Machine learning prediction of concrete frost resistance and optimization design of mix proportions

Abstract

This study explores nine machine learning (ML) methods, including linear, non-linear and ensemble learning models, using nine concrete parameters as characteristic variables. Including the dosage of cement (C), fly ash (FA), Ground granulated blast furnace slag (GGBS), coarse aggregate (G), fine aggregate (S), water reducing agent (WRA) and water (W), initial gas content (GC) and number of freeze-thaw cycles (NFTC), To predict relative dynamic elastic modulus (RDEM) and mass loss rate (MLR). Based on the linear correlation analysis and the evaluation of four performance indicators of R², MSE, MAE and RMSE, it is found that the nonlinear model has better performance. In the prediction of RDEM, the integrated learning GBDT model has the best prediction ability. The evaluation indexes were R² = 0.78, MSE = 0.0041, MAE = 0.0345, RMSE = 0.0157, SI = 0.0177, BIAS = 0.0294. In the prediction of MLR, ensemble learning Catboost algorithm model has the best prediction ability, and the evaluation indexes are R² = 0.84, MSE = 0.0036, RMSE = 0.0597, MAE = 0.0312, SI = 5.5298, BIAS = 0.1772. Then, Monte Carlo fine-tuning method is used to optimize the concrete mix ratio, so as to obtain the best mix ratio.

Keywords

Machine learning relative dynamic elastic modulus mass loss rate sensitivity analysis optimization design of mix proportions

1 Introduction

Freeze-thaw damage of concrete structures has been a key problem affecting the normal operation and service of projects in cold regions [1]. Concrete is the preferred construction material for engineering projects because of its simple preparation, low cost and excellent durability. However, most studies on the antifreeze properties of concrete require a large number of indoor and outdoor experiments [2 –4]. With the development of computer vision technology and artificial intelligence algorithm and the extension of application in civil engineering, the use of computer intelligence to identify and analyses microcrack information in concrete is attracting increasing attention. Therefore, use of ML can help predict the antifreeze performance of concrete and optimise the combination ratio of antifreeze concrete [2, 3].

The evaluation indexes of concrete’s antifreeze performance include RDEM, MLR and compressive strength loss rate. Most studies were based on experimental data to evaluate the effect of mixing ratio design [5 –8] and adding new materials [1 , 9] on concrete’s antifreeze performance. Han et al. [5] investigated the effect of adding iron tailing powder and fly ash on the antifreeze performance of concrete and found that the antifreeze performance of concrete significantly improves with the addition of iron tailing powder. Minsoo Kim et al. [5] studied the antifreeze properties of superabsorbent polymer concrete under freezing conditions and proposed the optimum amount of superabsorbent polymer to improve its antifreeze properties. Wang et al. [6] compared the effects of fine PS and FA admixture on the antifreeze, pore structure and fractal characteristics of hydraulic concrete, and they found that the compressive strength and antifreeze performance of fine PS concrete outperform that of FA concrete at 28 and 180 days at the same level of admixture. Tan et al. [7] investigated the effect of polypropylene fibre on the antifreeze performance of natural sand and mechanised sand concrete and found that 1.2% of the fibre volume is optimal in natural sand samples and 1.0% of the fibre volume in mechanised sand samples. Liu et al. [1] used the response surface method to study the effects of air attractant, defoaming agent and viscosity improver on the antifreeze properties of concrete, and they found that air attractant and defoaming agent are important factors affecting RDEM. The interaction between defoaming agent and viscosity improver has a significant effect on RDEM. Ding et al. [8] explored the effects of different proportions of fly ash substitution on the antifreeze structural properties of concrete under conditions of air exposure and accelerated carbonisation. On the basis of this experiment, the effect of adding air primer on the freeze-thaw durability of concrete was analysed. For FA concrete mixed with an aerator, the effect on its antifreeze performance can be regarded as insignificant due to the presence of air bubbles, fly ash and carbonisation. Ding et al. [9] also investigated the influence of carbonisation on the antifreeze and pore structure of concrete under different blast furnace slag displacement rates and found that the antifreeze resistance of A study on the change in frost resistance and pore structure of concrete containing blast furnace slag concrete decreases with the increase in replacement rates. Yu et al. [10] used the response surface method to study the effects of air attractant, defoaming agent and viscosity improver on the antifreeze properties of concrete. The interaction between defoaming agent and viscosity improver has a significant effect on the RDEM. Yuan et al. [11] compared ordinary concrete with phase change concrete, and they found that phase change concrete shows lower loss of mass, relative loss of elastic modulus and loss of strength than ordinary concrete. These findings provide strong support for the change in grouting ball admixture.

In recent decades, ML algorithms have been widely used in a wide range of industries and become an indispensable and important tool [12]. ML can effectively address the safety, operational and maintenance challenges of concrete buildings [13 –15]. Bhanu P. Koya et al. [16] used the ML model to predict six mechanical properties of concrete, namely, rupture modulus, compressive strength, elastic modulus, Poisson’s ratio, splitting tensile strength and coefficient of thermal expansion. Hamza Imran et al. [17] built an ML model to predict the compressive strength of environmentally friendly concrete mixtures. Zaher Mundher Yaseen et al. [18] studied the shear strength mechanism of FRP-reinforced concrete members, and the shear tests of FRP-reinforced concrete beams were performed. Kaffayatullah Khan et al. [19] established an ML model to investigate the effects of different input parameters on the properties of nano-silica-modified hybrid fibre-reinforced concrete. N Sharma et al. [20] used ML models to predict the compressive strength of concrete and to test the effectiveness of its composition. Surya Abisek Rajakarunakaran et al. [21] explored a regression model based on ML to predict the compressive strength of self-compacted concrete. Wang and Tak-Ming Chan [22] co-constructed an ML model to predict the ultimate strength of concrete tube slabs under eccentric loads. Using a beetle antenna search algorithm, Zhu et al. [23] developed an artificial intelligence-based cement slag concrete design method by adjusting the superparameters of random forests (RFs), decision trees (DTs), and support vector machines (SVMs). HaiVan Thi Mai et al. [24] developed and evaluated ML models to predict the compressive strength of fibre-reinforced self-compacted concrete and select optimal models from a variety of ML models. Using ML models trained on laboratory data, M.A. DeRousseau et al. [25] predicted the effectiveness of the compressive strength of concrete placed on site. Zhang et al. [26] used ML technology to predict the long-term prestressed loss of concrete cylindrical structures and established and corrected the finite element analysis model. Fadi Almohammed and Jatin Soni [4] constructed two ML models, stochastic forest and DT, to predict the tensile strength of basalt fiber concrete cracking subsequently, they selected the integrated algorithm model stochastic forest as the optimal model. Granata et al. [27] established a superimposed model of multi-layer perceptron (MLP), random forest (RF) and support vector regression (SVR), and applied it to the estimation of daily volumetric water content of soil. The results showed that compared with a single model, the superimposed model had the best prediction effect (R² = 0.962), and could keep fewer parameters. Advantages of short computation time. Shahcheraghi et al. [28] established a simple, fast and accurate optimal weight recovery model based on multiple adaptive regression spline (MARS) technology. The predicted results of the model were R² = 0.978 and RMSE = 2.538, which showed excellent prediction ability and could be used for rapid decision-making before, during and after mining operations. In the study of concrete durability, various ML algorithms were established to predict the chloride diffusion coefficient in concrete, and the optimal prediction model was selected by evaluating the model [29 –31]. Qiao et al. [32] established an interpretable ML model to predict freeze-thaw damage indexes of fibre-reinforced concrete by using a variety of classical models and integrated models. Xgboost has the best predictive ability, where R² = 0.965, RMSE = 0.019, and Xgboost has the best predictive ability. Practical methods for predicting and interpreting freeze-thaw damage of fiber reinforced concrete. Li et al. [33] Based on the multi-scale numerical simulation method of finite element model, combined with the machine learning model, an interpretable neural network model was established to provide a reliable and efficient continuous mapping between the design parameter space and the generated NFTC. Zhang et al. [34] adopted Bayesian optimity-random forest method to establish a three-stage frost resistance prediction model. The results show that the prediction ability of this model is superior to other algorithms, and it has better robustness and can accurately and quickly predict concrete frost resistance. Atasham ul haq et al. [35] established three machine learning methods, namely artificial neural network, random forest and support vector machine, and analyzed the influence of freeze-thaw cycle on concrete structure to quantify the progressive loss of compressive strength after freeze-thaw cycle. The results showed that the artificial neural network model could well predict the freeze-thaw resistance of concrete. R² = 0.924, higher than other models. Gao et al. [2] used a variety of ML models, such as RF, to predict the relative dynamic elastic modulus and mass loss of the RCC, whereas Chen et al. [36] used NSGA-II to optimise the stochastic forest model to fit better than 0.95. Interferences in the prediction process, such as noise, also affect the model performance. To address this issue, Wu et al. [3] developed a stochastic forest algorithm, while combined recursive feature elimination methods to eliminate coupling factors and noise to determine the optimal fit ratio design.

