# Towards Designing Durable Sculptural Elements: Ensemble Learning in Predicting Compressive Strength of Fiber-Reinforced Nano-Silica Modified Concrete

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}, RMSE, VAF, MAE, and bias, addressing an objective assessment of the predictive models’ performance and capabilities. The outcomes reveal that GWO-XGBoost, exhibiting R

^{2}values of (0.971 and 0.978) for the train and test stages, respectively, emerges as the best predictive model for estimating the compressive strength of fiber-reinforced nano-silica concrete (FrRNSC) compared to other models. Consequently, the proposed GWO-XGBoost algorithm proves to be an efficient tool for anticipating CSFrRNSC.

## 1. Introduction

_{2}, which results in a denser final product [25,26]. This causes the percentage of portlandite-Ca(OH)

_{2}to decrease in cementitious materials like cementitious materials. NS may be used to replace up to 4% of cement, according to previous research [27,28]. This can significantly improve the material’s durability and strength, particularly when subjected to challenging circumstances, such as corrosion and high temperatures. Even though there are many different uses, the best results are obtained when NS is used as a cement substitute in the range from 0.5 to 4%. This is because using excessive amounts of NS may result in the buildup of particles and a reduction in the material’s workability [29]. A wide variety of nanoparticles, including NS, are utilized as materials to be added to concrete in order to improve the macroscopic qualities and interpretation of the material. Nevertheless, the limited practical application of NS in constructions may be related to its much higher prices, which are about one thousand times more costly than normal cement [30,31].

_{2}O

_{3}, nano Al

_{2}O

_{3}, nano TiO

_{2}, and NS, have undergone scrutiny. Notably, NS stands out for its ability to significantly improve compressive strength (CS) in concrete. Additionally, it has been observed that NS reduces the initial and final setting times of concrete while augmenting its early-age strength. The nanostructure of NS plays a pivotal role, offering an unusually large SSA and functioning as an aggregate–cement binder [42]. The remarkable pozzolanic activity of NS is attributed to its nanoparticle size [43]. Furthermore, the ITZ, recognized as a weak phase in cementitious materials, experiences enhancement due to the packing of these tiny NPs in gaps and voids, thereby reducing permeability [44,45,46]. NS emerges as a highly active component that expedites the hydration process of cementitious materials [47], fostering the formation of more calcium silicate hydrate (C-S-H) gel [48], which is crucial for material strength [49]. The proportion of portlandite-Ca(OH)

_{2}decreases in cementitious materials as NS combines with Ca(OH)

_{2}to form a denser product [50,51]. Previous studies indicate that substituting NS for up to 4% of the cement in concrete enhances its mechanical strength and durability, especially under adverse conditions, like elevated temperatures and corrosion [52]. While NS has demonstrated efficacy in specific applications for cementitious materials, optimal utilization occurs in proportions ranging from 0.5% to 4% as a cement substitute. However, an excess amount of NS may lead to agglomeration, compromising workability due to improper dispersion [53]. The distinguishing characteristic of NP lies in their high volume-to-surface-area ratio, as depicted in Figure 2. Numerous NPs serve as nano-additives in cementitious composites to enhance their macroscopic characteristics and performance, with NS gaining prevalence among these NPs. Nevertheless, the limited practical adoption of NS in construction is primarily attributed to its high cost, which remains 1000 times more expensive than ordinary cement [54,55,56].

## 2. Materials and Methods

#### 2.1. AdaBoost

_{n}(x) reveals the overall model, F

_{n−}

_{1}(x) signifies the overall achieved in the previous iteration, y

_{i}indicate the estimation results of the ith tree, and h(x

_{i}) is the current generated tree.

#### 2.2. XGBoost

_{i}representing the measured values, ${\widehat{o}}_{i}$ as the estimated values, Ω as the regularization term, n denoting the number of constructed trees, and f representing the function.

#### 2.3. LightGBM

#### 2.4. CatBoost

_{1}, m

_{2}, …, m

_{n}). Each model, mn, is trained using the earliest I samplings in the permutation. Subsequently, in each repetition, the M

_{J}

_{−1}model is utilized to obtain the residual of the jth sample. This distinctive approach sets CatBoost apart in its tree construction methodology.

