Hybrid model-driven and data-driven method for predicting concrete creep considering uncertainty quantification

Abstract

Reasonable prediction of concrete creep is the basis of studying long-term deflection of concrete structures. In this paper, a hybrid model-driven and data-driven (HMD) method for predicting concrete creep is proposed by using the sequence integration strategy. Then, a novel uncertainty prediction model (UPM) is developed considering uncertainty quantification. Finally, the effectiveness of the proposed method is validated by using the North-western University (NU) database of creep, and the effect of uncertainty on prediction results are also discussed. The analysis results show that the proposed HMD method outperforms the model-driven and three data-driven methods, including the genetic algorithm-back propagation neural network (GA-BPNN), particle swarm optimization-support vector regression (PSO-SVR) and convolutional neural network only method, in accuracy and time efficiency. The proposed UPM of concrete creep not only ensures relatively good prediction accuracy, but also quantifies the model and measurement uncertainties during the prediction process. Additionally, although incorporating measurement uncertainty into concrete creep prediction can improve the prediction performance of UPM, the prediction interval of the creep compliance is more sensitive to model uncertainty than to measurement uncertainty, and the mean contribution of variance attributed to the model uncertainty to the total variance is about 90%.

Graphical abstract

Keywords

concrete creep / uncertainty prediction / hybrid method / data-driven / model-driven / convolutional neural network

1 Introduction

Concrete is widely used in civil engineering structures due to its favorable mechanical properties and cost-effectiveness [1–3]. However, concrete has an obvious time-dependent creep behavior, which not only leads to a continuous increase in the deflection of concrete flexural members [4,5], but also results in the relaxation or prestress loss of tendons in prestressed concrete structures [6,7]. These irreversible damages significantly affect the long-term performance and durability of concrete structures [8,9]. Therefore, a reasonable prediction of concrete creep is of utmost importance for accurately assessing the long-term service performance of concrete structures.

The creep behavior of concrete is a complex time-dependent process influenced by various internal and external factors, such as concrete composition, mechanical properties, external stresses (static or dynamic), and environmental factors [10,11]. Most of the existing prediction models of concrete creep are proposed based on the experimental and numerical analysis results [12,13]. For example, based on a practical prediction model that considers multiple influencing factors, Koh et al. [11] proposed a creep prediction method considering the effect of repeated loading, and found that the dynamic stress can accelerate the creep rate. In addition, various countries or organizations have developed their own standard models. Among these, the CEB-FIP 90 [14], ACI-209 [15], JTG 3362-2018 [16], AASHTO [17], and B4 [18] models are widely used. However, these models belong to the model-driven method, and are mostly semi-empirical and semi-theoretical models or purely empirical models [19]. This means that although the existing model-driven methods can effectively describe the physical meaning of each parameter, most of them are simplified, leading to limitations in prediction accuracy and applicability.

The rapid development of data-driven methods such as machine learning provides new ideas for predicting structural or material properties [20–22], and some scholars have gradually carried out the research on creep prediction based on data-driven method. At present, the research on data-driven creep prediction can be roughly divided into two categories. The first type is to establish the creep prediction models by using the processed North-western University (NU) database and different types of machine learning methods, and compare the prediction performance of these methods. Bal and Buyle-Bodin [23] and Gandomi et al. [24] proposed creep prediction models using artificial neural network (ANN) and Multi-Objective Genetic Programming respectively. Hodhod et al. [25] predicted the concrete creep compliance by using the Multi-Gene Genetic Programming and ANN. Li et al. [26] established a creep prediction model based on three machine learning methods including the back-propagation neural network (BPNN), support vector regression (SVR), and extreme learning machine, and found that the SVR-based model has high accuracy. Liang et al. [27] adopted three ensemble machine learning models, namely Random Forest (RF), Extreme Gradient Boosting Machine (XGBoost) and Light Gradient Boosting Machine, to predict the creep behavior of concrete, and identified five main factors that affect the creep behavior. Zhu and Wang [28] solved the problem of uneven distribution of NU data sets on the time scale by introducing the K-means clustering algorithm, and proposed a convolutional neural network (CNN) model for predicting concrete creep. The second type is to use the collected experimental data and machine learning methods to predict the creep behavior of new types of concrete such as high-performance concrete (HPC), recycled concrete, and sustainable concrete. Both Karthikeyan et al. [29] and Gedam et al. [30] used ANN to propose a creep prediction model for HPC, despite differences in database size and input parameters between the two studies. Liu et al. [31] developed the machine learning model for predicting the creep compliance of concrete containing supplementary cementitious materials. Feng et al. [32,33] studied the creep behaviors of recycled aggregate concrete (RAC) by using five typical ensemble machine learning models, and found that the XGBoost prediction model has better prediction performance. Rong et al. [34] studied the drying creep of RAC using the BPNN and support vector machine method. Sadowski et al. [35] and Yang et al. [36] employed the ANN to predict creep of containing ground granulated blast furnace slag concrete and seawater sand coral aggregate concrete, respectively. Bouras and Li [37] proposed a creep compliance prediction model of low-carbon concrete by using four supervised machine learning techniques, including the Gaussian process regression, ANN, and decision tree regression models. Additionally, Bouras and Li [38] has also established a short-term basic creep data set for concrete at high temperatures, which contains 132 groups of 1611 data, and proposed a prediction model for concrete creep at high temperatures using supervised machine learning methods. Although these studies can effectively improve prediction accuracy, they are all black-box models that lack the ability to provide a clear physical interpretation of model parameters, resulting in limited interpretability. This deficiency diminishes the credibility and applicability of such data-driven methods.