The previous studies used ML models to predict concrete mechanical properties such as compressive strength, tensile strength and ultimate strength of concrete members under eccentric loads. ML models have also been used to predict the chloride diffusion coefficient, an indicator of concrete durability, which provides insights into concrete durability. However, relatively little academic research has been performed on the use of ML models to predict the RDEM and MLR of concrete, and freeze-thaw damage remains the most important problem for building structures in cold regions. Therefore, this paper conducted in-depth research and analysis on the basis of predecessors, aiming to find out the frost resistance index of concrete in high cold environment and the corresponding prediction method, and introduced Monte Carlo fine-tuning method, combined with the optimal machine learning algorithm, took the concrete frost resistance and one-way cost as the objective function, and further optimized the design on the basis of the benchmark mix ratio. A set of optimal mix ratio is obtained, and the research results are applied to practical engineering design to solve engineering problems. The purpose of this study was to explore an economical, efficient, and practical way to study the antifreeze properties of concrete. A total of 7,088 datasets were collected with C, FA, GGBS, G, S, WRA, GC, W and NFTC as characteristic variables. ML models were developed using the concrete antifreeze indices RDEM and MLR as predictive labels for predicting the antifreeze properties of concrete, including RDEM and MLR, of which RDEM was not less than 60% and MLR was not more than 5%. The optimal model was selected by model evaluation index R², MSE, MAE and RMSE. The optimal match ratio was designed by the Monte Carlo fine-tuning method.

2 ML approach

In this study, we investigated 12 ML methods, including linear, nonlinear and integrated learning algorithms. Multivariate linear regression, quadratic polynomial regression (QPR) and locally weighted least square regression (LWLSR) were linear models. Nonlinear models included support vector machine regression (SVR), artificial neural network (ANN), decision tree (DT), random forest (RF), gradient boosting decision tree (GBDT), xgboost, adaboost and catboost. RF, GBDT, xgboost, adaboost and catboost also fell under the umbrella of integrated learning. below is a brief introduction to the 12 ML methods.

2.1 Multivariate linear regression

Multivariate linear regression establishes linear relationships between different variables, and the general form of the MLR model is given by the following Equation (1) [12, 25]: $\hat{Y} = \sum_{j = 1}^{m} a_{j} X_{j}$ (1) where $\hat{Y}$ is an output variable, X_js refers to independent input variables, α₀, α₁, α ; ₂... α_m is a regression coefficient and the multivariate linear regression algorithm is used to predict concrete performance [16, 36].

2.2 Quadratic polynomial regression

In the process of data analysis, if the data point is a curve, the linear fitting effect is poor, so the polynomial regression data points can be used for fitting. In this study, the second order polynomial data points are used for fitting, but when the factor y had a nonlinear relationship with the independent variable x, the linear relation could not satisfy the fitting state, so a suitable curve fitting cannot be found, such as Equation (2) [17]: $\hat{y} = w_{0} + w_{1} x + w_{2} x^{2} + w_{3} x^{3} + \dots + w_{s} x^{s}$ (2) where $\hat{y}$ is the output of the model; w₀, w₁, w₂... w_s is the regression coefficient and x is the input variable. The measured point is approximated by increasing the high order of x until the fitting result is satisfactory.

2.3 Locally weighted least square regression

By assigning a certain weight to each point near the point to be measured, that is, adding a kernel function matrix W, the final target function can be minimised, such as in Equation (3):

Target function: $\sum_{i} w_{i} {(y_{i} - {\hat{y}}_{i})}^{2}$ (3) weight matrix: $W = (\begin{matrix} ω_{1} & \dots & 0 & 0 \\ 0 & ω_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & ω_{n} \end{matrix})$ (4) kernel function (Gaussian kernel function): $ω (i, i) = exp (- {(x_{i} - x)}^{2} / 2 k^{2})$ (5) where x_i represents the first sample point, x is the prediction point and k is a key parameter.

As shown in Equation (5), as k approaches infinity, the weight of each sample point approaches 1. We first determined an initial value k.

2.4 Decision tree

The DT uses a tree structure model to make decisions by decomposing data and asking a series of questions, which are inferred from the characteristic variables in the training set. The DT is constructed by recursive partitioning from root nodes. The root node is divided into two nodes, which are left and right child nodes, and so on, until there is no more division. By using the concept of “information gain”, the nodes are divided so that the nodes with the maximum value of the function are segmented, as expressed in Equation (6) [16 , 37]. $IGf = IP - (\frac{N_{l}}{N_{p}} \times I_{l} + \frac{N_{r}}{N_{p}} \times I_{r})$ (6) where IGf is the information gain of the segmented features; IP is the information entropy of the parent node; I_l and I_r are the information entropy of the left and right nodes, respectively; N_P is the total sample number of the parent node; and N_l and N_r are the total sample number of the child node.

2.5 Random forest

In the framework of integrated learning, a stochastic forest is one in which multiple DTs are integrated into an unrelated forest and merged to predict the final outcome. On the basis of an extended variant of bagging idea, RF introduces a stochastic attribute selection mechanism to optimise the training process of the DT based on the DT-based learner. In the process of feature selection, a random method is used to divide each node. The error of each DT under different circumstances is compared to find the optimal solution. Finally, it is applied to node splitting [4 , 39]. Figure 1 shows schematic of RF.

Fig. 1

Structure of random forest.

2.6 Catboost algorithm

Catboost is a GBDT framework based on symmetric DT-based learner that supports category variables and handles category features efficiently and rationally. It is composed of categorical and boosting. In addition, by solving gradient deviation and predictive offset, the overfitting rate is reduced, and the accuracy and generalization of the algorithm are improved [40, 41].