#### 2.5. Grey Wolf Optimization Algorithm

**Figure 4.**Social hierarchy of wolves in GWO [102].

**Figure 5.**The location guiding and updating mechanism of the ω wolves adopted by GWO [102].

## 3. Data Analysis and Data Preparation

_{m}denotes the average value across all d data, and r

_{m}signifies the average value across all r data. A positive correlation is indicated by a value of p greater than zero (p > 0), signifying a positive linear relationship between the two parameters. A stronger positive correlation is indicated by a value of p closer to one (p ≃ 1). Conversely, if p is less than zero (p < 0), it implies a negative linear correlation between the two parameters. A stronger negative correlation is denoted by a value of p closer to minus one (p ≃ −1) [110,111].

_{r}and ${\widehat{X}}_{r}$ indicate measured and predicted CSFrRNSC, respectively. It should be noted that ${\widehat{X}}_{r}$ can be separately defined for each CatBoost, LightGBM, AdaBoost, and XGBoost models, as shown in Equations (11)–(14):

_{1}–h

_{6}are, respectively, n_estimators, learning_rate, gamma, max_depth, min_child_weight, and reg_alpha, which function as hyperparameters of the XGBoost method. In Equation (12), h

_{1}–h

_{4}are, respectively, the learning_rate, n_estimators, max_depth, and reg_alpha, which function as hyperparameters of the LightGBM method. In Equations (13) and (14), h

_{1}and h

_{2}are respectively learning_rate and n_estimators as hyperparameters of both CatBoost and AdaBoost models.

## 4. Prediction Results

#### 4.1. Hyperparameters Tunning

^{2}), root mean squared error (RMSE), variance accounted for (VAF), mean absolute error (MAE), and bias were utilized. In the context of the regression analysis, these three metrics commonly serve as benchmarks for assessing the performance of AI models and can be computed by applying Equations (15)–(19) [32,100,118,119,120,121,122,123,124]:

_{i}denotes the real CSFrRNSC, P

_{i}stands for the anticipated CSFrRNSC, ${\overline{P}}_{i}$ signifies the average of the anticipated CSFrRNSC, and n stands for the number of data samples. It is noteworthy that achieving values of one, zero, one hundred, zero, and zero for R

^{2}, RMSE, VAF, MAE, and bias, respectively, signifies optimal model capability and performance [125,126].

^{2}values for conventional and optimized modes in both training and testing phases, a model with the lowest RMSE and highest R

^{2}in both the training and testing phase is the GWO-XGBoost model.

^{2}value stands at 0.971, with an accompanying RMSE value of 1.933. On the testing subset, the R

^{2}value is 0.978, and the RMSE value is 2.129. Evidently, the developed GWO-XGBoost technique exhibits exceptional precision, effectively predicting the compressive strength of CSFrRNSC. Notably, the model’s performance on the testing subset surpasses that on the training subset to some extent.

#### 4.2. Results of Predictive Models

^{2}and VAF values and the lowest MAE, RMSE, and bias values. According to Table 5, the GWO-XGBoost model can predict CSFrRNSC better than other techniques. However, the optimized XGBoost model, i.e., GWO-XGBoost, has the most accurate results on the basis of RMSE 1.933 and 2.129 for training and testing phases, respectively. As shown in Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19, which illustrated the scatterplot of measured and predicted CSFrRNSC in both parts of the train and test, it can be shown that four trained systems anticipated the CSFrRNSC with an acceptable performance. But, the GWO-XGBoost technique with an R

^{2}of 0.978 for the testing phase was more accurate than the LightGBM, with an R

^{2}of 0.926; XGBoost, with an R

^{2}of 0.930; CatBoost, with an R

^{2}of 0.930; AdaBoost, with an R

^{2}of 0.905; GWO-LightGBM, with an R

^{2}of 0.937; GWO-CatBoost, with an R

^{2}of 0.924; and GWO-AdaBoost, with an R

^{2}of 0.929. Hence, the GWO-XGBoost technique is selected as a superior system in estimating CSFrRNSC. Figure 20 provides a visual representation of the comparison between the CSFrRNSC anticipated by the hybrid systems and the measured CSFrRNSC. From a logical standpoint, a majority of the projected CSFrRNSC values closely align with the actual measurements.

_{ij}is the sensitivity values of x

_{i}(input) and x

_{j}(output).

_{ij}values in ascending order is as follows: FV < Age < NS < w/b < CF < SP/B, with impact values of 0.718, 0.755, 0.821, 0.948, 0.954, and 0.958, respectively. The Rank value represents the rank of the influence degree of each parameter on CSFrRNSC. The smaller the ordinal number is, the higher the influence degree of the parameter on CSFrRNSC. Therefore, SP/B has the largest influence degree, and FV has the least influence degree.