As mentioned earlier, both model-driven and data-driven methods have obvious shortcomings, but they are naturally complementary. However, in the field of civil engineering, there has been limited research on combining these two methods to predict the properties of structures or materials. For example, Xia et al. [39] employed the data-driven method to obtain unknown model parameters, and subsequently established a physics-driven evaluation method for assessing the seismic performance of engineering structures. At present, the hybrid model-driven and data-driven (HMD) method is still to be studied, and its application in concrete creep prediction urgently requires exploration.

In the creep prediction of concrete in service, there are usually epistemic or aleatoric uncertainties in the calculation model and measurement parameters [40], which makes it difficult to accurately predict the concrete creep and its effect on the long-term performance of structure [41]. The existing probabilistic analysis of concrete creep are mostly carried out based on the model-driven methods. For example, by introducing the measurement uncertainty coefficient, internal uncertainty coefficient of creep phenomenon, and model uncertainty coefficient into the model recommended by different specifications, Keitel and Dimmig-Osburg [42] developed an uncertainty prediction method of concrete creep considering the effect of correlated input parameters. Criel et al. [3] proposed an approximate uncertainty quantification of concrete creep by using the Taylor series. Jin et al. [43] presented a probability prediction method for short-term concrete creep, considering the effect of model uncertainty by combining the predictive capability across various prediction models. However, the existing data-driven methods for predicting concrete creep can only obtain the point estimates of the prediction results, and cannot incorporate the effects of epistemic and aleatoric uncertainties. This limitation undoubtedly increases the contingency of the prediction results, potentially leading to significant deviations. Additionally, the uncertain prediction model of concrete creep based on HMD method has not been reported up to now. Therefore, to enhance the rationality of the prediction process, it is necessary to further consider the effect of epistemic and aleatoric uncertainties in the hybrid driven method for predicting concrete creep.

The purpose of this paper is to propose a HMD method for predicting concrete creep by using the sequence integration strategy, and then develop a novel uncertainty prediction model (UPM) considering the uncertainty of measurement and deep learning model. The novelty of this paper is that the proposed method cannot only take advantage of both data-driven and model-driven methods to improve the prediction efficiency and accuracy of concrete creep, but also incorporate the effect of aleatoric and epistemic uncertainties in the prediction process, thereby improving the rationality of prediction.

The rest of this paper is organized as follows. Section 2 proposes an HMD method for predicting concrete creep. Section 3 develops a novel UPM of concrete creep on the basis of the above HMD method. In Section 4, a case study is performed to illustrate the feasibility of the proposed approach. Section 5 summarizes the conclusion of this paper.

2 Hybrid model-driven and data-driven method

2.1 Model-driven method for predicting concrete creep

The common model-driven methods for predicting concrete creep include the CEB-FIP 90 model, ACI-209 model, JTG 3362-2018 model, AASHTO model, B4 model, etc. [44]. Here, the CEB-FIP 90 model is selected as the model-driven method for predicting concrete creep due to its frequent use and good applicability [45]. In the CEB-FIP 90 model, the creep compliance J(t, t₀) (designated as J_md here) is used to describe the time-dependent concrete creep, and it is expressed as [14]

(1)J_{\mathrm{m}\mathrm{d}}=J\left(t,t_0\right)=\frac{1}{E_{\mathrm{c}}\left(t_0\right)}+\frac{\phi \left(t,t_0\right)}{E_{\mathrm{c},28}},

where E_c(t₀) and E_c,28 are the elastic modulus at the concrete age of t₀ and 28 d, respectively. t₀ is the loading age, and φ(t, t₀) is the creep coefficient at time t, which can be calculated as

