Predicting the permeability and compressive strength of pervious concrete using ensemble machine learning model

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However, the current performance prediction models mainly relied on porosity, ignoring the influence of other pore structure parameters, resulting in insufficient prediction accuracy. The aim of this paper is to establish machine learning-based models for predicting permeability and compressive strength of pervious concrete. Firstly, six independent models, the multiple linear regression and the Stacking algorithm were applied to construct the ensemble model. Secondly, 90 groups of pervious concrete specimens with varying porosities and grades were prepared and tested to obtain the initial data set. Then, the initial data set was augmented, and the prediction models were trained. The results show that 6 input parameters were suitable to make the models gain a high prediction accuracy (0.93) with simplicity. Compared to independent models, the ensemble models showed the highest accuracy and stability. This was due to that the ensemble models both utilized the basic training set and the output of the first level learner. The ensemble models performed significantly better than empirical formulas in predicting both permeability and compressive strength. Compared with the permeability, the compressive strength of pervious concrete is more sensitive to porosity. Physical sciences/Engineering Physical sciences/Materials science Physical sciences/Mathematics and computing pervious concrete ensemble machine learning permeability compressive strength prediction Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1. Introduction Currently, rapid urbanization and high-density infrastructure development have caused a variety of ecological and environmental issues. Consequently, the concept of sustainable development is becoming increasingly important and being implemented, including low impact development (LID), green infrastructure (GI), best management practices (BMPs), sponge city (SC), etc. As an environmentally friendly paving material, pervious concrete is gradually being used in the construction of novel cities [ 1 – 2 ]. Compared to traditional dense concrete, pervious concrete has a rich pore structure (15%-30%) [ 3 – 4 ], which makes it play an active role in controlling surface rainwater runoff, improving the hydrological cycle of the city, realizing rainwater collection, infiltration and purification, and noise reduction [ 5 – 7 ]. Undoubtedly, the performance of pervious concrete is largely determined by its pore characteristics [ 8 – 9 ]. One of the most well-studied parameters is porosity. Generally, the relationship between permeability and porosity is generally positive, whereas the relationship between mechanical properties and porosity is negative [ 10 ]. Other pore features also have been reported to affect the water permeability and compressive strength of pervious concrete [ 11 – 13 ]. Yu et al. [ 14 ] found that water permeability increased with the increase of pore size, and it is more sensitive to the content of small pores. Pore distribution, pore diameter and tortuosity were also observed to have an obvious influence on permeability [ 15 – 16 ]. Liu [ 17 ] claimed that a more uniform pore distribution and larger pore spacing could significantly increase compressive strength. Unconfined compression strength was observed to increase with increasing pore surface area in another literature. Additionally, the closer the pore shape is to the sphere, the lower the unconfined compression strength [ 18 ]. When the porosity is constant, the increase in small pores will increase the compressive strength [ 19 ]. Based on the analysis of the influence of pore structure characteristics on the performance of pervious concrete, some empirical formulas between the performance of pervious concrete and pore structure parameters were established, which are summarized in Table 3 . These empirical formulas can accurately predict permeability and compressive strength, but each prediction formula was only applicable to specific conditions. Additionally, it can be observed from Table 3 that majority of the current prediction formulas only consider the general pore characteristics of porosity. However, the above analysis has shown that the performance of pervious concrete was not only affected by porosity, but also significantly affected by other pore characteristics. Therefore, the performance prediction of pervious concrete considering more pore structure parameters needs to be further studied. The pore structure of pervious concrete is very complex, and there are many characterization parameters. It is difficult to establish an accurate performance prediction formula based on pore structure parameters directly through theoretical derivation. Thus, this paper tries to establish a deep learning model for predicting pervious concrete performance using pore structure parameters. The object of this paper is to develop a Stacking ensemble model to predict permeability and compressive strength of pervious concrete by utilizing pore characterization parameters as inputs. First, six independent models (DT, BP, CNN, RF, GBDT and XGBoost) were selected as primary learners, the multiple linear regression as secondary learner, and the Stacking algorithm was applied to construct the ensemble model. Second, 90 groups of pervious concrete specimens with varying porosities and grades were prepared. 20 kinds of pore structure characterization parameters and the permeability and compressive strength were extracted to form the initial data set. Then, the initial data set was augmented, and the prediction models were trained. Afterward, the number of input parameters was determined, the stability of ensemble models was verified, and the prediction performance of ensemble models were evaluated. The flow chart of this paper was exhibited in Fig. 1 . 2 Deep learning models The performance prediction model for pervious concrete was built by Stacking, an effective integrated learning method. This method consists of two layers of learners. The first layer of learners implements several independent models. The second layer of learners integrates the independent models obtained in the previous step to obtain the final prediction results. In view of the different performance of the primary learner, this paper sets different weights according to the output results of each primary learner. In this paper, six independent models were selected in the first layer of the learner, including Decision Tree (DT), Back Propagation (BP) neural network, Convolutional Neural Network (CNN), Random Forest (RF), Gradient Boosting Decision Tree (GBDT), eXtreme Gradient Boosting (XGBoost). Multiple linear regression was used as the secondary learner in the second layer. 2.1 The first layer of the learner Decision Tree is a graphical method for the intuitive use of probability analysis. It is based on the known probability of occurrence in various situations [ 20 ]. It employs a decision tree to obtain the probability that the expected value of the net present value is greater than zero [ 21 ]. BP neural network is a multi-layer feedforward neural network trained according to the error back propagation algorithm. It is one of the most widely used neural network models [ 22 ]. CNN is a kind of feed forward neural networks FNN (Feed forward Neural Networks) with convolution calculation and deep structure [ 23 – 24 ]. RF is a classifier that uses multiple decision trees to train and predict samples [ 25 ]. GBDT is an iterative decision tree algorithm [ 26 ]. The algorithm consists of multiple decision trees, and the conclusions of all trees are summed up to make the final answer [ 27 ]. XGBoost is a network that iteratively trains multiple decision tree models and utilizes a gradient descent method to gradually optimize the predictive power of the model [ 28 – 29 ]. 2.2 The second layer of the learner Regression analysis provides a simple way to establish the functional relationship between variables, which is one of the most widely used statistical tools. It uses the regression coefficient to explain the relationship between independent variables and dependent variables, and selects the optimal regression coefficient through the gradient descent method [ 30 ]. In practice, the dependent variable is often affected by two or more independent variables. Therefore, predicting or estimating the dependent variable from the optimal combination of multiple independent variables is more accurate than from only one independent variable. The permeability and compressive strength of pervious concrete are affected by several pore structure parameters, such as porosity, pore size, and pore distribution, etc. Consequently, multivariate linear regression was used as a secondary learner to develop the prediction model between pore structure and macroscopic properties, in this paper. 