## 5. Conclusions

^{2}values of 0.971 and 0.978, RMSE values of 1.933 and 2.129, VAF values of 96.960 and 97.774, MAE values of 1.653 and 1.802, and bias values of 1.653 and 1.802 for the train and test stages, respectively, GWO-XGBoost emerges as the most efficient predictor for estimating the CSFrRNSC when compared to other models. In essence, the proposed GWO-XGBoost algorithm not only enhances accuracy but also establishes itself as a powerful and reliable tool for anticipating CSFrRNSC. This study contributes to the ongoing efforts in the field, providing valuable insights for the application of advanced optimization algorithms and ensemble learning techniques in the prediction of concrete compressive strength. While this research demonstrates the effectiveness of the GWO-XGBoost algorithm for predicting the compressive strength of FrRNSC in sculptural elements, it is important to mention two main limitations. First, the study primarily focuses on FrRNSC for sculptural elements. The findings may not be directly generalizable to other types of concrete or construction applications. Future research could explore the applicability of the proposed approach to a broader range of concrete formulations and use cases.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Particle sizes and SSA relevant to cementitious material [57].

**Figure 9.**The optimization of the hyperparameters relevant to GWO-LightGBM, GWO-XGBoost, GWO-CatBoost, and GWO-AdaBoost models.

**Figure 20.**Comparison of estimated and real CSFrRNSC on the train and test data points in the GWO-XGBoost model.

**Figure 21.**Revealing Taylor diagram for trained systems based on train (

**left**) and test (

**right**) sets.

**Figure 22.**Violin diagram relevant to developed systems in both train (

**left**) and test (

**right**) data points.

Variable | Unit | Notation | Min | Ave | Max | StD | Skewness | Kurtosis | |
---|---|---|---|---|---|---|---|---|---|

Input | Fiber volume | % | FV | 0 | 0.198 | 0.9 | 0.185 | 1.974 | 5.204 |

CA/FA | - | CF | 0.874 | 0.906 | 1.135 | 0.060 | 2.382 | 6.035 | |

w/b | - | w/b | 0.31 | 0.408 | 0.48 | 0.041 | 0.509 | 0.236 | |

Nano-silica | kg/m^{3} | NS | 0 | 21.214 | 49.6 | 17.303 | 0.317 | −1.084 | |

SP/B | - | SP/B | 0.005 | 0.017 | 0.025 | 0.006 | −1.176 | −0.396 | |

Age | day | Age | 7 | 41.651 | 120 | 38.252 | 0.785 | −0.930 | |

Output | Compressive strength of fiber-reinforced nano-silica concrete | MPa | CSFrRNSC | 19.1 | 66.483 | 91.2 | 17.829 | −0.838 | −0.243 |

Training Set | ||||||

Parameter | Min | Ave | Max | StD | Kurtosis | Skewness |

FV | 0 | 0.2 | 0.5 | 0.129 | 0.566 | 0.723 |

CF | 0.874 | 0.882 | 0.973 | 0.026 | 7.457 | 3.051 |

w/b | 0.39 | 0.398 | 0.48 | 0.025 | 7.323 | 3.035 |

NS | 0 | 23.710 | 49.6 | 18.282 | −1.318 | 0.110 |

SP/B | 0.005 | 0.019 | 0.02 | 0.004 | 7.323 | −3.035 |

Age | 7 | 38.756 | 90 | 35.202 | −1.349 | 0.663 |

CSFrRNSC | 42.1 | 74.114 | 91.2 | 11.111 | 0.129 | −0.838 |

Testing Set | ||||||

Parameter | Min | Ave | Max | StD | Kurtosis | Skewness |

FV | 0 | 0.9 | 0.193 | 0.295 | 2.140 | 1.876 |

CF | 0.905 | 1.135 | 0.977 | 0.074 | 0.778 | 1.334 |

w/b | 0.31 | 0.48 | 0.439 | 0.061 | 0.718 | −1.452 |

NS | 0 | 31.5 | 13.784 | 11.209 | −1.256 | 0.128 |

SP/B | 0.005 | 0.025 | 0.010 | 0.008 | −0.645 | 1.009 |

Age | 7 | 120 | 50.273 | 45.528 | −1.169 | 0.755 |

CSFrRNSC | 19.1 | 69.1 | 43.764 | 14.404 | −1.148 | 0.232 |

**Table 3.**Hyperparameters’ tunning in GWO-LightGBM, GWO-XGBoost, GWO-CatBoost, and GWO-AdaBoost techniques.

Technique | Optimizer | Hyperparameter | Optimum Values |
---|---|---|---|

LightGBM | GWO | learning_rate, n_estimators, max_depth, and reg_alpha | learning_rate = 0.005 n_estimators = 170 max_depth = 9 reg_alpha = 0.45 |