(2)\begin{array}{rl}\phi \left(t,t_0\right)=& {\phi }_{\mathrm{R}\mathrm{H}}\beta \left(f_{\mathrm{c}\mathrm{m}}\right)\beta \left(t_0\right){\beta }_{\mathrm{c}}\left(t-t_0\right)\\ =& \left[1+\frac{1-RH/100\mathrm{\%}}{0.46{\left(h_{\mathrm{t}}/100\right)}^{1/3}}\right]\frac{5.3}{\sqrt{f_{\mathrm{c}\mathrm{m}}/10}}\frac{1}{0.1+t_0^{0.2}}\\ & \cdot {\left[\frac{t-t_0}{{\beta }_{\mathrm{H}}+\left(t-t_0\right)}\right]}^{0.3},\end{array}

where φ_RH, β(f_cm), β(t₀), and β_c(t−t₀) are the parameters describing the effects of humidity, concrete compressive strength, loading age, and concrete age, respectively. RH is the relative environmental humidity, and f_cm is the concrete compressive strength at the age of 28 d. h_t is the nominal thickness of member and is calculated as 2A_c/u, in which A_c and u are the cross-sectional area and perimeter of the concrete component in contact with the atmosphere. β_H is a parameter related to RH, which can be calculated as

(3){\beta }_{\mathrm{H}}=min\left\{1.55\left[1+{(1.2RH)}^{18}\right]h_{\mathrm{t}}/100+250,1500\right\}.

2.2 Data-driven method for predicting concrete creep

The machine learning method is a data-driven method widely used in engineering field. As a representative machine learning method, CNN has been widely reported for its advantages in dealing with tabular data [46]. Moreover, compared with some traditional technologies, CNN has advantages such as good fault tolerance, parallel processing, strong self-learning ability, and fine adaptive performance [28,47]. Therefore, CNN is adopted here to construct the prediction model of concrete creep.

The general structure of CNN is mainly composed of input layer, convolution layer, pooling layer, flatten layer, and fully connection layer [48]. In this architecture, the convolution layer is responsible for extracting essential features from the input variables, the pooling layer is used to solve the over-fitting problem in the network structure, and the flatten layer is mainly used for data dimension conversion. The fully connected layer is the same structure as the traditional neural network, and it is usually used in the last few layers of CNN to convert the low-level information into the high-level features of structural response. For a one-dimensional (1-D) CNN structure, the calculations of convolution layer (Eq. (4)), pooling layer (Eq. (5)) and fully connected layer (Eq. (6)) can be expressed as

(4){\mathit{C}\mathit{L}}_j=f\left(\sum _i{\mathit{\text{z}}}_i\otimes {\mathit{w}}_{\text{c,}ij}+{\mathit{b}}_{\text{c,}j}\right),

(5)\mathit{P}\mathit{L}=down\left({\mathit{C}\mathit{L}}_j\right),

(6){\mathit{F}\mathit{C}\mathit{L}}_l=g\left(w_{\mathrm{f}\mathrm{c}\mathrm{l},l}{\mathit{F}\mathit{C}\mathit{L}}_{l-1}+{\mathit{b}}_{\mathrm{f}\mathrm{c}\mathrm{l},l-1}\right),

where CL_j, PL, and FCL_l are the operation output of the convolution layer, pooling layer, and fully connected layer, respectively. z_i is the input of the convolutional layer, w_c,ij is the convolutional kernel, b_c,j is the bias in the jth convolution layer, f() is the activation function in the convolution layer, and \otimes denotes the convolution operation. The down() is represents the pooling function. The most common pooling methods are maximum pooling and average pooling, and their detailed calculation method can be found in Ref. [49]. w_fcl,l and b_fcl,l−1 are the weight matrix and bias in the corresponding fully connected layer, respectively. FCL_l−1 is the output of the (l−1)th layer, and g() is the activation function in the fully connected layer.

For the CNN-based concrete creep prediction method, once the model parameters of CNN are fully determined by the training, testing and performance evaluation, and the set of input variables is recorded as z, the data-driven based prediction result (J_dd) of concrete creep compliance can be expressed as

(7)J_{\mathrm{d}\mathrm{d}}=CNN\{\mathit{\text{z}}\}=CNN\left\{{\mathit{\text{z}}}_1,{\mathit{\text{z}}}_2,\dots ,{\mathit{\text{z}}}_p\right\},

where z_i (i = 1,2,…,p) is the ith input variable.

2.3 Hybrid model-driven and data-driven method for predicting concrete creep

As previously described in the introduction, a reasonable integration of model-driven and data-driven methods is critical to taking advantage of these two prediction methods. The existing integration strategies mainly include three modes: sequential mode, parallel mode and physics guided data-driven mode [50]. The sequential mode usually takes the prediction result of model-driven method as an input parameter of data-driven method, thereby correcting the predictions obtained by the model-driven method. The parallel mode comprehensively processes the calculation results of the model-driven and data-driven methods by using different measures including stacking, weighted summation, and factor multiplication, and then take the processed prediction result as the final output result. The physics guided data-driven mode mostly incorporate the known physical information as constraint within the data-driven model, ensuring that the training results meet established physical information. This can usually be achieved by modifying the loss function during the training process of machine learning.