3 Dataset acquisition and model training 3.1 Initial dataset acquisition 3 kinds of porosity (15%, 20%, 25%) of 30 different gradations (Fig. 2 ) were designed, resulting in 90 groups of schemes. Afterwards, the pervious concrete samples (100mm × 100mm × 100mm) were prepared, and CT scanning was performed after 28 days of curing, and the permeability coefficient and compressive strength were measured. The 2D / 3D pore structure characteristics of 90 groups of samples were extracted by image processing technology (Table 1 ). The initial data set required for training the performance prediction model of pervious concrete with pore structure parameters as input was obtained. Table 1 The extracted pore structure parameters Property Names Abbreviations Mean value of face rate[ 31 ] P ave Maximum value of face rate[ 31 ] P max Mean value of shape factor(H/W)[ 32 ] SH ave Maximum value of shape factor(H/W)[ 32 ] SH max Mean value of coordination number[ 33 ] CO ave Maximum value of coordination number[ 33 ] CO max Kurtosis[ 34 ] K Skewness[ 34 ] Sk Fractal dimension[ 35 ] FD Mean value of surface curvature[ 36 ] C ave Tortuosity[ 15 ] T Mean free spacing[ 37 – 38 ] D mfs Pore area distribution parameters[ 39 ] A p Pore volume distribution parameters[ 39 ] V p Mean value of pore throat area[ 40 ] T m Pore throat distribution parameters[ 40 ] Ts Surface porosity of large pore[ 14 ] LP Surface porosity of medium pore[ 14 ] MP Surface porosity of small pores[ 14 ] SP Surface porosity of large and medium pore [ 14 ] LMP water permeability Q Compressive strength CS 3.2 Data set preprocessing In the initial data set, the values of the data varied considerably and were different in dimension. For better comprehensive and comparative evaluation of the data, each category of data needs to be processed to keep them in the same magnitude order. In addition, if the data with different dimension are analyzed directly, the validity and precision will be affected. The maximum standardization method was utilized to standardize the initial data set to accelerate the convergence of the model in this paper. $$\:{x}_{ij}^{{\prime\:}}=\frac{{x}_{ij}}{{max}\left\{{x}_{ij}\right\}}\:\:\:\:\:\:\:\left(i=\text{1,2},\dots\:,m;\:j=\text{1,2},\dots\:,n\right)$$ 1 3.3 Data augmentation based on Mixup method The increase in dataset size can improve the accuracy and robustness of the model. However, datasets are often inconvenient to obtain and costly to increase in dataset size. Fortunately, data augmentation techniques can be employed to effectively expand data set. Mixups combine multiple sets of samples from different data sets linearly to create new samples, expanding the data set (Fig. 3 ). For example, two samples x 1 and x 2 , their corresponding labels are y 1 and y 2 , respectively. By summarizing the two samples and the corresponding label data respectively at a set ratio, the new sample and label data are generated afterwards (Eq. 2). \(\:{x}^{{\prime\:}}\:=\:\lambda\:{x}_{1}+\left(1-\lambda\:\right){x}_{2}\) \(\:{y}^{{\prime\:}}\:=\:\lambda\:{y}_{1}+(1-\lambda\:){y}_{2}\) (2) where x is the input vector, y is the one-hot (one-hot) encoding of the label, λ is a random number between 0 and 1, and λ ~ Beta(α,α), i.e., λ obeys a Beta distribution with all parameters α, which denotes the weights of x 1 and x 2 in the new sample. Therefore, the loss function used in this paper can be expressed as Eq. 3 . $$\:loss=\:\lambda\:\ast\:criterion\:(outputs,\:a)+(1-\lambda\:\ast\:criterion\:(outputs,b)$$ 3 Mixup was employed to expand the original data set. Ultimately, 1800 sets of data were obtained to form the final data set. Then, the data set was divided into training set and testing set. Subsequently, model training was performed on training set. 3.4 Model training and evaluation 3.4.1 Model-evaluation index The model was evaluated by the following indexes, including the coefficient of determination ( R 2 ) and the accuracy (Accuracy). $$\:{R}^{2}=1-\frac{\sum\:_{i=1}^{n}{({y}_{i}-{\widehat{y}}_{i})}^{2}}{\sum\:_{i=1}^{n}{({y}_{i}-{\stackrel{-}{y}}_{i})}^{2}}$$ 4 $$\:Accuracy=\frac{1}{n}{\sum\:}_{i=1}^{n}\:\frac{\widehat{{y}_{i}}}{{y}_{i}}$$ 5 where y i represents the true value, \(\:{\widehat{y}}_{i}\:\) represents the predicted value, and \(\:{\stackrel{-}{y}}_{i}\:\) represents the average value of the true value. 3.4.2 Model parameter settings The specific parameter settings and parameter values of the six independent models used to build Stacking ensemble model are summarized in Table 2 . Table 2 The parameter settings of the independent model. Model Parameter settings and parameter values BP hidden_layer_sizes=(100,4), activation='relu', solver='adam', random_state = 15, alpha = 0.0001, max_iter = 500 DT splitter='best', criterion=’mse’, random_state = 15, max_depth = 5 CNN activation='relu', optimizer='adam', loss='mse', epochs = 500, batch_size = 10 RF n_estimators = 100, criterion='mse', random_state = 15, max_depth = 5 GBDT n_estimators = 100, learning_rate = 0.1, random_state = 15 XGboost n_estimators = 100, learning_rate = 0.05, random_state = 15, max_depth = 3 4 Results and discussions 4.1 Determination of model input parameters 4.1.1 Type of input parameters Pore structure characterization parameters such as porosity, pore size, and tortuosity (about 20 parameters have been reported) have been verified to affect water permeability and compressive strength of pervious concrete [ 41 – 43 ]. The objective of this paper is to establish a deep learning-based prediction models for water permeability performance and compressive strength using pore structure parameters as inputs. The prediction accuracy of the model depends on the input parameters. Firstly, the Pearson correlation coefficient method was used to investigate the correlation between the pore structure characterization parameters extracted in this study and the permeability or compressive strength. Afterwards, two groups of 10 pore structure characterization parameters were selected (If the correlation between the next parameter and the previous parameter exceeded 0.8, it was not selected for eliminating the high linear correlation between the parameters.) respectively for model training based on the degree of correlation from high to low. The results of Pearson correlation coefficient analysis are shown in Fig. 4 . It can be seen that the descending order of the correlation between pore structure characterization parameters and permeability was: A p > SP > D mfs > C ave > P ave > MP > LP > P max > T s > FD > LMP > CO ave > CO max > T > K > V p > S k > T m > SH ave > SH max . The descending order of the correlation between pore structure characterization parameters and compressive strength was: P ave > CO max > FD > P max > CO ave > LMP > T m > T > SP > LP > A p > MP > V p > K > SH ave > C ave > SH max > D mfs > T s > S k . After eliminating the parameters with a correlation of more than 0.8 with the previously selected variables, the parameters used to establish the permeability prediction model were: A p 、MP、LP、P max 、TS、CO ave 、T、K、V p and T m . The parameters used to establish the compressive strength prediction model were Pave、CO max 、T m 、T、A p 、MP、V p 、K、SH ave and SH max . 4.1.2 Number of input parameters Model prediction accuracy depends on input parameters, but it does not necessarily improve with more input parameters [ 44 ]. However, with the increase of input parameters, the model is more complex and difficult to operate [ 45 ]. Therefore, the optimal input parameters and quantities for the model need to be determined. According to the correlation degree from high to low (Fig. 3 (c)), 4,6,8 and 10 pore structure parameters were selected as inputs to train the permeability compressive strength prediction model. The prediction accuracy of the models was evaluated, and the results are shown in Fig. 5 . It can be seen from Fig. 4 that as the number of input parameters increased, the prediction accuracy of the model (permeability and compressive strength) increased slowly. When the input parameters beyond 6, the growth rate was significantly slowed down (except for the BP models). When the input parameter was 6, the accuracy of the permeability prediction model (ensemble models) and the compressive strength prediction model was as high as 0.93. Considering the accuracy and simplicity simultaneously, two groups of 6 parameters were determined respectively as input to predict permeability and compressive strength of pervious concrete finally. Namely, the input parameters for the permeability prediction model were A p , MP, LP, P max , TS and CO ave , and the input parameters for the compressive strength prediction model were Pave, CO max , T m , T, A p and MP. Additionally, the ensemble model is found to be more accurate than independent models. This indicates that the ensemble model exhibited better performance than the independent models, which will be discussed in detail in the following section. 4.2 Comparison between ensemble models and independent models To compare the performance of the ensemble model and the independent models in detail, a test set was randomly selected and predicted by each model. The prediction results are shown Fig. 6 and Fig. 7 . In Fig. 6 and Fig. 7 , models with data points nearer to 45 ° are more accurate. It is observed that the prediction results of the Stacking models were concentrated near the 45° line. And a considerable number of data points coincided with the 45° line. Moreover, almost all data points were located in the area surrounded by two baselines (± 25% or ± 15%). The prediction accuracy of the different independent models varied significantly. A considerable portion of the prediction points of the permeability prediction models (except CNN) were distributed in areas outside the ± 25% baseline. The predicted points of the compressive strength prediction models (except BP) mostly fell within the baseline (± 15%). However, the prediction points were more dispersed, with a significantly lower rate of prediction points near the 45° line than the Stacking model. Moreover, the Stacking models possessed the largest R 2 (the permeability prediction model and compressive strength prediction model are 0.92 and 0.93, respectively). This indicates that the Stacking models gained higher prediction accuracy than the independent models. This is due to the fact that the second level learner uses both the basic training set and the output from the first level learner. As a result, stacking models can make better predictions by keeping the information from the original training set. For permeability predicting, R 2 of CNN, XGBoost, and GBDT reached 0.90, 0.89, and 0.89, respectively, and for compressive strength predicting, R 2 of XGBoost and GBDT all achieved 0.89. Based on the values of R 2 , CNN, XGBoost, and GBDT models presented a competitive challenge to Stacking models. However, it can be seen in Fig. 5 and Fig. 6 that both CNN and GBDT have more discrete data points and even some predictions beyond the baseline compared to the Stacking models. 