2.7 Gradient boosting decision tree

Gradient lifting tree belongs to the boosting algorithm and improves the algorithm. At the beginning of the algorithm, we assigned equal weight value to each sample. By increasing the weight of misclassified points, we reduced the weight of correct classified points. After waiting for N iteration, we obtained the N simple base learner, as expressed in Equation (7). $f (x) = \sum_{m = 1}^{M} β_{m} b (x; γ_{m})$ (7) where f (x) is the model of strong learner; b (x ; γ_m) is a weak learning model; β_m is the weight factor of the weak learning model; and γ_m is the parameter of the weak learning model.

2.8 Xgboost algorithm

The XGboost algorithm belongs to the boosting integrated algorithm and the lifting tree model. A DT-based learner is adopted. The CART regression tree belongs to the binary tree type, in which each node only judges whether it is a binary tree or not and continues to divide the properties in the sample set by top-down splitting. (8) Weak learner integration is completed by continuously adding the CART tree and continuously splitting features, with the target function expressed as Equation (8) [42]: $L^{(k)} = \sum_{i = 1}^{n} l (y_{i}, y_{i}^{(k - 1)} + f_{k} (x_{i})) + Ω (f_{k})$ (8) where L^(k) is the target function, $\sum_{i = 1}^{n} l (y_{i}, y_{i}^{(k - 1)} + f_{k} (x_{i}))$ is the loss function, of which $l (y_{i}, y_{i}^{(k - 1)} + f_{k} (x_{i}))$ is the loss function of the k-1 iteration, $y_{i}^{(k - 1)}$ is the prediction value of the i-sample after the k-1 iteration, f_k (x_i) is the prediction of the k-tree for the i-sample and Ω (f_k) is the norm.

2.9 Adaboost algorithm

Adaboost uses the idea of forward step algorithm in regression problem, and base function uses regression tree T (x). Based on DT, loss function uses squared error loss L (y, f (x)) = (y - f (x)) ². According to the idea of forward step algorithm, our current addition model of m-round is f_m (x) = f_m-1 (x) + T_m (x). Assuming we pass through m-1 round, then our optimisation goal for m-round is expressed as follows Equation (9): $arg min T_{m} \sum_{i = 1}^{N} {(y_{i}, f_{m - 1} (x_{i}) + T_{m} (x_{i}))}^{2}$ (9) where the optimisation goal of m-wheel fits the residual of the current model f_m-1 (x).

2.10 Lightgbm algorithm

Lightgbm is an algorithm that uses a gradient fitting method based on the residuals of the previous weak learner to train the subsequent weak learner instead of training the subsequent weak learner on a weighted sample. Sam-pling method based on gradient was used to select samples with high gradient and exclude samples with low gradient during training. At the same time, Lightgbm promotes tree growth by segmenting data at the leaf nodes with the highest information gain, rather than at the same time at the same layer. Blade intelligence algorithms tend to select leaves that show the greatest variation in loss during growth for optimization [43]. Figure 2 shows the calculation structure of the Lightgbm algorithm.

Fig. 2

Lightgbm algorithm structure diagram [38].

2.11 Support vector machine regression

SVM regression is a method of mapping nonlinear regression in sample space to linear regression in high dimension using the kernel function. The purpose of SVR is to find an optimal classified superplane to minimise the error between all training samples and that optimal superplane. To obtain a model that can fit as closely as possible the training set sample, we determined the loss function by constructing a sample label and the loss function of the model predictor to minimize the loss function [22]. Figure 3 shows the calculation structure of SVR.

Fig. 3

Structure diagram of SVM algorithm.

2.12 Multilayer Perceptron

ANN consists of three layers: input layer, hidden layer, and output layer. The simplest ANN is a three-layer structure with only one hidden layer, and the layer is fully connected to the layer. ANN parameters include connection weights (w) and bias (b) between layers. Parameter optimisation by gradient descent: all parameters are randomly initialised, followed by iterative training, in which gradient and update parameters are continuously calculated until a condition is reached (e.g. errors and iterations) [44]. Figure 4 shows the computational structure of the ANN.

Fig. 4

Structure of artificial neural network algorithm.

3 Holistic approach

Figure 5 shows the overall approach of this study, which consists of four main steps: data collection and pre-processing, data visualisation and statistics, ML modelling and performance assessment, and matching ratio optimisation design. The model was built and analysed in Anaconda’s own Python programming environment, Jupyter Notebook, for data analysis and ML programming environments.

Fig. 5

Technology roadmap.

3.1 Data processing and analysis

3.1.1 Data collection and integration

The research data is stored in two databases: Database A contains 1215 groups of concrete mix ratio data, which were collected and sorted by the research group through traditional concrete freeze-thaw resistance experiments. Database B contains 5875 groups of concrete mix ratio data, extracted and collated from published literature. In the traditional experiments, RDEM and MLR are employed to assess potential damage to the concrete structure. When using machine learning algorithms for predicting freezing resistance of concrete, it is essential to consider the influence of various parameters such as C, FA, GGBS, S, G, GC, W and NFTC on its freezing resistance. This evaluation can be conducted using both RDEM and MLR methods. The minimum acceptable value for RDEM is 60%, while MLR should not exceed 5%. These criteria determine whether the concrete has been damaged and if it can no longer remain in service.

3.1.2 Data pre-processing

The information in database A and database B was integrated and nine input variables (including C, FA, GGBS, S, G, WRA, GC, W and NFTC) and two output variables (including MLR and RDEM) were extracted. The integrated data were stored in Excel tables, totalling 7,088 data sets, with all concrete components in the ingredients meeting international standards.

To facilitate robust model learning, we replaced the edited data by programming scripts, cleaned missing values and outliers and normalised the remaining data. Two different methods can be used to normalise data: one is to operate independently before the model is trained, and the nother is to operate during model training [13]. In this study, we used the function pre-processing in Python packet sklearn to convert data before model training, combined with MinMaxScaler () method, to optimise the training effect, such as Equation (10) [45]: $x^{'} = \frac{(x - {min}_{n}) ({max}_{n} - {min}_{n})}{{max}_{n} - {min}_{n}} + min_{n}$ (10) where x′ is the normalised value; x is the original value, $min_{n}$ is the sample minimum, $max_{n}$ is the sample maximum.

3.1.3 Data visualisation and statistics

By submitting the processed data to Python for data visualisation and statistical analysis, we can intuitively summarise and understand the distribution of the aggregated data, which are presented in Tables 1 and 2, showing the maximum, minimum, mean, median and standard deviation of the nine input variables and two output variables for the refrigeration evaluation index (RDEM and MLR). Figure 7(a) and 7(b) visualise the correlation between the nine input variables and RDEM and MLR, respectively, using a polynomial regression algorithm. Figure 8(a) and 8(b) visualise the correlation between the nine input variables and RDEM and MLR, respectively, using a locally weighted least square algorithm. Compared with the polynomial regression algorithm, the local weighted least square algorithm can fit the nonlinear relationship between input and output variables more accurately. At the same time, the optimal value range of input variables can be deduced by combining the scattering distribution and smoothing curve. By using the scattering distribution and the smooth fitting line between the nine input variables and the two output variables, the local weighted least square algorithm is used to achieve the smooth fitting line, thereby forming a bivariate graph. Each subgraph presents the scatter distribution and correlation of a single input and output variable, including linear, nonlinear and monotonous relationships.

