XGBoost builds on the Gradient Boosting Decision Tree (GBDT) algorithm, employing decision trees, particularly Classification and Regression Trees (CART), for classification and regression (Sagi and Rokach, 2021; Mohan et al., 2024). In GBDT, sequential base CART estimators are assigned weights adjusted during training, creating a robust ensemble model. For regression, a sample’s predicted value results from the weighted sum of each leaf node’s predictions in the decision trees, as shown in Equation 1. y ^ i = ∑ k K β k h k x i (1)

In the formula: y ^ i represents the predicted value; K denotes the total number of trees; β k is the weight of the k ^-th tree; h k x i represents the prediction result of the k ^-th tree; and x i represents the feature vector corresponding to the i ^-th sample.

The XGBoost model uses an iterative approach, combining weak learners and controlling complexity to discover complex nonlinear statistical relationships between the target variable and the observed features. Through this iterative process, the model continuously optimizes performance during training (Che and He, 2022). The objective function of the XGBoost model, combining the predicted results, as shown in Equations 2, 3. O bj = ∑ i = 1 M l y i , y ^ i + ∑ k = 1 K Ω f k (2) Ω f k = γ T + 1 2 λ ∑ k = 1 K ω k 2 (3)

In the formula: M represents the total number of samples in the dataset; yi ​ denotes the true value; l y i , y ^ i is the loss function, which measures the difference between the predicted and actual values; Ω f k represents the complexity of the model, which acts as a regularization term to help prevent overfitting; T is the number of leaf nodes; ω k is the weight of the leaf nodes; and γ and λ are pre-defined hyperparameters that control the number and scores of the leaf nodes.

The XGBoost model enhances the additive training process by iteratively advancing. In each iteration, it trains a new model and adds it to the previous ensemble, gradually reducing the loss function. As shown in Equations 4–6. y ^ i q = ∑ q Q f k x i = ∑ q Q − 1 f k x i + f q x i (4) ∑ k K Ω f k = ∑ q Q − 1 Ω f k + Ω f q (5) O bj = ∑ i = 1 M f q x i g i + 1 2 f q x i 2 h i + Ω f q (6)

In the formula: Q represents the number of iterations; y i q denotes the true value in the q ^-th iteration; y ^ i q BB is the predicted value in the q ^-th iteration; f q refers to the optimal tree in the q ^-th iteration; g i and h i are the first and second derivatives of the loss function l y i q , y ^ i q − 1 with respect to y i q − 1 ; and l y i q , y ^ i q − 1 is obtained by performing a Taylor expansion on f q x i .

Cross-validation is a common method to assess model performance on unseen data by partitioning the dataset into several exclusive subsets, then iteratively training and testing on these. In this study, we use k-fold cross-validation (k = 5), dividing the dataset into five equal subsets. Each subset serves as a validation set once, while the remaining k-1 subsets form the training set. This cycle repeats k times, and the average performance metrics across all iterations provide a stable and reliable assessment of model performance. As shown in Figure 1.

For the i ^-th fold, the model uses the i ^-th subset as the validation set, and the remaining k-1 subsets are combined to form the training set, yielding the performance metric Mi ​ for that fold. The overall model performance M is the average of the performance metrics from all k folds, denoted as M = 1 k ∑ i = 1 k M i . This average reflects the model’s comprehensive performance across the entire dataset and helps assess the model’s generalization ability. The key advantage of cross-validation is that it reduces the risk of overfitting by training and testing on different subsets, providing a more stable performance evaluation. This is especially useful when dealing with limited data, as cross-validation makes full use of every data point, improving the reliability of model assessment. In this study, cross-validation is applied to evaluate the performance of different optimization algorithms in predicting the mechanical properties of fiber-reinforced recycled aggregate concrete. By comparing the cross-validation results, we can objectively determine which algorithm performs best in practical applications.