Among these three methods, the sequential mode is suitable for correcting the calculation results of model-driven method with low accuracy, and can effectively reduce the dimension of input features in the data-driven method, thereby improving the accuracy and time efficiency of the model [50]. Additionally, unlike the sequential mode, the other two modes may require manual determination of the weights assigned to data-driven and model-driven effects, which undoubtedly reduces the reliability of the prediction model. Furthermore, the parallel mode and physics guided data-driven mode require higher computational resources and time than that of sequential mode, especially the latter. Moreover, the data in the creep database used has significant randomness, making it challenging for the training convergence of physics-guided data-driven models. In view of the above reasons, the sequential mode is adopted here to construct the HMD model for predicting concrete creep.

In the construction of the HMD model for predicting concrete creep in this paper, the calculation result obtained by using the model-driven method is used as an input parameter of the CNN model. Together with other parameters not considered in the model-driven method, they collectively form the reduced-dimensional input variable set to complete the HMD model training. For the convenience of subsequent representation, the input parameter set z in the data-driven method is divided into the variables x required for the model-driven calculation and other variables y. In this case, the calculation result obtained by using CEB-FIP 90 model (Eqs. (1)–(3)) can simply denoted as J_md = J(x). Then, by considering J_md and y as input parameters of the CNN model, the HMD model for predicting creep compliance can be proposed as follows

(8)\{\begin{array}{l}{J}_{\mathrm{m}\mathrm{d}}=J(\mathit{x}),\\ J_{\mathrm{h}\mathrm{d}}=CNN\left\{J_{\mathrm{m}\mathrm{d}},\mathit{y}\right\},\end{array}

where J_hd is the HMD-based prediction result of concrete creep compliance.

3 Uncertainty prediction model of concrete creep

In the HMD method for predicting creep compliance proposed in the previous section, the limitations of the database lead to the uncertainty of model itself. This model uncertainty (also called epistemic uncertainty) can be reduced by training the HMD model with a larger data set. Besides, the inherent noise in the measurement data set is inevitable, which indicates that the measurement uncertainty (also called aleatoric uncertainty) is also involved in the creep prediction. Therefore, it is necessary to further consider the effect of these two uncertainties on the prediction results.

In a general sense, a structural response can be assumed to be the sum of the prediction value and the measurement error [51]. In this context, the creep compliance of concrete is also expressed in this form as follows

(9)J(\mathit{\text{z}})=h(\mathit{\text{z}})+\epsilon (\mathit{\text{z}}),

where J(z) is the general expression of concrete creep compliance, h(z) is an invertible mathematical function for calculating creep compliance, and ε(z) is a zero mean Gaussian noise.

As previously described, since the model parameters in HMD have epistemic uncertainty, the corresponding prediction result J_hd(z) is no longer a deterministic value but rather a random variable, with its mean value is denoted as μ(z). To further quantify the effects of model and measurement uncertainties on the prediction result, the μ(z) is deducted from both sides of Eq. (9), then Eq. (9) can be rewritten as

(10)J(\mathit{\text{z}})-\mu (\mathit{\text{z}})=[h(\mathit{\text{z}})-\mu (\mathit{\text{z}})]+\epsilon (\mathit{\text{z}}),

where [J(z)−μ(z)] is the prediction uncertainty, which is composed of the model uncertainty part [h(z)−μ(z)] and the measurement uncertainty part ε(z).

The purpose of this section is to obtain the distribution of creep compliance considering the effect of model and measurement uncertainties. At present, the Bayesian method is widely used to find the joint or conditional probability distribution of structural responses under the condition of incomplete information. For example, Hamdia et al. [52] used the Bayesian method to quantify the selection probability of different models based on the limited experimental data, accounting for both model and parameter uncertainties. Inspired by the above idea, the conditional probability is also employed here to derive the probability distribution of concrete creep compliance. In this context, once μ(z) is calculated, the distribution P(J|z) of measurement J corresponding to a given influencing vector z can be derived from the distribution P[J−μ(z)|z]. However, it can be found by analyzing Eq. (10) that the distribution P[J−μ(z)|z] is closely related to the distribution of P[h−μ(z)|z] and ε(z) with a zero mean. Considering the fact that the prediction result deviation caused by the uncertainty of deep learning model can usually be regarded as Gaussian distribution [53], P[h−μ(z)|z] is assumed to follow the Gaussian distribution with a zero mean and variance of {\sigma }_{h-\mu }^2(\mathit{\text{z}}). In addition, ε(z) is a zero mean Gaussian noise as previously mentioned, and its variance is denoted as {\sigma }_{\epsilon }^2(\mathit{\text{z}}). In the following sections, μ(z), {\sigma }_{h-\mu }^2(\mathit{\text{z}}) and {\sigma }_{\epsilon }^2(\mathit{\text{z}}) will be calculated in turn, and finally the distribution P(J|z) will be derived.