Table 1
Statistical analysis of data variables of RDEM

Data variable Unit Statistic measures

Minimum Maximum Mean Median Standard Deviation

C kg/m³ 61 600 315 312 90

FA kg/m³ 0 266 55 61 49

GGBS kg/m³ 0 225 21 0 39

S kg/m³ 302 994 708 716 99

G kg/m³ 609 1703 1144 1116 151

WRA kg/m³ 0 17 3 2 3

GC % 0 9 4 5 2

W kg/m³ 70 240 154 155 26

NFTC time 0 400 96 75 92

RDEM % 40 113 89 92 12

Data variable	Unit	Statistic measures
C	kg/m³	61	600	315	312	90
FA	kg/m³	0	266	55	61	49
GGBS	kg/m³	0	225	21	0	39
S	kg/m³	302	994	708	716	99
G	kg/m³	609	1703	1144	1116	151
WRA	kg/m³	0	17	3	2	3
GC	%	0	9	4	5	2
W	kg/m³	70	240	154	155	26
NFTC	time	0	400	96	75	92
RDEM	%	40	113	89	92	12

Table 2

Statistical analysis of data variables of MLR

Data variable	Unit	Statistic measures
		Minimum	Maximum	Mean	Median	Standard Deviation
C	kg/m³	61	581	314	312	90
FA	kg/m³	0	266	54	61	48
GGBS	kg/m³	0	225	21	0	38
S	kg/m³	296	1079	709	720	101
G	kg/m³	609	1703	1142	1113	150
WRA	kg/m³	0	17	3	3	3
GC	%	0	9	4	5	2
W	kg/m³	70	240	153	154	26
NFTC	time	0	500	96	75	94
MLR	%	–2	8	1	0	1

Fig. 6

Data visualisation and analysis of RDEM and MLR (a. RDEM; b. MLR).

Fig. 7

Visualisation of datasets of RDEM and MLR using QPR (a. RDEM; b. MLR).

Fig. 8

Visualisation analysis of RDEM and MLR by using LWLSA (a. RDEM; b. MLR).

The correlation between input and output variables was estimated using the Pearson correlation coefficient, which can describe linear or monotonous relationship between two variables. Figure 6(a) and 6(b) show the intensity of linear correlation. The colour of the graph corresponds to the correlation coefficient, and the positive and negative symbols represent the positive and negative correlations, respectively. A consistent trend of positive correlation was found between two variables, that is, one variable had the same trend as the other, and vice versa. The opposite trend was noted exists between two variables that showed a negative correlation [46].

By analysing the visualisation of two types of bivariate graph and correlation diagram, the model can be predictive and the interaction between the two variables can be modelled, which can provide a reliable basis for the subsequent modelling process or model selection. According to the linear correlation diagram, the distribution of input variables was highly nonlinear, which provided a reliable basis for the selection of nonlinear models, and showed that the prediction performance of this model was obviously better than that of linear models. Visualization of Pearson correlation coefficients between RDEM and MLR correlation variables is presented in Fig. 6(a) and 6(b). The absolute values of the correlation coefficients between input and output variables were between 0 and 0.1, presenting a weak linear correlation. Thus, input and output variables were distributed according to nonlinear law and further validated by using the MLR algorithm of the linear model. Figure 9(a) and 9(b) demonstrate the visualisation of RDEM and MLR correlation coefficients, respectively, which predicted the accuracy of the RDEM and MLR, and the output value was 0.09.

Fig. 9

Prediction of RDEM and MLR by multiple linear regression (a. RDEM; b. MLR).

4 ML modelling and analysis

In a previous section, the data were visualised and statistically analysed and found to be characterised by a nonlinear distribution. Thus, the prediction performance of the nonlinear model was better than that of the linear model, and the validity of this inference was confirmed by the linear model multiple linear regression algorithm. Below, 9 nonlinear ML models was established to predict the correlation datasets of RDEM and MLR, the antifreeze indices of concrete.

4.1 Data segmentation

To assess the generalisability of the model and effective predictions in the sklearn module, the study used a stochastic approach to slice the dataset using the train test split () method [47]. The data were divided into training sets, which were used to train ML models, and test sets, which were used to assess model performance. 50/50, 70/30, 90/10, and other compartmentalisation methods are widely used when dividing data [48]. Proper disaggregation of the training set (test set) data will directly affect the evaluation of the model. If the training set is divided at a small capacity, the data of the training model will not be sufficient or distributed evenly, and the cleanliness requirements (not too much noise) will not be met to train a complete training model. Similarly, the small size of the partitioned test set also requires a balanced distribution and cleanliness, without which a complete test model cannot be predicted [49]. In this study, nine mix design parameters were used, which were C, W, FA, GGBS, S, G WRA, GC and NFTC. The total sample size of the concrete anti-freeze index RDEM data was 7,062, divided by 85/15 ratio, using 6,002 and 1,059 as training and test sets, respectively. In addition, the total sample size of the concrete antifreeze indicator MLR was 6,729 groups, 5,719 and 1,009 were used as training and test set data, respectively.

4.2 Model training

For ML models, each data set is trained using the fit () method, and the ML model programming script is provided in the appendix. K-fold cross-validation (10-fold) was used to train the mode, and the training of the model was evaluated to avoid overfitting [50]. In addition, the superparameters were adjusted by K-fold cross-validation to improve the model performance. By calling the python package and adjusting the hyperparameters, the training model of each ML model was optimised for optimal training. RF, adaboost, xgboost, GBDT and catboost algorithms all employ DT-based integration algorithms by modulating key superparameters such as n_estimators and max_depth, ANNs by regulating neuron counts and layers of hidden layers, and SVM by modulating superparametric training models such as C (penalty term coefficient), epsilon, and gamma (coefficient of nuclear function). Table 3 shows the combination of function packages and hyperparameters needed for training and prediction.

Table 3
ML methods and Python algorithm structure

ML methods Packages Train (“method”) Tuning hyperparameters

MLR sklearn LinearRegression ——

QPR sklearn PolynomialFeatures ——

LWLSA sklearn lrlw ——

Adaboost sklearn AdaBoostRegressor max_depth, n_estimators

Xgboost sklearn XGBRegressor max_depth,learning_rate,n_estimators, objective

RF sklearn RandomForestRegressor n_estimators, max_depth

DT sklearn DecisionTreeRegressor max_depth,splitter

MLP sklearn MLPRegressor hidden_layer_sizes, max_iter

SVR sklearn SVR C,epsilon,gamma

GBDT sklearn GradientBoostingRegressor loss,random_state,max_features,min_samples_leaf,learning_rate

Catboost catboost CatBoostRegressor iterations, learning_rate, depth, loss_function

Lightgbm lightgbm LGBMRegressor objective, num_leaves,learning_rate,n_estimators

ML methods	Packages	Train (“method”)	Tuning hyperparameters
MLR	sklearn	LinearRegression	——
QPR	sklearn	PolynomialFeatures	——
LWLSA	sklearn	lrlw	——
Adaboost	sklearn	AdaBoostRegressor	max_depth, n_estimators
Xgboost	sklearn	XGBRegressor	max_depth,learning_rate,n_estimators, objective
RF	sklearn	RandomForestRegressor	n_estimators, max_depth
DT	sklearn	DecisionTreeRegressor	max_depth,splitter
MLP	sklearn	MLPRegressor	hidden_layer_sizes, max_iter
SVR	sklearn	SVR	C,epsilon,gamma
GBDT	sklearn	GradientBoostingRegressor	loss,random_state,max_features,min_samples_leaf,learning_rate
Catboost	catboost	CatBoostRegressor	iterations, learning_rate, depth, loss_function
Lightgbm	lightgbm	LGBMRegressor	objective, num_leaves,learning_rate,n_estimators

4.3 Model testing

After the model was trained, the model was tested with an undisturbed test set, and the predictive performance of the model was evaluated by comparing the predictive and true values using predict () function output model results.