4.3 Stability analysis of the models Stability is an index that cannot be ignored when evaluating a model. To evaluate the stability of the model, 10 groups of data sets containing 100 sets of data were randomly selected from the test set as new test sets. Afterwards, the stability of the established model was evaluated on the new test sets. The prediction accuracy and the variance of accuracy are displayed in Fig. 8 . As can be seen in Fig. 8 (a), the prediction accuracy of the Stacking model (permeability) fluctuates very little on the 10 new test sets, ranging between 0.91 and 0.94. In contrast, the prediction accuracy of the six independent models exhibited obvious fluctuations to varying degrees. In addition, the variance of the prediction accuracy of the Stacking model is 0.0085, which was significantly smaller than that of the independent models. It can be observed from Fig. 8 (b) that the prediction accuracy of the Stacking model (compressive strength) also fluctuated little, ranging from 0.92 to 0.94. Obviously, the prediction accuracy of CNN, RF, DT, and BP models fluctuated greatly, and the corresponding variances were significantly greater than the Stacking model. Interestingly, the fluctuation of prediction accuracy of XGBboost and GBDT models was not obvious, and the corresponding variances were close to that of Stacking model. However, the prediction accuracies of XGBboost and GBDT models were lower than 0.9, which is lower than that of the Stacking model of 0.93. Overall, the Stacking models showed the highest prediction accuracy and the smallest variance of prediction accuracy. Namely, the Stacking models had the highest accuracy and stability. 4.4 Comparison of Stacking models with empirical formulations Widely agreement has been reached on the fact that pore structure plays a dominant role in the permeability and compressive strength of pervious concrete. A variety of empirical formulas for predicting permeability and compressive strength have been established by the researchers, which are summarized in Table 3 . To compare the prediction accuracy of the Stacking models established in this paper and the empirical formulas, 10 sets of data were randomly selected from the test set. Afterwards, the Stacking models and the empirical formulas were used to predict permeability and compressive strength respectively. The ratio of predicted value to measured value are shown in Fig. 9 . It can be seen from Fig. 9 that the ratio of the predicted value of the Stacking model to the measured value was very close to 1 (both for permeability and compressive strength), ranging from 0.86 to 1.14. However, for most of the empirical formulas, the ratio of predicted value to measured value was distributed outside 0.7–1.3. Strikingly, a considerable part of the ratio was smaller than 0.5 or larger than 1.5. Intriguingly, when the empirical formula predicted the compressive strength, a considerable part of the ratio of the predicted value to the measured value was distributed between 0.7 and 1.3, and the ratio is rarely smaller than 0.5 or larger than 1.5. Although the empirical formulas established by different researchers were based on significant differences in test conditions (different in coarse aggregate types, cement types, and additives), the accuracy of empirical formulas in predicting compressive strength was significantly higher than that of permeability predicting. Therefore, it can be inferred that although the compressive strength is affected by the raw materials, the porosity plays a vital role in determining the compressive strength. Additionally, although porosity largely determines the water permeability, other pore structure characterization parameters also have a significant impact. Overall, the accuracy of the Stacking models was significantly better than the empirical formulas for predicting both permeability property and compressive strength. Moreover, compared with the permeability, the compressive strength of pervious concrete is more sensitive to porosity. Table 3 Empirical formulas for permeability and compressive strength. Permeability Compressive strength Code Empirical formula Reference Code Empirical formula Reference 1 k = 0.0286 ϕ 2t 0.721 [ 41 ] Kuang et al.,2011 1 σ =−1.2863 ϕ t + 46.692 [ 51 ] Bhutta et al., 2012 2 k = 1.93 e 0.0755 ϕ t [ 33 ] Sriravindrarajah et al., 2012 2 σ = 70.2e − 0.066 ϕ t [ 33 ] Sriravindrarajah et al., 2012 3 k = 0.03 ϕ 2t − 0.676 ϕ t + 4.615 [ 46 ] Liu et al., 2018 3 σ=(-581.9(w/c)2 + 301.8w/c-57.4) ln( ϕ t ) + 83.4 [ 47 ] Li et al., 2019 4 k = 2.98e0.06ϕe − 3.05 [ 47 ] Li et al., 2019 4 σ = 72.9 − 18.4ln( ϕ e ) [ 48 ] Tan et al., 2020 5 k = 0.57 e 0.98 ϕ e [ 48 ] Tan et al., 2020 6 k = 0.8511 ϕ t − 0.864 [ 49 ] Zhang et al., 2020 7 k = 1.19 ϕ t − 12.74 [ 50 ] Singh and Madasamy, 2022 where ϕt denotes total porosity and ϕe denotes effective porosity. 5 Conclusions In this paper, six independent models, the multiple linear regression and the Stacking algorithm were used to construct the ensemble model. 90 groups of pervious concrete specimens with varying porosities and grades were prepared to obtain the initial data set for permeability and compressive strength predicting models training. The models were evaluated and compared with empirical formulations. The main conclusions are as follows: (1) It was found that 6 input parameters were suitable to make the model gain a high prediction accuracy (0.93) with simplicity simultaneously. The input parameters for the permeability prediction model were A p , MP, LP, P max , TS and CO ave , and for the compressive strength prediction model were Pave, CO max , T m , T, A p and MP, respectively. (2) The ensemble models for predicting permeability and compressive strength exhibited higher accuracy than the independent models, the R 2 reached 0.92 and 0.93 respectively. This was due to that the ensemble models both utilized the basic training set and the output of the first level learner. (3) Although the identification accuracy of XGBboost and GBDT models was high (reached 0.89) on individual data set. However, the stability of its recognition accuracy was lower than that of the ensemble models on different datasets. Therefore, the ensemble models had the highest accuracy and stability. (4) The ensemble models performed significantly better than empirical formulas in predicting both permeability and compressive strength. In addition, compared with the permeability, the compressive strength of pervious concrete is more sensitive to porosity. Declarations Author Contribution Fan Yu: Conceptualization, Methodology, Writing- Original Draft, Writing - review & editing. Wei Chu: Writing - Original Draft, Writing - review & editing. Rui Zhang: Funding acquisition, Resources, Investigation, Supervision, Writing - review & editing. Zhang Gao: Writing - review & editing. Yunan Yang: Funding acquisition, Writing - review & editing. Acknowledgements The authors gratefully acknowledge the financial support provided by Natural Science Research Project of Yichang (A22-3-003), Research Fund for Excellent Dissertation of China Three Gorges University (Grant No. 2021BSPY005). and the Open Fund of Badong National Observation and Research Station of Geohazards (No. BNORSG202313). Data Availability The datasets used and/or analysed during the current study available from the corresponding author on reasonable request. References Chen X, Yang Y, Zhang C, et al. Valorization of construction waste materials for pavements of sponge cities: A review[J]. Construction and Building Materials, 2022, 356: 129247. Shen S, Burton M, Jobson B, Haselbach L (2012). Pervious concrete with titanium dioxide as a photo catalyst compound for a greener urban road environment. Transportation Research Board 91th Annual Meeting. January, Washington D.C. Guneyisi, E., Gesoglu, M., Kareem, Q., Ipek, S., 2016. 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Mater. 6 (4) (2012) 239–246, https://doi.org/10.1007/s40069-012-0024-x. Bouri E, Jalkh N. Spillovers of joint volatility-skewness-kurtosis of major cryptocurrencies and their determinants[J]. International Review of Financial Analysis, 2023, 90: 102915. Shan, J.G., Zhang, Y., Wu, S.Y., Lin, Z.S., Li, L., Wu, Q.L., 2022. Pore characteristics of pervious concrete and their influence on permeability attributes. Constr. Build. Mater. 327(2022). Zhu X. Image filtering by combination of the curvature-driven and edge-stop nonlinear diffusion[J]. Acta Photonica Sinica, 2008, 37(3): 609. M.S. Sumanasooriya, N. Neithalath, Stereology-and morphology-based pore structure descriptors of enhanced porosity (Pervious) concretes, ACI Mater. J. 106 (5) (2009) 429-438. O. Deo, N. Neithalath, Compressive behavior of pervious concretes and a quantification of the influence of random pore structure features, Mat. Sci. Eng. A 528 (1) (2010) 402-412. Neithalath N, Sumanasooriya M S, Deo O. Characterizing pore volume, sizes, and connectivity in pervious concretes for permeability prediction[J]. Materials characterization, 2010, 61(8): 802-813. J.G. Shan, Y. Zhang, S.Y. Wu, et al., Pore characteristics of pervious concrete and their influence on permeability attributes, Constr. Build. Mater. 327 (2022). Kuang, X., Sansalone, J., Ying, G., Ranieri, V., 2011. Pore-structure models of hydraulic conductivity for permeable pavement. J. Hydrol. 399, 148-157. Zhong, R., Xu, M., Netto, R.V., Wille, K., 2016. Influence of pore tortuosity on hydraulic conductivity of pervious concrete: characterization and modeling. Constr. Build. Mater. 125, 1158-1168. Zhong, R., Wille, K., 2016. Linking pore system characteristics to the compressive behavior of pervious concrete. Cement. Concrete. Comp. 70, 130-138. S.S. Matin, L. Farahzadi, S. Makaremi, S.C. Chelgani, G. Sattari, Variable selection and prediction of uniaxial compressive strength and modulus of elasticity by random forest, Appl. Soft Comput. (2017) Deo O, Neithalath N. Compressive behavior of pervious concretes and a quantification of the influence of random pore structure features[J]. Materials Science and Engineering: A, 2010, 528(1): 402-412. R.T. Liu, H.J. Liu, F. Sha, et al., Investigation of the porosity distribution, permeability, and mechanical performance of pervious concretes, Processes 6 (7) (2018). L.G. Li, J.J. Feng, J. Zhu, et al., Pervious concrete: effects of porosity on permeability and strength, Mag. Concr. Res. 73 (2019) 1-35. Y. Tan, Y.T. Zhu, H.L. Xiao, Evaluation of the hydraulic, physical, and mechanical properties of pervious concrete using iron tailings as coarse aggregates, Appl. Sci. 10 (8) (2020) 2691. Y. Zhang, H. Li, A. Abdelhady, et al., Comparative laboratory measurement of pervious concrete permeability using constant-head and falling-head permeameter methods, Construct. Build. Mater. 