3.1 Estimation of μ(z)

Since the HMD method proposed in the previous section can only obtain the deterministic output based on the mapping relationship, the Monte–Carlo (MC) dropout approach is adopted to account for model uncertainty. This technique is mainly used to randomly turn off some neurons in the network during the test phase, resulting in different weight parameters of each run and simulating the output results of different network structures. As a result, the weight uncertainty is propagated to the output uncertainty. Detailed information of MC dropout approach can be found in Ref. [54].

Given a trained HMD structure with a database (z,J) and structure parameter k = {w_c,b_c,w_fcl,b_fcl}, the posterior distribution of creep compliance J* for a new input z*can be expressed as [55]

(11)p\left(J^{\ast }|{\mathit{\text{z}}}^{\ast },\mathit{J},\mathit{\text{z}}\right)=\int p\left(J^{\ast }|{\mathit{\text{z}}}^{\ast },\mathit{k}\right)p(\mathit{k}|\mathit{J},\mathit{\text{z}})\mathrm{d}\mathit{k}.

Since the posterior distribution p(\mathit{k}|\mathit{J},\mathit{\text{z}}) usually lacks an analytical solution and is challenging to solve, a variational distribution q_θ(k) is used to estimate p(\mathit{k}|\mathit{J},\mathit{\text{z}}). Consequently, solving the posteriori distribution p(\mathit{k}|\mathit{J},\mathit{\text{z}}) is transformed into an optimization problem. Here, the Kullback–Leibler (KL) divergence, which measures the closeness between the variational distribution q_θ(k) and posterior distribution p(\mathit{k}|\mathit{J},\mathit{\text{z}}), is selected as the minimization objective. To facilitate this optimization process, minimizing the KL divergence is equivalent to maximizing the evidence lower bound (ELBO), which can be expressed as [56]

(12)ELBO=\int q_{\theta }(\mathit{k})\log p(\mathit{J}|\mathit{\text{z}},\mathit{k})\mathrm{d}\mathit{k}-KL\left[q_{\theta }(\mathit{k})\parallel p(\mathit{k})\right],

where the first term in Eq. (12) represents the expected log-likelihood of the posterior distribution, and the second term is KL divergence between q_θ(k) and p(k).

The Bernoulli distribution is used as the variational distribution in the MC dropout, and q_{\mathit{\text{θ}}}(\hat{k}) is defined as the variational distribution on a matrix whose columns are randomly set to zero. In which \hat{\mathit{k}} is the Bernoulli distributed parameters of the fully connected layer in HMD with dropout, and it can be expressed as

(13)\hat{\mathit{k}}=\left\{{\mathit{w}}_{\mathrm{c}},{\mathit{b}}_{\mathrm{c}},{\hat{\mathit{w}}}_{\mathrm{f}\mathrm{c}\mathrm{l}},{\mathit{b}}_{\mathrm{f}\mathrm{c}\mathrm{l}}\right\},

where {\hat{\mathit{w}}}_{\mathrm{f}\mathrm{c}\mathrm{l}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\eta )\theta, η is a vector composed of Bernoulli distribution random variables in the corresponding fully connected layer.

In this case, by further writing ELBO as a loss function in the training process of the CNN model, the maximization objective can be replaced using Eq. (12) as follows [54]

(14)\begin{array}{rl}& L\left({\mathit{w}}_{\text{c}},{\mathit{b}}_{\text{c}},{\mathit{w}}_{\text{fcl}},{\mathit{b}}_{\text{fcl}}\right)\\ =& \frac{1}{N}\sum _{n=1}^N\text{log}p\text{(}\mathit{J}|\mathit{\text{z}},\hat{\mathit{k}}\text{)}-KL\left[q_{\mathit{\text{θ}}}\text{(}\mathit{k}\text{)}\parallel p\text{(}\mathit{k}\text{)}\right]\\ =& \frac{1}{N}\sum _{n=1}^N{\Vert J_n-{\hat{J}}_n\Vert }^2+\sum _{i=1}^l\left({\lambda }_i{\Vert {\mathit{w}}_{\text{fcl},i}\Vert }^2+{\lambda }_{i+l}{\Vert {\mathit{b}}_{\text{fcl},i}\Vert }^2\right),\end{array}

where {\hat{J}}_n is the calculated value when the Bernoulli distributed parameters \hat{\mathit{k}} is adopted in Eq. (13), and λ representsthe weight attenuation of each parameter after weight regularization.

Here, the gradient-based learning method is used to solve the optimization problem in Eq. (14). Then, the mean of prediction creep compliance J* for a new input z* can be obtained by averaging the outputs of p\left(J|{\mathit{\text{z}}}^{\ast },\hat{\mathit{k}}\right) after M sampling of the structure parameters \hat{\mathit{k}}, which is expressed as

(15){\mu }_{\text{h}}(\mathit{\text{z}})=p\left(J^{\ast }|{\mathit{\text{z}}}^{\ast },\mathit{J},\mathit{\text{z}}\right)=\frac{1}{M}\sum _{s=1}^{M}p\left(J^{\ast }|{\mathit{\text{z}}}^{\ast },{\hat{\mathit{k}}}_s\right).