5 Model performance assessment

5.1 Performance indicators

To evaluate the predictive performance of ML models, functions such as mean square error (MSE), mean absolute error (MAE), mean absolute error (RMSE), fit optimization (R²), scatter index (SI) and deviation value (BIAS) were used to evaluate the predictive performance of models such as Equations (11)–(16) [51 –54]: $R^{2} = 1 - \frac{RSS}{TSS} = 1 - \frac{\sum_{i = 1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}}{\sum_{i = 1}^{n} (y_{i} - y)^{2}}$ (11) $RMSE (%) = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {({\hat{y}}_{i} - y_{i})}^{2}}$ (12) $MAE (%) = \frac{1}{n} \sum_{i = 1}^{n} | {\hat{y}}_{i} - y_{i} |$ (13) $MSE (%) = \frac{SSE}{n} = \frac{1}{n} \sum_{i = 1}^{n} w_{i} {(y_{i} - {\hat{y}}_{i})}^{2}$ (14) $BIAS = \frac{1}{n} \sum_{n}^{i = 1} ({\hat{y}}_{i} - y_{i})$ (15) $S I = \frac{\sqrt{(1 / n) \sum_{n}^{i = 1} {(({\hat{y}}_{i} - \bar{\hat{y}}) - (y_{i} - \bar{y}))}^{2}}}{(1 / n) \sum_{n}^{i = 1} y_{i}}$ (16) where y_i and y represent observations and mean observations, respectively, ${\hat{y}}_{i}$ is prediction; RSS and TSS are residual square sum and total square sum, is the mean of the predicted values, $\bar{y}$ is the mean of the actual observations. In this study, the Python packet sklearn. metrics were invoked using the mean absolute error (), mean squared error (), mean absolute error () and r2 _ score () function.

5.2 Training performance

Figures 10(a) and 10(b) show a box diagram of the training performance of each ML model on a dataset of RDEM and MLR, including model evaluation indicators R², MAE and RMSE. In non-linear models, box diagrams are trained datasets by k-fold cross-validation. By contrast, whereas in this study, 10-fold cross-validation was performed by cross_val_score () method. The training performance of the integrated learning model (Catboost, GBDT, RF, Xgboost, Lightgbm and Adaboost) exceeded that of conventional nonlinear models (MLP, DT and SVM) [55].

Fig. 10

Box diagram of the training performance of 9 ML models on a dataset of RDEM and MLR (a. RDEM; b. MLR).

5.3 Test performance

The ML model is used to test the prediction of the RDEM and MLR on the dataset. The better the prediction performance of the ML model, the smaller the difference between the prediction and the measured value [56], that is, the points with the measured value as the horizontal coordinate, the predicted point as the longitudinal coordinate is closely distributed near the 45 ° line through the origin, and the residual value is symmetrically distributed near the 0 ° horizontal line.

5.3.1 ML model predicts RDEM

Table 4 shows the predicted performance statistics of nine ML models for the RDEM of the concrete antifreeze index. Figure 11 shows the predicted maps with measured values as transverse coordinates and predicted values as longitudinal coordinates, Fig. 12 shows the residual maps with predicted values as transverse coordinates and residual values as longitudinal coordinate

Table 4
Statistical results of RDEM prediction performance

ML methods Predictive performance evaluation index

R² MSE MAE RMSE SI BIAS

DT 0.6200 0.0113 0.0533 0.0179 0.0202 0.0074

RF 0.7600 0.0074 0.0470 0.0212 0.0239 1.0713

GBDT 0.7800 0.0041 0.0345 0.0157 0.0177 0.0294

Adaboost 0.7600 0.0071 0.0460 0.0209 0.0235 0.0073

Catboost 0.7700 0.0065 0.0443 0.0807 0.0909 0.2800

Lightgbm 0.7700 0.0067 0.0479 0.0261 0.0294 0.0069

MLP 0.6100 0.0116 0.0695 0.0403 0.0454 0.0094

XGboost 0.7600 0.0070 0.0450 0.0185 0.0208 0.2695

SVR 0.7400 0.0075 0.0457 0.0189 0.0213 0.2744

RBF 0.5341 0.0117 0.0775 0.1081 0.1218 0.0118

ML methods	Predictive performance evaluation index
DT	0.6200	0.0113	0.0533	0.0179	0.0202	0.0074
RF	0.7600	0.0074	0.0470	0.0212	0.0239	1.0713
GBDT	0.7800	0.0041	0.0345	0.0157	0.0177	0.0294
Adaboost	0.7600	0.0071	0.0460	0.0209	0.0235	0.0073
Catboost	0.7700	0.0065	0.0443	0.0807	0.0909	0.2800
Lightgbm	0.7700	0.0067	0.0479	0.0261	0.0294	0.0069
MLP	0.6100	0.0116	0.0695	0.0403	0.0454	0.0094
XGboost	0.7600	0.0070	0.0450	0.0185	0.0208	0.2695
SVR	0.7400	0.0075	0.0457	0.0189	0.0213	0.2744
RBF	0.5341	0.0117	0.0775	0.1081	0.1218	0.0118

Fig. 11

Distribution of true and predicted values of RDEM.

Fig. 12

Distribution of predicted and residual values of RDEM data.

5.3.2 ML model predicts MLR

Table 5 shows the predictive performance statistics of nine ML algorithms for the MLR of concrete antifreeze indices. Figure 13 shows the predictive maps with measured values as transverse coordinates and predicted values as longitudinal coordinates, whereas Fig. 14 shows the residual maps with predicted values as transverse coordinates and residual values as longitudinal coordinates

Table 5
Statistical results of quality loss rate prediction performance

ML methods Predictive performance evaluation index

R² MSE MAE RMSE SI BIAS

DT 0.7000 0.0067 0.0378 0.0090 0.8336 0.8336

RF 0.8200 0.0040 0.0325 0.0115 1.0652 0.4650

GBDT 0.8200 0.0041 0.0345 0.0157 1.4542 0.0090

Adaboost 0.8000 0.0045 0.0328 0.0100 0.9262 0.0042

Catboost 0.8400 0.0036 0.0312 0.0597 5.5298 0.1772

Lightgbm 0.8200 0.0039 0.0353 0.0173 1.6024 0.0039

MLP 0.7500 0.0056 0.0463 0.0251 2.3249 0.0051

XGboost 0.8100 0.0042 0.0318 0.0096 0.8892 0.1863

SVR 0.8300 0.0036 0.0322 0.0114 1.0559 0.2406

RBF 0.5978 0.0087 0.0697 0.0934 8.6506 0.0086

ML methods	Predictive performance evaluation index
DT	0.7000	0.0067	0.0378	0.0090	0.8336	0.8336
RF	0.8200	0.0040	0.0325	0.0115	1.0652	0.4650
GBDT	0.8200	0.0041	0.0345	0.0157	1.4542	0.0090
Adaboost	0.8000	0.0045	0.0328	0.0100	0.9262	0.0042
Catboost	0.8400	0.0036	0.0312	0.0597	5.5298	0.1772
Lightgbm	0.8200	0.0039	0.0353	0.0173	1.6024	0.0039
MLP	0.7500	0.0056	0.0463	0.0251	2.3249	0.0051
XGboost	0.8100	0.0042	0.0318	0.0096	0.8892	0.1863
SVR	0.8300	0.0036	0.0322	0.0114	1.0559	0.2406
RBF	0.5978	0.0087	0.0697	0.0934	8.6506	0.0086

Fig. 13

Distribution of measured and predicted values of MLR data.