The experiment utilized C30 concrete mixed with P.O 42.5 cement. Crushed pebbles (5–20 mm) were used as coarse aggregate, and natural river sand (0.075–4.750 mm) served as fine aggregate, with a fineness modulus of 2.8. A polycarboxylate superplasticizer, at 1.0% of the cement weight, improved workability. Key parameters of the steel and basalt fibers are shown in Table 1, where d is the fiber diameter, l is the fiber length, and l/d is the ratio affecting concrete performance. Fiber density (ρ) was crucial for determining volume percentage, while fcu referred to tensile strength.

The experimental design of this study carefully considered various factors affecting the compressive strength of basalt fiber-reinforced recycled aggregate concrete (BFRC), including fiber type, volume fraction, curing age, and testing conditions. A C30 concrete mix was used, with P.O 42.5 cement, crushed pebbles (5–20 mm) as coarse aggregate, and river sand (0.075–4.75 mm) as fine aggregate. Each specimen was molded into a standard 100 mm cubic mold and cured in a controlled environment at a constant temperature of 20°C ± 2°C and relative humidity of 95% for 3, 7, 14, and 28 days. Compressive strength testing followed the “Standard for Test Methods of Mechanical Properties of Ordinary Concrete” (GB/T 50,081–2019), with each specimen tested on a hydraulic press at a loading rate of 0.5 MPa/s. Additionally, in line with the “Test Methods for Steel Fiber Reinforced Concrete” standards, variables such as curing age, fiber content, and fiber type were integrated into the design. To ensure stability and repeatability of results, a total of 40 experimental groups were set, each containing three specimens, amounting to 120 specimens in total. The concrete mix ratios are provided in Table 2, where c represents material quantities and Bs denotes the sand rate. During each test, stress-strain behavior was also monitored alongside compressive strength to provide deeper insights into the material’s ductility and toughness under load (Saud et al., 2020).

This study primarily analyzes the impact of different types and volume fractions of steel fibers on the compressive strength of basalt fiber-reinforced concrete (BFRC) at curing ages of 3, 7, 14, and 28 days. According to previous research, the optimal content of basalt fibers in BFRC is 2.6 kg/m³ (Shahjalal et al., 2023; Sharma et al., 2022; Yang et al., 2021; Nikolenko et al., 2021), which was set as a fixed value in this study. The reference group is BFRC, labeled as BF. In the specific experiments, the volume fractions of hooked-end steel fibers were set at 0.8%, 1.0%, and 1.2%, corresponding to the specimen labels SF0.8, SF1.0, and SF1.2. The volume fractions of wavy steel fibers were similarly set at 0.8%, 1.0%, and 1.2%, with the corresponding labels WF0.8, WF1.0, and WF1.2. The volume fractions of copper-coated steel fibers were consistent with the previous ones, labeled as RPC0.8, RPC1.0, and RPC1.2.

The compressive strength of cubic specimens was measured according to the “Standard for Test Methods of Mechanical Properties of Ordinary Concrete” (GB/T 50,081–2019). The relationship between different types of steel fiber volume fractions, curing age (ta), and compressive strength (fcu) is shown in Figure 2.

This study primarily investigates the effects of different types and volume fractions of steel fibers on the compressive strength of basalt fiber-reinforced concrete (BFRC) at curing ages of 3, 7, 14, and 28 days. Based on prior research, the optimal content of basalt fibers in BFRC is determined to be 2.6 kg/m³, which was thus set as a constant parameter in this study. The control group, representing BFRC without steel fibers, is labeled as BF. For the experiments, hooked-end steel fibers were added at volume fractions of 0.8%, 1.0%, and 1.2%, labeled as SF0.8, SF1.0, and SF1.2, respectively. Likewise, the volume fractions of wavy steel fibers were set at 0.8%, 1.0%, and 1.2%, with the labels WF0.8, WF1.0, and WF1.2, while copper-coated steel fibers followed the same volume fractions and were labeled as RPC0.8, RPC1.0, and RPC1.2.