263 (2020). S.B. Singh, M. Madasamy, Investigation of aggregate size effects on properties of basalt and carbon fibre-reinforced pervious concrete, Road Mater. Pavement 23 (6) (2022) 1305-1328. M.A.R. Bhutta, K. Tsuruta, J. Mirza, Evaluation of high-performance porous concrete properties, Construct. Build. Mater. 31 (2012) 67-73. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 02 Jul, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 15 Apr, 2025 Reviews received at journal 11 Apr, 2025 Reviews received at journal 06 Apr, 2025 Reviewers agreed at journal 27 Mar, 2025 Reviewers agreed at journal 27 Mar, 2025 Reviewers agreed at journal 27 Mar, 2025 Reviewers invited by journal 27 Mar, 2025 Editor assigned by journal 27 Mar, 2025 Editor invited by journal 26 Mar, 2025 Submission checks completed at journal 26 Mar, 2025 First submitted to journal 17 Mar, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6243923","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":442778824,"identity":"629612e9-58d5-42ab-bdfb-14678c4a4b25","order_by":0,"name":"Fan Yu","email":"","orcid":"","institution":"China Three Gorges University","correspondingAuthor":false,"prefix":"","firstName":"Fan","middleName":"","lastName":"Yu","suffix":""},{"id":442778825,"identity":"98b9d311-a86e-47d1-8f5a-4b443cf8789c","order_by":1,"name":"Wei Chu","email":"","orcid":"","institution":"China Three Gorges University","correspondingAuthor":false,"prefix":"","firstName":"Wei","middleName":"","lastName":"Chu","suffix":""},{"id":442778826,"identity":"91d48185-65b6-4907-af2e-7cb07f89baf3","order_by":2,"name":"Rui Zhang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtElEQVRIiWNgGAWjYDACCQbGB1CmAdFamA0YEkjUwiZBmhb52b3PKn7+qE1sYG/eJsFQc4ewFsY5x81u9iQcT2zgOVYmwXDsGWEtzBJpbLcZEo4lNkjkmEkwNhwmrIUNqKUYrEX+DZFaeIBamBkSaoC28BCpRUIijVmyJ+2AcRtPWrFFwjEitMjPSGP88MOmTraf/fDGGx9qiNACBYcZ2EBUAtEaGBjqSFA7CkbBKBgFIw4AAJHaM2BoG6jjAAAAAElFTkSuQmCC","orcid":"","institution":"China Three Gorges University","correspondingAuthor":true,"prefix":"","firstName":"Rui","middleName":"","lastName":"Zhang","suffix":""},{"id":442778827,"identity":"3e2f1279-cfdb-47a8-a3fb-1164c8e4e278","order_by":3,"name":"Zhang Gao","email":"","orcid":"","institution":"China Three Gorges University","correspondingAuthor":false,"prefix":"","firstName":"Zhang","middleName":"","lastName":"Gao","suffix":""},{"id":442778828,"identity":"81f5e27a-bec0-4313-baf3-b180aed9576d","order_by":4,"name":"Yunan Yang","email":"","orcid":"","institution":"China Three Gorges University","correspondingAuthor":false,"prefix":"","firstName":"Yunan","middleName":"","lastName":"Yang","suffix":""}],"badges":[],"createdAt":"2025-03-17 11:08:16","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6243923/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6243923/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-025-08479-0","type":"published","date":"2025-07-02T15:58:04+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":80626084,"identity":"524a44ca-61aa-4948-9eb3-095f6e84bc84","added_by":"auto","created_at":"2025-04-15 10:47:22","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":1364787,"visible":true,"origin":"","legend":"\u003cp\u003eThe flow chart of this paper.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-6243923/v1/1d7bf245c43e75014ae295b6.png"},{"id":80626086,"identity":"f8893a0d-9e97-46a3-8682-a7ebfacbc1f2","added_by":"auto","created_at":"2025-04-15 10:47:22","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":991668,"visible":true,"origin":"","legend":"\u003cp\u003eGradation design scheme.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6243923/v1/46c3f62852983a3527944e6f.png"},{"id":80625766,"identity":"be762f1f-79d6-4fe0-b3b3-f41945821f32","added_by":"auto","created_at":"2025-04-15 10:39:22","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":199743,"visible":true,"origin":"","legend":"\u003cp\u003eMixup Data Enhancement Logic Schematic\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-6243923/v1/5a45d7dd6449f435db871936.png"},{"id":80626859,"identity":"052e2d67-5d37-40cb-aad2-648add33c984","added_by":"auto","created_at":"2025-04-15 10:55:22","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":1487973,"visible":true,"origin":"","legend":"\u003cp\u003ePearson correlation coefficient analysis results.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-6243923/v1/8e35594623fe3dd86848c146.png"},{"id":80625768,"identity":"3258a007-117b-45fa-af09-09d154877929","added_by":"auto","created_at":"2025-04-15 10:39:22","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":615517,"visible":true,"origin":"","legend":"\u003cp\u003eEffect of the number of input parameters on model accuracy.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-6243923/v1/cfe42200a3bedef37d5e84cb.png"},{"id":80625776,"identity":"c2736b15-0d65-4ee3-8f6c-bbfcc370049f","added_by":"auto","created_at":"2025-04-15 10:39:22","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":461172,"visible":true,"origin":"","legend":"\u003cp\u003ePredictions of permeability made by different models.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-6243923/v1/b8101ee053d68e178107fd58.png"},{"id":80625770,"identity":"a3c194ea-a26a-404a-bb90-c71147bee2da","added_by":"auto","created_at":"2025-04-15 10:39:22","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":449214,"visible":true,"origin":"","legend":"\u003cp\u003ePredictions of compressive strength made by different models.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-6243923/v1/a63bde7ab0251894c5a26f2d.png"},{"id":80626861,"identity":"ed2c306e-334e-4ce1-908a-4db1b43ab636","added_by":"auto","created_at":"2025-04-15 10:55:22","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":391815,"visible":true,"origin":"","legend":"\u003cp\u003eThe results of model predictive stability analysis.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-6243923/v1/aa6e2d8ef9a6d7af20214a3a.png"},{"id":80625778,"identity":"34804726-fdb5-4e16-8554-e87e7e94401a","added_by":"auto","created_at":"2025-04-15 10:39:22","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":179522,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of Stacking models with empirical formulations.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-6243923/v1/21fe44e009f76611c3b050a0.png"},{"id":86179091,"identity":"ad6a2b7d-de16-44ae-911f-6cacd9b3dce0","added_by":"auto","created_at":"2025-07-07 16:15:36","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":7033605,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6243923/v1/532038df-0c8a-4869-8477-2001bdc51987.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Predicting the permeability and compressive strength of pervious concrete using ensemble machine learning model","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eCurrently, rapid urbanization and high-density infrastructure development have caused a variety of ecological and environmental issues. Consequently, the concept of sustainable development is becoming increasingly important and being implemented, including low impact development (LID), green infrastructure (GI), best management practices (BMPs), sponge city (SC), etc. As an environmentally friendly paving material, pervious concrete is gradually being used in the construction of novel cities [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Compared to traditional dense concrete, pervious concrete has a rich pore structure (15%-30%) [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], which makes it play an active role in controlling surface rainwater runoff, improving the hydrological cycle of the city, realizing rainwater collection, infiltration and purification, and noise reduction [\u003cspan additionalcitationids=\"CR6\" citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. Undoubtedly, the performance of pervious concrete is largely determined by its pore characteristics [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. One of the most well-studied parameters is porosity. Generally, the relationship between permeability and porosity is generally positive, whereas the relationship between mechanical properties and porosity is negative [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Other pore features also have been reported to affect the water permeability and compressive strength of pervious concrete [\u003cspan additionalcitationids=\"CR12\" citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. Yu et al. [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] found that water permeability increased with the increase of pore size, and it is more sensitive to the content of small pores. Pore distribution, pore diameter and tortuosity were also observed to have an obvious influence on permeability [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. Liu [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] claimed that a more uniform pore distribution and larger pore spacing could significantly increase compressive strength. Unconfined compression strength was observed to increase with increasing pore surface area in another literature. Additionally, the closer the pore shape is to the sphere, the lower the unconfined compression strength [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. When the porosity is constant, the increase in small pores will increase the compressive strength [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eBased on the analysis of the influence of pore structure characteristics on the performance of pervious concrete, some empirical formulas between the performance of pervious concrete and pore structure parameters were established, which are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. These empirical formulas can accurately predict permeability and compressive strength, but each prediction formula was only applicable to specific conditions. Additionally, it can be observed from Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e that majority of the current prediction formulas only consider the general pore characteristics of porosity. However, the above analysis has shown that the performance of pervious concrete was not only affected by porosity, but also significantly affected by other pore characteristics. Therefore, the performance prediction of pervious concrete considering more pore structure parameters needs to be further studied.