XGBoost | GWO | n_estimators, learning_rate, gamma, max_depth, min_child_weight, and reg_alpha | n_estimators = 100 learning_rate = 0.25 gamma = 0.6 max_depth = 2 min_child_weight = 5 reg_alpha = 1 |

CatBoost | GWO | learning_rate and n_estimators | learning_rate = 0.003 n_estimators = 300 |

AdaBoost | GWO | learning_rate and n_estimators | learning_rate = 0.001 n_estimators = 500 |

Technique | Train Phase | Test Phase | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

MAE | R^{2} | RMSE | VAF | Bias | MAE | R^{2} | RMSE | VAF | Bias | |

CatBoost | 3.107 | 0.914 | 3.691 | 89.484 | 3.107 | 3.184 | 0.930 | 3.838 | 92.782 | 3.184 |

AdaBoost | 3.164 | 0.910 | 3.628 | 89.632 | 3.164 | 4.294 | 0.905 | 4.858 | 88.709 | 4.294 |

LightGBM | 2.674 | 0.929 | 3.040 | 92.593 | 2.674 | 3.506 | 0.926 | 4.102 | 91.715 | 3.506 |

XGBoost | 2.129 | 0.954 | 2.499 | 94.987 | 2.129 | 3.746 | 0.930 | 4.232 | 91.387 | 3.746 |

GWO-CatBoost | 2.880 | 0.923 | 3.322 | 91.114 | 2.880 | 3.273 | 0.924 | 3.971 | 92.325 | 3.273 |

GWO-AdaBoost | 2.669 | 0.938 | 3.106 | 93.090 | 2.669 | 3.569 | 0.929 | 4.101 | 91.789 | 3.569 |

GWO-LightGBM | 2.455 | 0.948 | 2.911 | 94.499 | 2.455 | 3.036 | 0.937 | 3.606 | 93.600 | 3.036 |

GWO-XGBoost | 1.653 | 0.971 | 1.933 | 96.960 | 1.653 | 1.802 | 0.978 | 2.129 | 97.774 | 1.802 |

Technique | Train Phase | Test Phase | Total Rate | Rank | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | R^{2} | RMSE | VAF | Bias | MAE | R^{2} | RMSE | VAF | Bias | |||

CatBoost | 2 | 2 | 1 | 1 | 2 | 6 | 5 | 6 | 6 | 6 | 37 | 6 |

AdaBoost | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 12 | 8 |

LightGBM | 4 | 4 | 5 | 4 | 4 | 4 | 3 | 3 | 3 | 4 | 38 | 5 |

XGBoost | 7 | 7 | 7 | 7 | 7 | 2 | 6 | 2 | 2 | 2 | 49 | 3 |

GWO-CatBoost | 3 | 3 | 3 | 3 | 3 | 5 | 2 | 5 | 5 | 5 | 37 | 6 |

GWO-AdaBoost | 5 | 5 | 4 | 5 | 5 | 3 | 4 | 4 | 4 | 3 | 42 | 4 |

GWO-LightGBM | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 7 | 65 | 2 |

GWO-XGBoost | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 80 | 1 |

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**MDPI and ACS Style**

Wang, R.; Zhang, J.; Lu, Y.; Huang, J.
Towards Designing Durable Sculptural Elements: Ensemble Learning in Predicting Compressive Strength of Fiber-Reinforced Nano-Silica Modified Concrete. *Buildings* **2024**, *14*, 396.
https://doi.org/10.3390/buildings14020396

**AMA Style**

Wang R, Zhang J, Lu Y, Huang J.
Towards Designing Durable Sculptural Elements: Ensemble Learning in Predicting Compressive Strength of Fiber-Reinforced Nano-Silica Modified Concrete. *Buildings*. 2024; 14(2):396.
https://doi.org/10.3390/buildings14020396

**Chicago/Turabian Style**

Wang, Ranran, Jun Zhang, Yijun Lu, and Jiandong Huang.
2024. "Towards Designing Durable Sculptural Elements: Ensemble Learning in Predicting Compressive Strength of Fiber-Reinforced Nano-Silica Modified Concrete" *Buildings* 14, no. 2: 396.
https://doi.org/10.3390/buildings14020396