3.2 Estimation of {\sigma }_{h-\mu }^2(\mathit{\text{z}}) and {\sigma }_{\epsilon }^2(z)

Since P[(h−μ)|z] follows a zero mean Gaussian distribution with a variance of {\sigma }_{h-\mu }^2(\mathit{\text{z}}), the variance {\sigma }_{h-\mu }^2(\mathit{\text{z}}) is equal to the variance of J_hd(z). As discussed in Subsection 3.1, we have obtained the mean value of J_hd(z) and the prediction results of each sampling path by using the MC dropout approach to conduct M sampling of structural parameters of HMD. Thus, {\sigma }_{h-\mu }^2(\mathit{\text{z}}) can be calculated using the definition of variance as

(16){\sigma }_{\text{h-μ}}^2\text{(}\mathit{\text{z}}\text{)}=\frac{1}{M}\sum _{s=1}^M{\left[p\left(J^{\ast }|{\mathit{\text{z}}}^{\ast },{\hat{\mathit{k}}}_s\right)-\mu \text{(}\mathit{\text{z}}\text{)}\right]}^2.

To further obtain the variance {\sigma }_{\epsilon }^2(\mathit{\text{z}}), the following equation related to mean and variance can be derived from Eq. (10) as follows [57]

(17)E[J(\mathit{\text{z}})-\mu (\mathit{\text{z}})]=E[h(\mathit{\text{z}})-\mu (\mathit{\text{z}})]+E[\epsilon (\mathit{\text{z}})]=0,

(18)\begin{array}{rl}\mathrm{V}\mathrm{a}\mathrm{r}[J(\mathit{\text{z}})-\mu (\mathit{\text{z}})]=& \mathrm{V}\mathrm{a}\mathrm{r}[h(\mathit{\text{z}})-\mu (\mathit{\text{z}})]+\mathrm{V}\mathrm{a}\mathrm{r}[\epsilon (\mathit{\text{z}})]\\ & +2\mathrm{C}\mathrm{o}\mathrm{v}[h(\mathit{\text{z}})-\mu (\mathit{\text{z}}),\epsilon (\mathit{\text{z}})]\\ =& {\sigma }_{h-\mu }^2(\mathit{\text{z}})+{\sigma }_{\epsilon }^2(\mathit{\text{z}}).\end{array}

Since h(z)−μ(z) is independent of the measurement noise ε(z), \mathrm{C}\mathrm{o}\mathrm{v}[h(\mathit{\text{z}})-\mu (\mathit{\text{z}}),\epsilon (\mathit{\text{z}})]=0 in Eq. (18). Additionally, according to the definition of variance, \mathrm{V}\mathrm{a}\mathrm{r}[J(\mathit{\text{z}})-\mu (\mathit{\text{z}})] can also be expressed as

(19)\mathrm{V}\mathrm{a}\mathrm{r}[J(\mathit{\text{z}})-\mu (\mathit{\text{z}})]=E\left\{{[J(\mathit{\text{z}})-\mu (\mathit{\text{z}})]}^2\right\}=[J(\mathit{\text{z}})-\mu (\mathit{\text{z}}{)]}^2.

Thus, when the training database D\{\left[{\mathit{\text{z}}}_n,J_n\right],n=1,2,\dots ,N\} is given, the variance {\sigma }_{\epsilon }^2\left({\mathit{\text{z}}}_n\right) can be obtained based on Eqs. (18) and (19), which is expressed as

(20){\sigma }_{\epsilon }^2\left({\mathit{\text{z}}}_n\right)=max\left\{{\left[J_n-\mu \left({\mathit{\text{z}}}_n\right)\right]}^2-{\sigma }_{h-\mu }^2\left({\mathit{\text{z}}}_n\right),0\right\}.

After obtaining the data set \left\{\left[{\mathit{\text{z}}}_n,{\sigma }_{\epsilon }^2\left({\mathit{\text{z}}}_n\right)\right],n=1,2,\dots ,N\right\}, as long as the output in the proposed HMD method is changed to {\sigma }_{\epsilon }^2, the general variance {\sigma }_{\epsilon }^2\left(\mathit{\text{z}}\right) can be obtained by the training model.