\u003c/p\u003e \u003cp\u003eThe pore structure of pervious concrete is very complex, and there are many characterization parameters. It is difficult to establish an accurate performance prediction formula based on pore structure parameters directly through theoretical derivation. Thus, this paper tries to establish a deep learning model for predicting pervious concrete performance using pore structure parameters.\u003c/p\u003e \u003cp\u003eThe object of this paper is to develop a Stacking ensemble model to predict permeability and compressive strength of pervious concrete by utilizing pore characterization parameters as inputs. First, six independent models (DT, BP, CNN, RF, GBDT and XGBoost) were selected as primary learners, the multiple linear regression as secondary learner, and the Stacking algorithm was applied to construct the ensemble model. Second, 90 groups of pervious concrete specimens with varying porosities and grades were prepared. 20 kinds of pore structure characterization parameters and the permeability and compressive strength were extracted to form the initial data set. Then, the initial data set was augmented, and the prediction models were trained. Afterward, the number of input parameters was determined, the stability of ensemble models was verified, and the prediction performance of ensemble models were evaluated. The flow chart of this paper was exhibited in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"2 Deep learning models","content":"\u003cp\u003eThe performance prediction model for pervious concrete was built by Stacking, an effective integrated learning method. This method consists of two layers of learners. The first layer of learners implements several independent models. The second layer of learners integrates the independent models obtained in the previous step to obtain the final prediction results. In view of the different performance of the primary learner, this paper sets different weights according to the output results of each primary learner. In this paper, six independent models were selected in the first layer of the learner, including Decision Tree (DT), Back Propagation (BP) neural network, Convolutional Neural Network (CNN), Random Forest (RF), Gradient Boosting Decision Tree (GBDT), eXtreme Gradient Boosting (XGBoost). Multiple linear regression was used as the secondary learner in the second layer.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 The first layer of the learner\u003c/h2\u003e \u003cp\u003eDecision Tree is a graphical method for the intuitive use of probability analysis. It is based on the known probability of occurrence in various situations [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]. It employs a decision tree to obtain the probability that the expected value of the net present value is greater than zero [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. BP neural network is a multi-layer feedforward neural network trained according to the error back propagation algorithm. It is one of the most widely used neural network models [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. CNN is a kind of feed forward neural networks FNN (Feed forward Neural Networks) with convolution calculation and deep structure [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]. RF is a classifier that uses multiple decision trees to train and predict samples [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. GBDT is an iterative decision tree algorithm [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. The algorithm consists of multiple decision trees, and the conclusions of all trees are summed up to make the final answer [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]. XGBoost is a network that iteratively trains multiple decision tree models and utilizes a gradient descent method to gradually optimize the predictive power of the model [\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e].\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 The second layer of the learner\u003c/h2\u003e \u003cp\u003eRegression analysis provides a simple way to establish the functional relationship between variables, which is one of the most widely used statistical tools. It uses the regression coefficient to explain the relationship between independent variables and dependent variables, and selects the optimal regression coefficient through the gradient descent method [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. In practice, the dependent variable is often affected by two or more independent variables. Therefore, predicting or estimating the dependent variable from the optimal combination of multiple independent variables is more accurate than from only one independent variable. The permeability and compressive strength of pervious concrete are affected by several pore structure parameters, such as porosity, pore size, and pore distribution, etc. Consequently, multivariate linear regression was used as a secondary learner to develop the prediction model between pore structure and macroscopic properties, in this paper.\u003c/p\u003e \u003c/div\u003e"},{"header":"3 Dataset acquisition and model training","content":"\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Initial dataset acquisition\u003c/h2\u003e \u003cp\u003e3 kinds of porosity (15%, 20%, 25%) of 30 different gradations (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) were designed, resulting in 90 groups of schemes. Afterwards, the pervious concrete samples (100mm \u0026times; 100mm \u0026times; 100mm) were prepared, and CT scanning was performed after 28 days of curing, and the permeability coefficient and compressive strength were measured. The 2D / 3D pore structure characteristics of 90 groups of samples were extracted by image processing technology (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). The initial data set required for training the performance prediction model of pervious concrete with pore structure parameters as input was obtained.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThe extracted pore structure parameters\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eProperty Names\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAbbreviations\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMean value of face rate[\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eP\u003csub\u003eave\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMaximum value of face rate[\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eP\u003csub\u003emax\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMean value of shape factor(H/W)[\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSH\u003csub\u003eave\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMaximum value of shape factor(H/W)[\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSH\u003csub\u003emax\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMean value of coordination number[\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCO\u003csub\u003eave\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMaximum value of coordination number[\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCO\u003csub\u003emax\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKurtosis[\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eK\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSkewness[\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSk\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFractal dimension[\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFD\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMean value of surface curvature[\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eC\u003csub\u003eave\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTortuosity[\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eT\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMean free spacing[\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eD\u003csub\u003emfs\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePore area distribution parameters[\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eA\u003csub\u003ep\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePore volume distribution parameters[\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eV\u003csub\u003ep\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMean value of pore throat area[\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eT\u003csub\u003em\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePore throat distribution parameters[\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTs\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSurface porosity of large pore[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLP\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSurface porosity of medium pore[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMP\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSurface porosity of small pores[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSP\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSurface porosity of large and medium pore [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLMP\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ewater permeability\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eQ\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCompressive strength\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCS\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Data set preprocessing\u003c/h2\u003e \u003cp\u003eIn the initial data set, the values of the data varied considerably and were different in dimension. For better comprehensive and comparative evaluation of the data, each category of data needs to be processed to keep them in the same magnitude order. In addition, if the data with different dimension are analyzed directly, the validity and precision will be affected. The maximum standardization method was utilized to standardize the initial data set to accelerate the convergence of the model in this paper.