As shown in Figure 2, although steel fibers are primarily intended to reinforce tensile strength, their inclusion also enhances the compressive strength of BFRC within a specific range. This effect occurs because an optimal amount of steel fibers improves the bond between the concrete matrix and fibers, effectively inhibiting crack propagation. Dispersed steel fibers within the concrete can help reduce stress concentration at crack tips and defect points, thereby enhancing compressive strength. However, when the fiber volume becomes excessive, the increased surface area of steel fibers leads to insufficient cement mortar to fully encapsulate both aggregates and fibers, resulting in compromised overall integrity and a decline in compressive strength, demonstrating an initial increase followed by a decrease (Zhao et al., 2022; Weli et al., 2020; Wang et al., 2021; Wang et al., 2022; Zhang and Liu, 2023).

The copper-coated steel fibers form a dense oxide layer, providing a good bond with the concrete and reducing susceptibility to corrosion during curing, which achieves the highest compressive strength at 28 days. In contrast, the hooked-end and wavy steel fibers, lacking corrosion resistance treatment, exhibited rust on the fiber surfaces as the curing period progressed. This rust led to volume expansion from Fe(OH)₂ and Fe(OH)₃ formation, which, in turn, weakened compressive strength, causing the 14-day compressive strength to exceed that of the 28-day mark.

As shown in Figure 3, the concrete specimen was subjected to compression, radial cracking appeared around the specimen, while the middle portion, due to the weaker hoop effect, experienced localized lateral spalling of the concrete and the formation of fine cracks, as shown in Figure 4. Upon reaching the ultimate load, the specimen failed with a muffled sound, and through-cracks appeared. However, no fiber breakage was observed. At the location of the longitudinal cracks, the overlapping of steel fibers was clearly visible. Unlike typical BFRC, the specimen did not shatter, and its integrity was well-maintained during the failure process, exhibiting characteristics of plastic failure.

Nuclear magnetic resonance (NMR) uses the CPMG pulse sequence test on fully saturated water samples to obtain decay signals from spin echoes, which are then transformed into T₂ spectra using Fourier transform. The distribution of the T₂ spectrum reflects the pore size. In this experiment, the MesoMR-60 NMR instrument was used to analyze the concrete structure. Figure 5 shows the T₂ spectrum obtained from the nuclear magnetic resonance test.

Figure 5A shows the T2 relaxation time spectra of BFRC with different volume fractions of end-hooked steel fibers at the 28-day curing period, where Is represents signal intensity. It can be seen that the internal pore distribution of the concrete exhibits a positive correlation. As the volume fraction of steel fibers increases, the spectrum area first decreases and then increases. The SF1.0 group has the smallest spectrum area, confirming that a 1.0% volume fraction is the optimal dosage for end-hooked steel fibers. The introduction of steel fibers results in a significant narrowing of the main peak and a slight reduction in the area of the secondary peak. However, when the volume fraction of steel fibers exceeds 1.0%, the internal pores in the concrete increase, and the spectrum area expands. This may be due to excessive fiber content, leading to a larger specific surface area, which prevents full encapsulation of the fibers by the cement paste.

Figure 5B studies the changes in the internal micro-pore structure of BFRC with a 1.0% volume fraction of end-hooked steel fibers as the curing period progresses. At 3 days of curing, the spectrum presents a “three-peak” structure, while at 7, 14, and 28 days, it transitions to a “two-peak” structure, indicating that the internal pore structure and quantity evolve over time, which in turn affects the compressive strength. Since the spectrum area is proportional to the volume of fluid in the concrete, it also reflects changes in pore volume. The number of peaks corresponds to the distribution characteristics of the pore structure. At the 3-day curing period, the main peak reaches the highest value, and the spectrum area is the largest, indicating that the hydration reaction of the cement is not yet complete, leaving many unfilled pores, which leads to lower compressive strength. As the curing time extends, harmful pores (“three-peak” structure) gradually disappear, and the area of harmless pores (“two-peak” structure) decreases with time. This is due to the progressive filling of pores by cement hydration products, increasing the density and improving the compressive strength. However, during the 14- to 28-day curing period, steel fibers undergo corrosion, and the resulting iron compounds expand in volume, weakening the bond between the steel fibers and the cementitious matrix. This not only fails to prevent the expansion of micro-cracks but also interferes with the formation and filling of cement hydration products, leading to a decrease in density and an increase in pore size.