3.3 Estimation of distribution P(J|z)

After obtaining the Gaussian distribution of h(z)−μ(z) and ε(z), the distribution P[J−μ(z)|z, μ(z)] and P[J|z, μ(z)] can be derived in turn. The mean and variance of P[J|z, μ(z)] are μ(z) and {\sigma }_{h-\mu }^2(\mathit{\text{z}})+{\sigma }_{\epsilon }^2(\mathit{\text{z}}), respectively. Since h(z)−μ(z) follows a Gaussian distribution with a zero mean and variance {\sigma }_{h-\mu }^2(\mathit{\text{z}}), μ(z) can be regarded as an estimate of h(z). As a result, P[J|z, h(z)] can be approximated to the P[J|z, μ(z)]. Therefore, given that h(z) is an invertible function, the distribution P(J|z) can be expressed as

(21)P(J|\mathit{\text{z}})\approx P[J|h(\mathit{\text{z}})]\approx P[J|\mu (\mathit{\text{z}})]=N\left(\mu (\mathit{\text{z}}),{\sigma }_{h-\mu }^2(\mathit{\text{z}})+{\sigma }_{\text{z}}^2(\mathit{\text{z}})\right).

3.4 Performance evaluation of uncertainty prediction model

Different from the deterministic prediction method of concrete creep compliance, the prediction result of the UPM proposed in this paper is a probability distribution, which can also be considered as a probability interval (PI) with a certain confidence. Reliability and clarity are two important dimensions of reasonable evaluation PIs. The former can be evaluated by the prediction interval coverage probability (PICP), while the latter can be evaluated by the prediction interval normalized averaged width (PINAW) [58]. The PICP index quantitatively describes the reliability of the proposed method by counting the number of actual target values covered by the prediction interval. The larger the PICP value is, the more test samples fall into the prediction interval, which also indicates that the proposed method is more reliable. The PICP index can be calculated as [58]

(22)PICP=\frac{1}{n_{\mathrm{t}}}\sum _{l=1}^{n_{\mathrm{t}}}{R}_l,R_l=\{\begin{array}{ll}1,& L_l⩽J_{\mathrm{m},l}⩽U_l,\\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\end{array}

where n_t is the number of test samples, L_l and U_l are the lower and upper bound of the lth prediction interval of creep compliance, respectively.

The PINAW index quantitatively characterizes the clarity of the prediction results of the proposed method by calculating the percentage of prediction interval width. The larger the value of PINAW, the wider the prediction interval, which also indicates that the greater the dispersion of the prediction results. The PINAW index can be calculated as [58]

(23)PINAW=\frac{{\displaystyle \frac{1}{n_{\mathrm{t}}}}{\displaystyle \sum _{l=1}^{n_{\mathrm{t}}}\left(U_l-L_l\right)}}{J_{\mathrm{m},\mathrm{m}\mathrm{a}\mathrm{x}}-J_{\mathrm{m},\mathrm{m}\mathrm{i}\mathrm{n}}},

where J_m,max and J_m,min are the maximum and minimum values of the measured creep compliance.

It can be seen from the above introduction that PICP and PINAW are a pair of contradictory indexes. According to their definitions, the larger the PICP is, the larger the PINAW is. Meanwhile, the lower the PICP is, the lower the PINAW is. To solve this problem, the coverage width-based criterion (CWC) index that comprehensively consider the reliability and clarity is also adopted to evaluate the rationality of the proposed UPM, which can be expressed as [59]

(24)

where β is the nominal confidence level, which is usually set as 95%. ρ is a penalty factor, which is generally set to a large value, such as 50 [59]. In general, the smaller the value of CWC, the better the interval prediction result, that is, the better the performance of the corresponding model.

3.5 Calculation flowchart of proposed prediction method

The detailed calculation flowchart of the proposed uncertainty prediction method for concrete creep can be divided into two parts. The first part is mainly to obtain the HMD model for predicting concrete creep compliance. The second part is mainly to calculate the creep compliance considering the effect of model and measurement uncertainties, and then evaluate the model performance. The calculation flowchart of the proposed prediction method is described in Fig.1.

4 Case study

The effectiveness of the proposed method is assessed by using the NU database of concrete creep. First, the HMD-based modeling process of UPM of concrete creep is introduced. Then, the performance evaluation of the HMD method is presented. After that, the prediction result of UPM is discussed. Finally, the effect of uncertainty on the prediction result is analyzed.

4.1 HMD-based modeling process of uncertainty prediction model

4.1.1 Input variable selection and data preprocessing

Since the NU database of concrete creep is the largest open database to date, it is selected here for training and testing the proposed method. Properly selecting input variables is critical to improve the accuracy and computational efficiency of the prediction model. Based on the analysis results of the correlation coefficient matrix of influencing variables provided in Ref. [60] and the input variables selected in Refs. [26,28], it is found that 11 influencing variables including the water−cement ratio (w/c), aggregate−cement ratio (a/c), cement content (c), environmental temperature (T), volume−surface ratio (V/S), concrete compressive strength at 28 d (f_cm), elastic modulus (E_c,28), relative humidity (RH), specimen height (h), loading age (t₀), and loading time (dt) exhibit a strong correlation with concrete creep.