Fig. 14

Distribution of predicted and residual values of MLR data.

5.3.3 ML model predictions

A high degree of symmetry can be observed in most data points near zero horizontal lines, whereas a very small number of points is scattered around the periphery. In addition, in the projection picture, most data points are closely distributed within a confidence interval of 10%, indicating that the results of these samples are close to the true values and have a high fit. In addition, some data points are located outside the 20% confidence interval, which may be due to a combination of noise not being cleaned thoroughly in the data, hyperparametric optimisation in the model and weight learning. The influence of different algorithms on each evaluation index was analysed by comparative experiment. The prediction performance of nonlinear models was better than that of linear models in terms of RDEM prediction, which was validated in Section 2.1. It is worth noting that the prediction ability of gradient lifting regression tree is the best, namely R² = 0.78, MSE = 0.0041, MAE = 0.0345, RMSE = 0.0157, SI = 0.0177, BIAS = 0.0294. In terms of MLR prediction, Catboost has the best prediction ability, namely R² = 0.84, MSE = 0.0036, RMSE = 0.0597, MAE = 0.0312, SI = 5.5298, BIAS = 0.1772. Although the RMSE parameter (0.0597) predicted by Catboost model is higher than that of other models and has forecasting inaccuracy, it cannot be used as a basis for judgment. Catboost model certainly has high forecasting ability in MLR data.

5.4 Optimal model performance

According to the training and testing results, it can be concluded that the gradient lifting decision tree performs best in the prediction of RDEM. In the training, the average values of R², MAE and RMSE of the gradient lifting decision tree are 0.72, 0.0074 and 0.0261, respectively. In the test, compared with the other 8 models, GBDT model had the best prediction ability, R² (0.78), MSE (0.0041), MAE (0.0345), RMSE (0.0223), SI (0.0177), BIAS (0.0294) (see Table 4). In addition, Catboost has the best performance in MLR prediction, and the mean values of R², RMSE and MAE of Catboost in training are 0.81,0.0113 and 0.004, respectively. In the test, compared with the other 8 models, the prediction ability of Catboost model is the best, R² (0.84), MSE (0.0036), RMSE (0.0597), MAE (0.0312), SI (0.0177), BIAS (0.0294) (see Table 5).

5.5 Comparison with previous studies

Compared with previous research results, it is found that due to the difference in the number of sample data used in the prediction process, the selection of hyperparameters, and the way of data pre-processing, the prediction results are very different, and the prediction ability of machine learning models is also very different. In the subsequent research, we will continue to optimize the data processing, debug the model hyperparameter combination, and improve the model prediction ability. Table 6 shows the comparison between the predictive ability of the algorithm used in this study and previous studies.

Table 6
Comparison of predictive ability of machine learning with previous studies