The concept of fluid saturation was introduced to analyze the internal pore structure changes of BFRC. Bound fluid exists within the micro-pores of the concrete, and the greater the bound fluid saturation, the more micro-pores in the concrete, indicating higher compactness.

Figure 6A shows the trend of saturation (S) and porosity (θ _pd ) of BFRC with different volume fractions of end-hooked steel fibers at the 28-day curing period. As the volume fraction of steel fibers increases, the saturation of the bound fluid in the concrete initially rises and then falls. It can be observed that the SF1.0 group exhibits the highest bound fluid saturation, with an 18.53% increase compared to the control group. Porosity is an important indicator that reflects the ratio of pore volume to the volume of the concrete matrix. The smaller the porosity, the fewer internal pores, and the more compact the structure. The SF1.0 group’s porosity is 0.242% lower than the control group.

Figure 6B shows the changes in porosity and saturation over time for BFRC with a 1% volume fraction of end-hooked steel fibers. Similar to Figure 6A, bound fluid saturation increases first and then decreases as curing time progresses. At the 14-day curing period, due to the hydration reaction of cement filling the pores, porosity decreased by 0.121% compared to the 3-day period, while bound fluid saturation increased by 35.88%. By the 28-day curing period, rust spots had appeared on the surface of the steel fibers, leading to changes in the pore structure. Porosity slightly increased by 0.012%, and bound fluid saturation decreased by 1.61%.

The pore distribution within the concrete can be determined through the transverse relaxation time (T₂). The internal pores are classified into three types: gel pores (T₂ < 1 m), capillary pores (1 m < T₂ < 100 m), and non-capillary pores (T₂ > 100 m). The presence of gel and capillary pores has no significant impact on the compressive strength of the concrete and is therefore referred to as harmless pores. In contrast, non-capillary pores have larger pore sizes, and the greater the number of non-capillary pores, the more they negatively impact the compressive strength of the concrete, thus being referred to as harmful pores. Figure 7 shows the pore size distribution.

From Figure 7, it can be observed that after 28 days of hydration, the proportion of harmless pores in BFRC reaches 64.6%. With the increase in steel fiber content, the proportion of harmless pores initially increases and then decreases. When the volume content of hooked-end steel fibers is 1.0%, the proportion of harmless pores reaches a maximum of 81.9%, indicating that the internal structure of the concrete is most compact at this point. However, when the steel fiber content increases to 1.2%, the proportion of harmful pores rises from 18.1% to 21.4%. This phenomenon is primarily due to the excessive amount of steel fibers, which increases the specific surface area, preventing the cement paste from fully enveloping the fibers. This is consistent with the trend observed in the bound fluid saturation, further confirming that the rusting of steel fibers impacts the internal pore structure of the concrete. As the number of harmful pores increases, the compressive strength of the concrete decreases.

The dataset used in this study consists of 120 experimental data points, primarily aimed at predicting the compressive strength of fiber-reinforced recycled aggregate concrete (BFRC). Each data point includes multiple input feature variables, such as the type of steel fiber, fiber volume content, curing age, water-cement ratio, and sand ratio. The dataset also includes the corresponding output target variable, which is the compressive strength of the concrete.

To ensure the reliability and generalization capability of the model, we divided the 120 data points into a training set and a test set at an 8:2 ratio, yielding 96 points for training and 24 for testing. For model optimization and to avoid overfitting, we applied five-fold cross-validation to the training data. In this process, the training data was divided into five subsets, with each subset used once as a validation set while the other four subsets trained the model. This procedure fine-tuned model parameters and ensured a robust evaluation of model performance. Figure 8 shows the data distribution (Figure 8A) and a correlation heatmap (Figure 8B) illustrating the relationships among variables.