\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{x}_{ij}^{{\\prime\\:}}=\\frac{{x}_{ij}}{{max}\\left\\{{x}_{ij}\\right\\}}\\:\\:\\:\\:\\:\\:\\:\\left(i=\\text{1,2},\\dots\\:,m;\\:j=\\text{1,2},\\dots\\:,n\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Data augmentation based on Mixup method\u003c/h2\u003e \u003cp\u003eThe increase in dataset size can improve the accuracy and robustness of the model. However, datasets are often inconvenient to obtain and costly to increase in dataset size. Fortunately, data augmentation techniques can be employed to effectively expand data set. Mixups combine multiple sets of samples from different data sets linearly to create new samples, expanding the data set (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). For example, two samples x\u003csub\u003e1\u003c/sub\u003e and x\u003csub\u003e2\u003c/sub\u003e, their corresponding labels are y\u003csub\u003e1\u003c/sub\u003e and y\u003csub\u003e2\u003c/sub\u003e, respectively. By summarizing the two samples and the corresponding label data respectively at a set ratio, the new sample and label data are generated afterwards (Eq.\u0026nbsp;2).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{x}^{{\\prime\\:}}\\:=\\:\\lambda\\:{x}_{1}+\\left(1-\\lambda\\:\\right){x}_{2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{y}^{{\\prime\\:}}\\:=\\:\\lambda\\:{y}_{1}+(1-\\lambda\\:){y}_{2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e(2)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere x is the input vector, y is the one-hot (one-hot) encoding of the label, λ is a random\u003c/p\u003e \u003cp\u003enumber between 0 and 1, and λ\u0026thinsp;~\u0026thinsp;Beta(α,α), i.e., λ obeys a Beta distribution with all parameters α, which denotes the weights of x\u003csub\u003e1\u003c/sub\u003e and x\u003csub\u003e2\u003c/sub\u003e in the new sample.\u003c/p\u003e \u003cp\u003eTherefore, the loss function used in this paper can be expressed as Eq.\u0026nbsp;\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:loss=\\:\\lambda\\:\\ast\\:criterion\\:(outputs,\\:a)+(1-\\lambda\\:\\ast\\:criterion\\:(outputs,b)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eMixup was employed to expand the original data set. Ultimately, 1800 sets of data were obtained to form the final data set. Then, the data set was divided into training set and testing set. Subsequently, model training was performed on training set.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Model training and evaluation\u003c/h2\u003e \u003cdiv id=\"Sec10\" class=\"Section3\"\u003e \u003ch2\u003e3.4.1 Model-evaluation index\u003c/h2\u003e \u003cp\u003eThe model was evaluated by the following indexes, including the coefficient of determination (\u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sup\u003e) and the accuracy (Accuracy).\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:{R}^{2}=1-\\frac{\\sum\\:_{i=1}^{n}{({y}_{i}-{\\widehat{y}}_{i})}^{2}}{\\sum\\:_{i=1}^{n}{({y}_{i}-{\\stackrel{-}{y}}_{i})}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:Accuracy=\\frac{1}{n}{\\sum\\:}_{i=1}^{n}\\:\\frac{\\widehat{{y}_{i}}}{{y}_{i}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003ey\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e represents the true value, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\widehat{y}}_{i}\\:\\)\u003c/span\u003e\u003c/span\u003erepresents the predicted value, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\stackrel{-}{y}}_{i}\\:\\)\u003c/span\u003e\u003c/span\u003erepresents the average value of the true value.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section3\"\u003e \u003ch2\u003e3.4.2 Model parameter settings\u003c/h2\u003e \u003cp\u003eThe specific parameter settings and parameter values of the six independent models used to build Stacking ensemble model are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThe parameter settings of the independent model.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eModel\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParameter settings and parameter values\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ehidden_layer_sizes=(100,4), activation='relu', solver='adam', random_state\u0026thinsp;=\u0026thinsp;15, alpha\u0026thinsp;=\u0026thinsp;0.0001, max_iter\u0026thinsp;=\u0026thinsp;500\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDT\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003esplitter='best', criterion=\u0026rsquo;mse\u0026rsquo;, random_state\u0026thinsp;=\u0026thinsp;15, max_depth\u0026thinsp;=\u0026thinsp;5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCNN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eactivation='relu', optimizer='adam', loss='mse', epochs\u0026thinsp;=\u0026thinsp;500, batch_size\u0026thinsp;=\u0026thinsp;10\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRF\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003en_estimators\u0026thinsp;=\u0026thinsp;100, criterion='mse', random_state\u0026thinsp;=\u0026thinsp;15, max_depth\u0026thinsp;=\u0026thinsp;5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGBDT\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003en_estimators\u0026thinsp;=\u0026thinsp;100, learning_rate\u0026thinsp;=\u0026thinsp;0.1, random_state\u0026thinsp;=\u0026thinsp;15\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eXGboost\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003en_estimators\u0026thinsp;=\u0026thinsp;100, learning_rate\u0026thinsp;=\u0026thinsp;0.05, random_state\u0026thinsp;=\u0026thinsp;15, max_depth\u0026thinsp;=\u0026thinsp;3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"4 Results and discussions","content":"\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Determination of model input parameters\u003c/h2\u003e \u003cdiv id=\"Sec14\" class=\"Section3\"\u003e \u003ch2\u003e4.1.1 Type of input parameters\u003c/h2\u003e \u003cp\u003ePore structure characterization parameters such as porosity, pore size, and tortuosity (about 20 parameters have been reported) have been verified to affect water permeability and compressive strength of pervious concrete [\u003cspan additionalcitationids=\"CR42\" citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e]. The objective of this paper is to establish a deep learning-based prediction models for water permeability performance and compressive strength using pore structure parameters as inputs. The prediction accuracy of the model depends on the input parameters. Firstly, the Pearson correlation coefficient method was used to investigate the correlation between the pore structure characterization parameters extracted in this study and the permeability or compressive strength. Afterwards, two groups of 10 pore structure characterization parameters were selected (If the correlation between the next parameter and the previous parameter exceeded 0.8, it was not selected for eliminating the high linear correlation between the parameters.) respectively for model training based on the degree of correlation from high to low.\u003c/p\u003e \u003cp\u003eThe results of Pearson correlation coefficient analysis are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. It can be seen that the descending order of the correlation between pore structure characterization parameters and permeability was: A\u003csub\u003ep\u003c/sub\u003e \u0026gt; SP\u0026thinsp;\u0026gt;\u0026thinsp;D\u003csub\u003emfs\u003c/sub\u003e \u0026gt; C\u003csub\u003eave\u003c/sub\u003e \u0026gt; P\u003csub\u003eave\u003c/sub\u003e \u0026gt; MP\u0026thinsp;\u0026gt;\u0026thinsp;LP\u0026thinsp;\u0026gt;\u0026thinsp;P\u003csub\u003emax\u003c/sub\u003e \u0026gt; T\u003csub\u003es\u003c/sub\u003e \u0026gt; FD\u0026thinsp;\u0026gt;\u0026thinsp;LMP\u0026thinsp;\u0026gt;\u0026thinsp;CO\u003csub\u003eave\u003c/sub\u003e \u0026gt; CO\u003csub\u003emax\u003c/sub\u003e \u0026gt; T \u0026gt; K\u0026thinsp;\u0026gt;\u0026thinsp;V\u003csub\u003ep\u003c/sub\u003e \u0026gt; S\u003csub\u003ek\u003c/sub\u003e \u0026gt; T\u003csub\u003em\u003c/sub\u003e \u0026gt; SH\u003csub\u003eave\u003c/sub\u003e \u0026gt; SH\u003csub\u003emax\u003c/sub\u003e. The descending order of the correlation between pore structure characterization parameters and compressive strength was: P\u003csub\u003eave\u003c/sub\u003e \u0026gt; CO\u003csub\u003emax\u003c/sub\u003e \u0026gt; FD\u0026thinsp;\u0026gt;\u0026thinsp;P\u003csub\u003emax\u003c/sub\u003e \u0026gt; CO\u003csub\u003eave\u003c/sub\u003e \u0026gt; LMP\u0026thinsp;\u0026gt;\u0026thinsp;T\u003csub\u003em\u003c/sub\u003e \u0026gt; T \u0026gt; SP\u0026thinsp;\u0026gt;\u0026thinsp;LP\u0026thinsp;\u0026gt;\u0026thinsp;A\u003csub\u003ep\u003c/sub\u003e \u0026gt; MP\u0026thinsp;\u0026gt;\u0026thinsp;V\u003csub\u003ep\u003c/sub\u003e \u0026gt; K \u0026gt; SH\u003csub\u003eave\u003c/sub\u003e \u0026gt; C\u003csub\u003eave\u003c/sub\u003e \u0026gt; SH\u003csub\u003emax\u003c/sub\u003e \u0026gt; D\u003csub\u003emfs\u003c/sub\u003e \u0026gt; T\u003csub\u003es\u003c/sub\u003e \u0026gt; S\u003csub\u003ek\u003c/sub\u003e. After eliminating the parameters with a correlation of more than 0.8 with the previously selected variables, the parameters used to establish the permeability prediction model were: A\u003csub\u003ep\u003c/sub\u003e、MP、LP、P\u003csub\u003emax\u003c/sub\u003e、TS、CO\u003csub\u003eave\u003c/sub\u003e、T、K、V\u003csub\u003ep\u003c/sub\u003e and T\u003csub\u003em\u003c/sub\u003e. The parameters used to establish the compressive strength prediction model were Pave、CO\u003csub\u003emax\u003c/sub\u003e、T\u003csub\u003em\u003c/sub\u003e、T、A\u003csub\u003ep\u003c/sub\u003e、MP、V\u003csub\u003ep\u003c/sub\u003e、K、SH\u003csub\u003eave\u003c/sub\u003e and SH\u003csub\u003emax\u003c/sub\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section3\"\u003e \u003ch2\u003e4.1.2 Number of input parameters\u003c/h2\u003e \u003cp\u003eModel prediction accuracy depends on input parameters, but it does not necessarily improve with more input parameters [\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e]. However, with the increase of input parameters, the model is more complex and difficult to operate [\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e]. Therefore, the optimal input parameters and quantities for the model need to be determined. According to the correlation degree from high to low (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e (c)), 4,6,8 and 10 pore structure parameters were selected as inputs to train the permeability compressive strength prediction model. The prediction accuracy of the models was evaluated, and the results are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eIt can be seen from Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e that as the number of input parameters increased, the prediction accuracy of the model (permeability and compressive strength) increased slowly. When the input parameters beyond 6, the growth rate was significantly slowed down (except for the BP models). When the input parameter was 6, the accuracy of the permeability prediction model (ensemble models) and the compressive strength prediction model was as high as 0.93. Considering the accuracy and simplicity simultaneously, two groups of 6 parameters were determined respectively as input to predict permeability and compressive strength of pervious concrete finally. Namely, the input parameters for the permeability prediction model were A\u003csub\u003ep\u003c/sub\u003e, MP, LP, P\u003csub\u003emax\u003c/sub\u003e, TS and CO\u003csub\u003eave\u003c/sub\u003e, and the input parameters for the compressive strength prediction model were Pave, CO\u003csub\u003emax\u003c/sub\u003e, T\u003csub\u003em\u003c/sub\u003e, T, A\u003csub\u003ep\u003c/sub\u003e and MP. Additionally, the ensemble model is found to be more accurate than independent models. This indicates that the ensemble model exhibited better performance than the independent models, which will be discussed in detail in the following section.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Comparison between ensemble models and independent models\u003c/h2\u003e \u003cp\u003eTo compare the performance of the ensemble model and the independent models in detail, a test set was randomly selected and predicted by each model. The prediction results are shown Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, models with data points nearer to 45 \u0026deg; are more accurate. It is observed that the prediction results of the Stacking models were concentrated near the 45\u0026deg; line. And a considerable number of data points coincided with the 45\u0026deg; line. Moreover, almost all data points were located in the area surrounded by two baselines (\u0026plusmn;\u0026thinsp;25% or \u0026plusmn;\u0026thinsp;15%). The prediction accuracy of the different independent models varied significantly. A considerable portion of the prediction points of the permeability prediction models (except CNN) were distributed in areas outside the \u0026plusmn;\u0026thinsp;25% baseline. The predicted points of the compressive strength prediction models (except BP) mostly fell within the baseline (\u0026plusmn;\u0026thinsp;15%). However, the prediction points were more dispersed, with a significantly lower rate of prediction points near the 45\u0026deg; line than the Stacking model. Moreover, the Stacking models possessed the largest R\u003csup\u003e2\u003c/sup\u003e (the permeability prediction model and compressive strength prediction model are 0.92 and 0.93, respectively). This indicates that the Stacking models gained higher prediction accuracy than the independent models. This is due to the fact that the second level learner uses both the basic training set and the output from the first level learner. As a result, stacking models can make better predictions by keeping the information from the original training set. For permeability predicting, R\u003csup\u003e2\u003c/sup\u003e of CNN, XGBoost, and GBDT reached 0.90, 0.89, and 0.89, respectively, and for compressive strength predicting, R\u003csup\u003e2\u003c/sup\u003e of XGBoost and GBDT all achieved 0.89. Based on the values of R\u003csup\u003e2\u003c/sup\u003e, CNN, XGBoost, and GBDT models presented a competitive challenge to Stacking models. However, it can be seen in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e that both CNN and GBDT have more discrete data points and even some predictions beyond the baseline compared to the Stacking models.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Stability analysis of the models\u003c/h2\u003e \u003cp\u003eStability is an index that cannot be ignored when evaluating a model. To evaluate the stability of the model, 10 groups of data sets containing 100 sets of data were randomly selected from the test set as new test sets. Afterwards, the stability of the established model was evaluated on the new test sets. The prediction accuracy and the variance of accuracy are displayed in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eAs can be seen in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e (a), the prediction accuracy of the Stacking model (permeability) fluctuates very little on the 10 new test sets, ranging between 0.91 and 0.94. In contrast, the prediction accuracy of the six independent models exhibited obvious fluctuations to varying degrees. In addition, the variance of the prediction accuracy of the Stacking model is 0.0085, which was significantly smaller than that of the independent models. It can be observed from Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e (b) that the prediction accuracy of the Stacking model (compressive strength) also fluctuated little, ranging from 0.92 to 0.94. Obviously, the prediction accuracy of CNN, RF, DT, and BP models fluctuated greatly, and the corresponding variances were significantly greater than the Stacking model. Interestingly, the fluctuation of prediction accuracy of XGBboost and GBDT models was not obvious, and the corresponding variances were close to that of Stacking model. However, the prediction accuracies of XGBboost and GBDT models were lower than 0.9, which is lower than that of the Stacking model of 0.93. Overall, the Stacking models showed the highest prediction accuracy and the smallest variance of prediction accuracy. Namely, the Stacking models had the highest accuracy and stability.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Comparison of Stacking models with empirical formulations\u003c/h2\u003e \u003cp\u003eWidely agreement has been reached on the fact that pore structure plays a dominant role in the permeability and compressive strength of pervious concrete. A variety of empirical formulas for predicting permeability and compressive strength have been established by the researchers, which are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. To compare the prediction accuracy of the Stacking models established in this paper and the empirical formulas, 10 sets of data were randomly selected from the test set. Afterwards, the Stacking models and the empirical formulas were used to predict permeability and compressive strength respectively. The ratio of predicted value to measured value are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eIt can be seen from Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e that the ratio of the predicted value of the Stacking model to the measured value was very close to 1 (both for permeability and compressive strength), ranging from 0.86 to 1.14. However, for most of the empirical formulas, the ratio of predicted value to measured value was distributed outside 0.7\u0026ndash;1.3. Strikingly, a considerable part of the ratio was smaller than 0.5 or larger than 1.5. Intriguingly, when the empirical formula predicted the compressive strength, a considerable part of the ratio of the predicted value to the measured value was distributed between 0.7 and 1.3, and the ratio is rarely smaller than 0.5 or larger than 1.5. Although the empirical formulas established by different researchers were based on significant differences in test conditions (different in coarse aggregate types, cement types, and additives), the accuracy of empirical formulas in predicting compressive strength was significantly higher than that of permeability predicting. Therefore, it can be inferred that although the compressive strength is affected by the raw materials, the porosity plays a vital role in determining the compressive strength. Additionally, although porosity largely determines the water permeability, other pore structure characterization parameters also have a significant impact.\u003c/p\u003e \u003cp\u003eOverall, the accuracy of the Stacking models was significantly better than the empirical formulas for predicting both permeability property and compressive strength. Moreover, compared with the permeability, the compressive strength of pervious concrete is more sensitive to porosity.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eEmpirical formulas for permeability and compressive strength.