Reference algorithm R² MAE MSE RMSE

Ke et al. (2023) [57] Catboost 0.9520 4.9570

RF 0.9090 6.8400

Huang et al. (2023) [58] RF 0.5768 1.7408 2.5709

Rajczakowska et al. (2023) [59] SVR 0.8520 0.0731 0.0092 0.0960

ANN 0.8990 0.0598 0.0063 0.0795

Wang et al. (2023) [60] SVR 0.9700 143907.0000

RF 0.9300 228953.0000

ANN 0.9600 174585.0000

Tran et al. (2022) [31] SVR 0.5897 5.8178×10^-12 8.1323×10^-12

RF 0.8413 3.1813×10^-12 4.0811×10^-12

Xgboost 0.8558 2.7616×10^-12 3.9311×10^-12

Yaseen et al. (2023) [18] RF 0.87907 46.7718 54.2559

Vu et al. (2015) [61] ANN 0.9100 117.5100

Pham et al. (2015) [62] ANN 0.7600 6.7400

SVR 0.7900 6.0700

El-Mir et al. (2023) [63] SVR 0.9400 3.7200 4.6600

Khan et al. (2023) [19] Xgboost 0.7600 0.0700 0.0700

Feng et al. (2020) [64] ANN 0.9600 2.6000 3.300

SVR 0.9680 1.7300 2.9500

Kazemi et al. (2023) [65] Xgboost 0.9980 1.3300 0.0300 1.7190

RF 0.9930 2.3100 0.0960 3.0920

Adaboost 0.9880 3.5100 0.1510 3.8910

SVR 0.9500 7.0940 0.6510 8.0670

ANN 0.9990 0.7690 0.0080 0.8860

Liu et al. (2023) [66] RF 0.9320 3.7360

GBDT 0.9600 2.7960

Xgboost 0.9630 2.7130

Zheng et al. (2023) [67] RF 0.9220 0.0560 2.7700

DT 0.8780 1.8840 5.9690

Adaboost 0.8970 1.715 4.5980

GBDT 0.9430 0.8720 1.20900

Xgboost 0.9430 0.7810 1.2130

Lightgbm 0.94200 0.9040 1.2850

Catboost 0.9320 0.5510 0.6180

Al Fuhaid et al. (2023) [68] ANN 0.9510 1.90500 1.6300

SVR 0.9580 1.6400 1.3370

Cook et al. (2019) [69] SVR 0.7581 1.2353 1.5245

RF 0.9990 0.0753 0.0977

Zhang et al. (2023) [70] Xgboost 0.8067 4.7576 7.1011

RF 0.7770 5.3456 7.6264

SVR 0.4114 9.2291 12.3920

GBDT 0.7907 4.4714 7.3897

Zhang et al. (2023) [26] ANN 0.9600 4.0000 5.6400

Rajakarunakaran et al. (2022) [71] DT 0.9100 1.9100 17.0100 4.1200

RF 0.9400 1.86000 10.6100 3.2600

Bernardo et al. (2023) [72] DT 0.9730 10.713 17.749

Reference	algorithm	R²	MAE	MSE	RMSE
Ke et al. (2023) [57]	Catboost	0.9520			4.9570
	RF	0.9090			6.8400
Huang et al. (2023) [58]	RF	0.5768	1.7408		2.5709
Rajczakowska et al. (2023) [59]	SVR	0.8520	0.0731	0.0092	0.0960
	ANN	0.8990	0.0598	0.0063	0.0795
Wang et al. (2023) [60]	SVR	0.9700			143907.0000
	RF	0.9300			228953.0000
	ANN	0.9600			174585.0000
Tran et al. (2022) [31]	SVR	0.5897	5.8178×10^-12		8.1323×10^-12
	RF	0.8413	3.1813×10^-12		4.0811×10^-12
	Xgboost	0.8558	2.7616×10^-12		3.9311×10^-12
Yaseen et al. (2023) [18]	RF	0.87907	46.7718		54.2559
Vu et al. (2015) [61]	ANN	0.9100			117.5100
Pham et al. (2015) [62]	ANN	0.7600			6.7400
	SVR	0.7900			6.0700
El-Mir et al. (2023) [63]	SVR	0.9400	3.7200		4.6600
Khan et al. (2023) [19]	Xgboost	0.7600	0.0700		0.0700
Feng et al. (2020) [64]	ANN	0.9600		2.6000	3.300
	SVR	0.9680		1.7300	2.9500
Kazemi et al. (2023) [65]	Xgboost	0.9980	1.3300	0.0300	1.7190
	RF	0.9930	2.3100	0.0960	3.0920
	Adaboost	0.9880	3.5100	0.1510	3.8910
	SVR	0.9500	7.0940	0.6510	8.0670
	ANN	0.9990	0.7690	0.0080	0.8860
Liu et al. (2023) [66]	RF	0.9320			3.7360
	GBDT	0.9600			2.7960
	Xgboost	0.9630			2.7130
Zheng et al. (2023) [67]	RF	0.9220	0.0560		2.7700
	DT	0.8780	1.8840		5.9690
	Adaboost	0.8970	1.715		4.5980
	GBDT	0.9430	0.8720		1.20900
	Xgboost	0.9430	0.7810		1.2130
	Lightgbm	0.94200	0.9040		1.2850
	Catboost	0.9320	0.5510		0.6180
Al Fuhaid et al. (2023) [68]	ANN	0.9510	1.90500		1.6300
	SVR	0.9580	1.6400		1.3370
Cook et al. (2019) [69]	SVR	0.7581	1.2353		1.5245
	RF	0.9990	0.0753		0.0977
Zhang et al. (2023) [70]	Xgboost	0.8067	4.7576		7.1011
	RF	0.7770	5.3456		7.6264
	SVR	0.4114	9.2291		12.3920
	GBDT	0.7907	4.4714		7.3897
Zhang et al. (2023) [26]	ANN	0.9600	4.0000		5.6400
Rajakarunakaran et al. (2022) [71]	DT	0.9100	1.9100	17.0100	4.1200
	RF	0.9400	1.86000	10.6100	3.2600
Bernardo et al. (2023) [72]	DT	0.9730	10.713		17.749

5.6 Feature importance

Figures 15 and 16 respectively show the feature importance of ML’s optimal model in predicting RDEM and MLR. The bar chart shows each feature. The most important feature ranks at the top, that is, the most important feature contributes the most to the prediction of the model, while the bottom bar represents the least important feature. In the prediction of the RDEM, the three most important features are NFTC, G and GC, in which NFTC shows the highest importance and the largest contribution to the prediction of the RDEM. In the prediction of MLR, the three most important features are NFTC, WRA and G, and NFTC also shows the highest importance and the greatest contribution to the prediction of MLR.

Fig. 15

GBDT algorithm feature importance ranking.

Fig. 16

Catboost algorithm feature importance ranking.

5.7 Sensitivity analysis

The partial dependency graph (PDP) is a comprehensive approach that examines the linearity, monotonicity, or complexity of the relationship between labels and features. It utilizes all the data in a dataset to infer the relationship between features and predicted values. The analysis principle states that the significance of a feature increases with a larger change in PDP, while it decreases with a smaller change in PDP [73]. For numerical features, feature importance is defined as the standard deviation of $f_{S} (x_{s}^{k})$ of each value ( $x_{s}^{k}$ ) of feature (x_s), as shown in Equation (17): $I (x_{s}) = \sqrt{\frac{1}{K - 1} \sum_{K}^{k = 1} {(f_{s} (x_{s}^{k}) - \frac{1}{K} \sum_{K}^{k = 1} f_{s} (x_{s}^{k}))}^{2}}$ (17)

Where, $x_{s}^{k}$ is the k independent values of characteristic x_s.

As can be seen from Fig. 17, when the content of C ranges from 70 to 260 kg/m³, RDEM first slowly rises to a certain level and remains stable, indicating that the content of C has little influence on RDEM, and excessive C has no positive influence on the change of RDEM. When the content of S is greater than 100 kg/m³, RDEM begins to decrease gradually. The variation law of S, W, G and NFTC is similar to FA, and the optimal range is 500∼550 kg/m³, respectively. 100∼110 kg/m³, 1000∼1050 kg/m³ and 70∼80 times; When GGBS content is in the range of 0∼50 kg/m³, RDEM gradually decreases and remains stable. When GGBS content is greater than 140 kg/m³, RDEM gradually increases, and the optimal range is 45∼50 kg/m³. When WRA ranges from 0 to 4 kg/m³, RDEM increases gradually, when WRA ranges from 4 to 10.5 kg/m³, RDEM decreases gradually, and when WRA exceeds 10.5 kg/m³, RDEM increases gradually. RDEM always increases with the increase of GC, which indicates that the influence of GC on RDEM is positive in the range of 0∼8%.

Fig. 17

Partial dependence analysis of RDEM data based on GBDT algorithm.

As can be seen from Fig. 18, the content of C in the range of 80∼400 kg/m³, MLR decreases with the increase of C, and the change law of S is the same, but the change law of G is opposite. When GC content is in the range of 0∼5.5%, MLR first increases and then decreases with GC, which is the same as that of W and GGBS. The optimal range is 0∼2%, 70∼80 kg/m³ and 120∼140 kg/m³, respectively, while when FA is in the range of 0∼170 kg/m³, MLR first decreases and then increases. The optimal range of NFTC is 80∼100 kg/m³ and 90∼110 times. WRA is always level in the range of 0∼13 kg/m³, indicating that WRA content has no effect on MLR.

Fig. 18

Partial dependency analysis of MLR data based on Catboost algorithm.

Combined with RDEM and MLR partial dependence analysis, the optimal content of each material can be obtained, respectively, as C: [260, 280] kg/m³, FA: [80, 100] kg/m³, GGBS: [0, 50] kg/m³, S: [500, 600] kg/m³, G: [1000, 1100] kg/m³, WRA: [3, 4] kg/m³, GC: [2, 4] kg/m³, W: [100, 110] kg/m³.

6 Optimal design of concrete combination ratio

Based on the optimized model and durability design code of railway concrete structure [74], the mix proportions of RDEM and MLR were optimized [75], and the design operation steps were optimized and adjusted:

On the basis of the concrete proportions database, the optimal concrete ratio needed by users was selected according to the durability design specification [75] and the principle of lowest cost, and the unit price was calculated as the initial unit price.

Fine-tuning using the Monte Carlo method is described in section 5.2;

Under the premise of conforming to the specification, the best model was used to predict the antifreeze performance of concrete. If the specification was met, the next step was taken; otherwise, steps b and c were repeated;

For the calculation of concrete specific unit price, if it was lower than the unit price specified in step a, the next operation was carried out; otherwise, steps (2) –(4) were repeat;

After assessing the number of iterations, the output of the optimal match proportions was obtained; otherwise, the unit price in step a was updated and step (2) –(5) were repeat.

6.1 Concrete ratio unit price

According to the market findings, the concrete materials used in this paper included C, FA, GGBS, S, G, WRA and W in seven different quantities. The average price of each raw material is given in Table 7.

Table 7
Average prices of raw materials

Raw materials (m³/kg) C FA GGBS S G W WRA

Unit price (yuan) 0.429 0.23 0.22 0.064 0.006 0.003 1.752

Raw materials (m³/kg)	C	FA	GGBS	S	G	W	WRA
Unit price (yuan)	0.429	0.23	0.22	0.064	0.006	0.003	1.752

Concrete unilateral cost by Equation (18): $\begin{matrix} {Cost = C}_{C} P_{C} + C_{FA} P_{FA} + C_{GGBS} P_{GGBS} \\ + C_{S} P_{S} + C_{G} P_{G} + C_{W} P_{W} {+ C}_{WRA} P_{WRA} \end{matrix}$ (18) where C_C, C_FA, C_GGBS, C_S, C_G, C_W and C_WRA refer to the amount of each raw material per cubic meter of concrete (m³/kg) and cost for concrete (Yuan/m³).