The correlation analysis in Figure 8B highlights the relationships between key variables. Cement content has a strong positive correlation (0.5) with compressive strength, signifying that as cement content increases, compressive strength also improves. Water content exhibits a moderate positive correlation (0.3) with compressive strength, indicating that although water contributes to strength, its impact is less significant than that of cement. Similarly, the water-reducing agent shows a correlation of 0.28 with compressive strength, suggesting that it enhances mechanical properties by lowering the water-cement ratio. Conversely, sand and aggregate contents have negative correlations of −0.28 and −0.38, respectively, with compressive strength, suggesting that higher sand and aggregate proportions may reduce concrete strength. The highest correlation, 0.8, is observed between curing age and compressive strength, consistent with the known effect of increased strength over time.

In the process of model evaluation, MSE (Mean Squared Error), MAE (Mean Absolute Error), MAPE (Mean Absolute Percentage Error), and R ² (Coefficient of Determination) are commonly used performance metrics (Song et al., 2021; Chen et al., 2023). MSE amplifies larger errors and is therefore sensitive to extreme values, helping to capture the model’s response to outliers. MAE, on the other hand, calculates the absolute value of the error and is not affected by the direction of the error, making it more suitable for reflecting the overall prediction accuracy of the model, especially when focusing on the overall error magnitude. Additionally, MAPE expresses the error as a percentage, making it easier to compare data across different scales, and it is suitable for datasets with varying units or magnitudes. R ² provides a relative measure, assessing the model’s explanatory power based on the linear relationship between predicted and actual values. For regression tasks, an R ² value close to one indicates strong explanatory power, while an R ² value near 0 suggests poor model performance, indicating that the model fails to capture the relationship between the data.

To ensure comprehensive model evaluation, multiple metrics are typically combined to analyze model performance holistically. For example, while both MSE and MAE can measure error, MSE emphasizes the impact of large errors, whereas MAE focuses on the average level of error. MAPE provides information about the relative size of the error, making it more comparable across different datasets. R ², on the other hand, offers an intuitive evaluation of the model’s goodness-of-fit, allowing for a quick assessment of overall performance. The calculation formulas for each metric are as shown in Equations 26–29: M S E = 1 n ∑ i = 1 n y i − y ^ i 2 (26) MAE = 1 n ∑ i = 1 n y i − y ^ i (27) MAPE = 1 n ∑ i = 1 n y i − y ^ i y i × 100 % (28) R 2 = 1 − ∑ i = 1 n y i − y ^ i 2 ∑ i = 1 n y i − y ¯ 2 (29)

To ensure the generalization ability of the model across different datasets, this study utilized five-fold cross-validation for model training and parameter optimization. Cross-validation allows the model’s performance to be validated on various data splits, effectively reducing overfitting and enhancing the model’s robustness. Figure 9 shows the cross-validation loss functions for each optimized model.

The above figure demonstrates the mean squared error (MSE) variations for five different optimization algorithms applied to the XGBoost model (SOA-XGBoost, TSA-XGBoost, MA-XGBoost, GWO-XGBoost, and SSA-XGBoost) over 200 iterations. As shown in the figure, the MSE values of the three optimized models, SOA-XGBoost, TSA-XGBoost, and MA-XGBoost, are significantly lower than those of the traditional GWO-XGBoost and SSA-XGBoost, indicating faster convergence and reduced errors. This further verifies the advantage of the new optimization algorithms in enhancing the prediction accuracy of the model. The MSE values of the traditional optimization algorithms, GWO-XGBoost and SSA-XGBoost, are initially higher and tend to stabilize over time, but their final error values remain higher than those of the newer optimization algorithms. The optimal parameter combinations for the models, based on the minimum error, are provided in Table 3.