimization-based data generation strategy.

The results show that machine learning models can learn the optimum design space derived from metaheuristic optimization outputs with high accuracy. In particular, Gradient Boosting-based models (Gradient Boosting, XGBoost, and LightGBM) were found to perform better than other models in both optimum wall thickness and total cost estimation. However, it was clearly demonstrated that model performance varies depending on both the support condition and the estimated output parameter, and that no single machine learning model provides the best result for all scenarios.

In estimating optimum wall thickness, a nonlinear relationship structure was observed, particularly under fixed support conditions, thus limiting linear regression-based models. In contrast, tree-based ensemble methods and Gradient Boosting algorithms were able to represent the complex relationship structure more effectively. In total cost estimation, a more regular relationship structure was observed under hinged support conditions, while a more complex effect on cost was seen under fixed support conditions.

The most significant advantage of the proposed approach is its substantial computational efficiency compared to traditional iterative optimization processes. The ability to directly estimate optimum design parameters without the need for any iterative analysis process provides an effective decision support tool for engineers, especially during the preliminary design phases. In this respect, this study contributes to sustainable and cost-effective design practices.

Unlike classical optimization approaches (e.g., gradient-based methods or genetic algorithms), which must be executed separately for each new design case, the proposed machine learning framework provides a reusable surrogate model of the optimal design space. Although a single optimization run is feasible, practical engineering applications often require hundreds of parametric analyses, sensitivity studies, or iterative design cycles. In such situations, the trained ML model offers orders-of-magnitude faster predictions while preserving the accuracy of the original optimization results.

The primary motivation for using machine learning is not to replace classical optimization for a single design case but to provide a computational surrogate that enables rapid evaluation of optimum design parameters for many scenarios. While a single optimization run is feasible, real-world engineering applications often require hundreds of parametric studies, sensitivity analyses, or design iterations. In such cases, the proposed ML-based surrogate model offers orders-of-magnitude faster predictions compared to traditional iterative optimization methods.

The main novelty of this study is the development of a reliable machine learning surrogate model trained on optimization-based, physically consistent optimal design data for axially symmetric cylindrical RC walls under different support conditions.

In conclusion, this study demonstrates that the integrated use of metaheuristic optimization and machine learning methods offers a powerful and practical tool for the optimum design of reinforced concrete liquid storage structures. In future studies, extending the proposed framework to include different geometry types, loading conditions, and material uncertainties will further enhance the generalizability of the method.