6.2 Concrete ratio fine-tuning

Subject to the relevant norms [74] and the principle of minimum cost, the optimal proportions was determined and fine-tuned as follows:

According to user demand, the highest-rated concrete production ratio was initially shifted in the database through the design specification [74] for durability of railway concrete structures, The production cost of this ratio was calculated as the default optimal proportions per unit of production cost of Cost _ min, with d set at 0 and maximum effective drop point dmax set at 50,000 times.

If order $W_{i}^{n}$ is the random amount of raw material in the n search, then: $W_{i}^{n} = s_{i} + (i_{Q_{3}} - i_{Q_{1}}) ξ_{i}^{n} i = 1, 2, \dots, m$ (19)

where S_i is the amount of each material used in the baseline ratio, I_Q3 is the upper quartile of each material used, I_Q1 is the lower quartile of each material used and $ξ^{n} = {[ξ_{1}^{n}, ξ_{2}^{n}, \dots, ξ_{m - 1}^{n}, ξ_{m}^{n}]}^{T}$ is a random number of m generated by a random function and normalized, of which m is the number of units entered.

For the benchmark ratio, the amount of various materials related to the parameters of the production ratio can be selected for floating random fine-tuning to check that the fine-tuned proportions meets the specification requirements. If it does, the prediction is made using a trained concrete antifreeze performance prediction model; if not, step (2) is returned.

Calculate and judge whether the concrete antifreeze of the ratio meets the requirements of concrete antifreeze design. If so, calculate the target function according to Equation (15) and determine the new production cost of concrete antifreeze proportions, otherwise return to step (2) and continue fine-tuning.

Compare the cost of the new concrete mix with the cost of the current optimal mix. If the new concrete mix is cheaper, make min_Cost = cost and update the mix to the amount of each material used for the current mix. Otherwise, return to step (2) and continue fine-tuning.

Determine if the iteration has reached the maximum number of iterations of 10,000. If so, then the output will match the amount of concrete per cubic meter to the user; otherwise, return to step (2) and continue fine-tuning.

6.3 Co-ordination ratio optimisation results

Assuming the user needs concrete with a C40 strength rating, this step is followed, based on the optimal ML model, with the lowest unilateral cost as the optimisation goal, and the specification requirements for antifreeze concrete as constraints. The benchmark ratio is selected from a known database and optimised using Monte Carlo fine-tuning methods. The unilateral cost of concrete is calculated in Table 8.

Table 8
Comparison of concrete combination ratio optimization results

Raw materials C FA GGBS S G W WRA RDEM MLR UC

Base mix ratio 247.00 82.00 82.00 686.00 1119.00 144.00 4.11 91.60 3.50 195.26

Optimized ratio 232.04 87.88 89.65 655.64 1165.64 154.35 3.45 71.45 2.94 194.94

Raw materials	C	FA	GGBS	S	G	W	WRA	RDEM	MLR	UC
Base mix ratio	247.00	82.00	82.00	686.00	1119.00	144.00	4.11	91.60	3.50	195.26
Optimized ratio	232.04	87.88	89.65	655.64	1165.64	154.35	3.45	71.45	2.94	194.94

7 Conclusion

This study explored nine ML methods for predicting the freezing resistance of concrete. A total of 7,088 sets of concrete mix design and experimental data of RDEM and MLR were collected as samples, C, FA, GGBS, S, G, WRA, GC, W and NFTC were used as input variables. RDEM and MLR were predicted. The model performance evaluation indexes (R², MSE, MAE and RMSE) were used to evaluate the predicted results, and the best ML model was obtained.

The prediction results indicate that the nonlinear model outperforms the linear model in terms of prediction performance. The scatterplot of data visualization reveals a scattered distribution pattern, which lacks linearity and is closely associated with factors such as the inherent characteristics of the data, hyperparameter optimization, weight learning, and training and prediction processes. Among various models, the gradient boosting decision tree exhibits superior goodness of fit (R²) and accuracy (lower MSE, MAE, and RMSE values) in predicting RDEM outcomes within the nonlinear model. Specifically, R² = 0.78, MSE = 0.0041, MAE = 0.0345, RMSE = 0.0157; SI = 0.0177; BIAS = 0.0294. The Catboost based MLR prediction also demonstrates higher goodness of fit (R²) and accuracy (smaller MSE, MAE, and RMSE values), namely R² = 0.84, MSE = 0.0036, RMSE = 0.0597, MAE = 0.0312, SI = 5.5298, BIAS = 0.1772.

Through the optimal ML model, with the lowest single cost as the optimal objective and the specification requirement as the constraint condition, the Monte Carlo method was used to obtain the optimal single cost proportions of antifreeze concrete by randomly fine-tuning the base proportions. In this study, a large number of noise in the dataset interfered with the prediction performance of the model. The noise in the data will continue to be cleaned and the model’s hyperparameters will be adjusted to improve the prediction performance of the ML.

Conflict of interest

The authors declared that there is no conflict of interest.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China [Nos. 52368032 and 51808272], the China Postdoctoral Science Foundation [No. 2023M741455], the Natural Science Foundation of Gansu Province [No. 21JR7RA330], the Tianyou Youth Talent Lift Program of Lanzhou Jiaotong University, the Gansu Province Youth Talent Support Project [No. GXH20210611-10].

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Machine learning prediction of concrete frost resistance and optimization design of mix proportions

Abstract

Keywords

1 Introduction

2 ML approach

2.1 Multivariate linear regression

2.7 Gradient boosting decision tree

3.1.1 Data collection and integration

3.1.2 Data pre-processing

4.1 Data segmentation

4.2 Model training

5 Model performance assessment

5.1 Performance indicators

5.3.1 ML model predicts RDEM

5.4 Optimal model performance

5.5 Comparison with previous studies

6.1 Concrete ratio unit price

Table 7 Average prices of raw materials Raw materials (m3/kg) C FA GGBS S G W WRA Unit price (yuan) 0.429 0.23 0.22 0.064 0.006 0.003 1.752

Table 8 Comparison of concrete combination ratio optimization results Raw materials C FA GGBS S G W WRA RDEM MLR UC Base mix ratio 247.00 82.00 82.00 686.00 1119.00 144.00 4.11 91.60 3.50 195.26 Optimized ratio 232.04 87.88 89.65 655.64 1165.64 154.35 3.45 71.45 2.94 194.94

Conflict of interest

Footnotes

Acknowledgments

References

Table 7
Average prices of raw materials

Raw materials (m³/kg) C FA GGBS S G W WRA

Unit price (yuan) 0.429 0.23 0.22 0.064 0.006 0.003 1.752

Table 8
Comparison of concrete combination ratio optimization results

Raw materials C FA GGBS S G W WRA RDEM MLR UC

Base mix ratio 247.00 82.00 82.00 686.00 1119.00 144.00 4.11 91.60 3.50 195.26

Optimized ratio 232.04 87.88 89.65 655.64 1165.64 154.35 3.45 71.45 2.94 194.94