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colspan=\"3\" nameend=\"c3\" namest=\"c1\"\u003e \u003cp\u003ePermeability\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"3\" nameend=\"c6\" namest=\"c4\"\u003e \u003cp\u003eCompressive strength\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCode\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEmpirical formula\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eReference\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eCode\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eEmpirical formula\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eReference\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ek\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.0286\u003cem\u003eϕ\u003c/em\u003e\u003csub\u003e2t\u003c/sub\u003e 0.721\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e] Kuang et al.,2011\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003eσ\u003c/em\u003e=\u0026minus;1.2863\u003cem\u003eϕ\u003c/em\u003e\u003csub\u003et\u003c/sub\u003e \u0026thinsp;+\u0026thinsp;46.692\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e] Bhutta et al., 2012\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ek\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.93\u003cem\u003ee\u003c/em\u003e0.0755\u003cem\u003eϕ\u003c/em\u003e\u003csub\u003et\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e] Sriravindrarajah et al., 2012\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003eσ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;70.2e\u0026thinsp;\u0026minus;\u0026thinsp;0.066\u003cem\u003eϕ\u003c/em\u003e\u003csub\u003et\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e] Sriravindrarajah et al., 2012\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ek\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.03\u003cem\u003eϕ\u003c/em\u003e\u003csub\u003e2t\u003c/sub\u003e\u0026thinsp;\u0026minus;\u0026thinsp;0.676\u003cem\u003eϕ\u003c/em\u003e\u003csub\u003et\u003c/sub\u003e\u0026thinsp;+\u0026thinsp;4.615\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e] Liu et al., 2018\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eσ=(-581.9(w/c)2\u0026thinsp;+\u0026thinsp;301.8w/c-57.4) ln(\u003cem\u003eϕ\u003c/em\u003e\u003csub\u003et\u003c/sub\u003e)\u0026thinsp;+\u0026thinsp;83.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e] Li et al., 2019\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ek\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2.98e0.06ϕe\u0026thinsp;\u0026minus;\u0026thinsp;3.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e] Li et al., 2019\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eσ\u0026thinsp;=\u0026thinsp;72.9\u0026thinsp;\u0026minus;\u0026thinsp;18.4ln(\u003cem\u003eϕ\u003c/em\u003e\u003csub\u003ee\u003c/sub\u003e )\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e] Tan et al., 2020\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ek\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.57\u003cem\u003ee\u003c/em\u003e0.98\u003cem\u003eϕ\u003c/em\u003e\u003csub\u003ee\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e] Tan et al., 2020\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ek\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.8511\u003cem\u003eϕ\u003c/em\u003e\u003csub\u003et\u003c/sub\u003e\u0026thinsp;\u0026minus;\u0026thinsp;0.864\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e] Zhang et al., 2020\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ek\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.19\u003cem\u003eϕ\u003c/em\u003e\u003csub\u003et\u003c/sub\u003e\u0026thinsp;\u0026minus;\u0026thinsp;12.74\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e] Singh and Madasamy, 2022\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere ϕt denotes total porosity and ϕe denotes effective porosity.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"5 Conclusions","content":"\u003cp\u003eIn this paper, six independent models, the multiple linear regression and the Stacking algorithm were used to construct the ensemble model. 90 groups of pervious concrete specimens with varying porosities and grades were prepared to obtain the initial data set for permeability and compressive strength predicting models training. The models were evaluated and compared with empirical formulations. The main conclusions are as follows:\u003c/p\u003e \u003cp\u003e(1) It was found that 6 input parameters were suitable to make the model gain a high prediction accuracy (0.93) with simplicity simultaneously. The input parameters for the permeability prediction model were A\u003csub\u003ep\u003c/sub\u003e, MP, LP, P\u003csub\u003emax\u003c/sub\u003e, TS and CO\u003csub\u003eave\u003c/sub\u003e, and for the compressive strength prediction model were Pave, CO\u003csub\u003emax\u003c/sub\u003e, T\u003csub\u003em\u003c/sub\u003e, T, A\u003csub\u003ep\u003c/sub\u003e and MP, respectively.\u003c/p\u003e \u003cp\u003e(2) The ensemble models for predicting permeability and compressive strength exhibited higher accuracy than the independent models, the R\u003csup\u003e2\u003c/sup\u003e reached 0.92 and 0.93 respectively. This was due to that the ensemble models both utilized the basic training set and the output of the first level learner.\u003c/p\u003e \u003cp\u003e(3) Although the identification accuracy of XGBboost and GBDT models was high (reached 0.89) on individual data set. However, the stability of its recognition accuracy was lower than that of the ensemble models on different datasets. Therefore, the ensemble models had the highest accuracy and stability.\u003c/p\u003e \u003cp\u003e(4) The ensemble models performed significantly better than empirical formulas in predicting both permeability and compressive strength. In addition, compared with the permeability, the compressive strength of pervious concrete is more sensitive to porosity.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eFan Yu: Conceptualization, Methodology, Writing- Original Draft, Writing - review \u0026amp; editing. Wei Chu: Writing - Original Draft, Writing - review \u0026amp; editing. Rui Zhang: Funding acquisition, Resources, Investigation, Supervision, Writing - review \u0026amp; editing. Zhang Gao: Writing - review \u0026amp; editing. Yunan Yang: Funding acquisition, Writing - review \u0026amp; editing.\u003c/p\u003e\u003ch2\u003eAcknowledgements\u003c/h2\u003e \u003cp\u003eThe authors gratefully acknowledge the financial support provided by Natural Science Research Project of Yichang (A22-3-003), Research Fund for Excellent Dissertation of China Three Gorges University (Grant No. 2021BSPY005). and the Open Fund of Badong National Observation and Research Station of Geohazards (No. BNORSG202313).\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe datasets used and/or analysed during the current study available from the corresponding author on reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eChen X, Yang Y, Zhang C, et al. Valorization of construction waste materials for pavements of sponge cities: A review[J]. 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Pavement 23 (6) (2022) 1305-1328.\u003c/li\u003e\n \u003cli\u003eM.A.R. Bhutta, K. Tsuruta, J. Mirza, Evaluation of high-performance porous concrete properties, Construct. Build. Mater. 31 (2012) 67-73.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"pervious concrete, ensemble machine learning, permeability, compressive strength, prediction","lastPublishedDoi":"10.21203/rs.3.rs-6243923/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6243923/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eDeveloping the relationship of pore characteristics and performance is vital for predicting the properties of pervious concrete. However, the current performance prediction models mainly relied on porosity, ignoring the influence of other pore structure parameters, resulting in insufficient prediction accuracy. The aim of this paper is to establish machine learning-based models for predicting permeability and compressive strength of pervious concrete. Firstly, six independent models, the multiple linear regression and the Stacking algorithm were applied to construct the ensemble model. Secondly, 90 groups of pervious concrete specimens with varying porosities and grades were prepared and tested to obtain the initial data set. Then, the initial data set was augmented, and the prediction models were trained. The results show that 6 input parameters were suitable to make the models gain a high prediction accuracy (0.93) with simplicity. Compared to independent models, the ensemble models showed the highest accuracy and stability. This was due to that the ensemble models both utilized the basic training set and the output of the first level learner. The ensemble models performed significantly better than empirical formulas in predicting both permeability and compressive strength. Compared with the permeability, the compressive strength of pervious concrete is more sensitive to porosity.\u003c/p\u003e","manuscriptTitle":"Predicting the permeability and compressive strength of pervious concrete using ensemble machine learning model","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-04-15 10:39:17","doi":"10.21203/rs.3.rs-6243923/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-04-15T06:08:51+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-04-11T16:41:39+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-04-07T02:05:41+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"97095011867434630358255381579739915248","date":"2025-03-27T15:43:26+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"82791584749277779395758342740432601266","date":"2025-03-27T15:06:12+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"235983667192087520583713392817426713871","date":"2025-03-27T14:50:49+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-03-27T14:46:36+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-03-27T14:43:28+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-03-26T18:16:20+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-03-26T06:43:01+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-03-17T10:55:11+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"e03479ba-d198-413f-b676-3369dda470d7","owner":[],"postedDate":"April 15th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":47138088,"name":"Physical sciences/Engineering"},{"id":47138089,"name":"Physical sciences/Materials science"},{"id":47138090,"name":"Physical sciences/Mathematics and computing"}],"tags":[],"updatedAt":"2025-07-07T16:05:02+00:00","versionOfRecord":{"articleIdentity":"rs-6243923","link":"https://doi.org/10.1038/s41598-025-08479-0","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-07-02 15:58:04","publishedOnDateReadable":"July 2nd, 2025"},"versionCreatedAt":"2025-04-15 10:39:17","video":"","vorDoi":"10.1038/s41598-025-08479-0","vorDoiUrl":"https://doi.org/10.1038/s41598-025-08479-0","workflowStages":[]},"version":"v1","identity":"rs-6243923","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6243923","identity":"rs-6243923","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}