According to the proposed HMD method, the calculation result of the model-driven method needs to be used as one of the input variables for the CNN model. Since the selected model-driven method has already considered six important influencing variables including f_cm, E_c,28, RH, h, t₀, and dt, these parameters are no longer considered during the subsequent training of the CNN model. Instead, they are replaced by the calculation result (J_md) of the model-driven method. Thus, only J_md is combined with the remaining five important influencing variables to form the input variables. That is, only w/c, a/c, c, V/S, T, and J_md are selected as input variables. This process reduces the dimension of input variables and improve computational efficiency. Additionally, only the concrete creep compliance is selected as the output variable. Consequently, the HMD of concrete creep compliance is expressed as

(25)J_{\mathrm{h}\mathrm{d}}=CNN\left\{w/c,a/c,c,V/S,T,J_{\mathrm{m}\mathrm{d}}\right\}.

The data of the 11 influencing variables mentioned above are either incompletely provided or not considered by different test groups in the NU database, indicating that some data in this database are unavailable. To address this issue, the data groups with missing items and obvious data anomalies are excluded from consideration. After the preliminary processing of the NU database, the database used for model training in this study is obtained, and the detailed information of the database can be found in Electronic Supplementary Material. The ranges of input and output variables are summarized in Tab.1, and the histograms of input and output variable distributions are also given in Fig.2. In addition, the data post-preliminary processing should be standardized to eliminate dimensional differences between variables, and the z-score standardization algorithm [61] is used here.

4.1.2 Model structure

The proposed model structure of the 1-D CNN-based HMD method sequentially contains one convolution layer, one max-pooling layer, one flatten layer, and three fully connected layers. To study the uncertainty of prediction results by using the MC dropout approach, a dropout layer is also added after the first and second fully connection layers. The detailed introduction of model architecture of 1-D CNN-based HMD is presented in Tab.2.

4.1.3 Model training

The proposed model is constructed on the deep learning framework Keras and is trained and tested on an Intel Core i7-10510U CPU @1.80 GHz with 16 GB RAM. For partitioning the filtered database, 75% of data (3985 sets) is randomly selected for model training, while the remaining 25% (1328 sets) is used for model testing. It is worth noting that the random partitioning rather than cross validation is chosen here for two main reasons. First, the scale of the creep database used here is relatively large within civil engineering. In such cases, compared to cross validation method, random partitioning conserves computational resources and cost. Moreover, the data within the creep database is acquired through diverse methodologies by scholars from different countries and regions. This acquisition method coupled with the inherent complexity of creep behavior contributes to significant randomness in the data. In this case, the random partitioning can already provide a reliable estimate of the model’s generalization performance.

In the model construction process, the ReLU function is chosen as the activation function due to its faster training speed and absence of gradient saturation issues when the input is positive. The Adam optimization algorithm is adopted because of its simple use and faster gradient convergence [62]. The learning rate is set to 0.001, the mean square error (MSE) is adopted for the loss function, the batch size for the training process is set to 128, and a moderate dropout rate of 0.2 is selected in the uncertainty prediction. Under the aforementioned conditions, the variation of loss during the six times training are shown in Fig.3. As indicated, all training losses decrease rapidly at the initial stage, and then gradually stabilize when the epoch reaches approximately 1200. Thus, in this paper, the target epoch is set to 1200 considering the balance between the prediction accuracy and calculation cost.

4.2 Performance evaluation of hybrid model-driven and data-driven method

For the deterministic HMD prediction model, we adopt a multi-metrics combination strategy to conduct model performance evaluation to overcome the shortcomings of evaluating based on a single metric. Considering that the prediction performance of HMD model is affected by the database scale and there is no zero value in the prediction response, the mean absolute error (MAE) and mean absolute percentage error (MAPE) are selected to evaluate the overall and relative errors of HMD model, respectively. In addition, the R² is selected because it can effectively evaluate the rationality of material performance prediction models, and the root mean squared error (RMSE) is also adopted to consider the impact of measured outliers in the database on model performance. In summary, four metrics including MAE, MAPE, R², and RMSE are simultaneously selected here to evaluate the prediction performance of the proposed HMD method for concrete creep. The detailed selection criteria for performance metrics can be found in Refs. [63, 64]. The calculation methods of the above four metrics can be found in Refs. [65,66]. Due to the random selection of training and testing data, the performance evaluation indexes vary for each run of the model. To address this issue, 20 model runs are conducted, and the average value of these 20 runs’ results is considered as the overall model performance. This approach enhances the robustness of the model.

To demonstrate the superiority of the proposed HMD method, the prediction performance of different machine learning methods is compared. At present, a series of machine learning methods have been developed in the field of concrete structures [67,68]. Among these, particle swarm optimization-BPNN (PSO-BPNN) and GA-SVR are selected for comparison because they can objectively obtain the optimal model hyperparameters and have been widely used [69]. Furthermore, to visually illustrate the importance of hybrid-driven model, both the CNN only method and model-driven method are also selected for comparison. The key parameters of the three data-driven methods (including the GA-BPNN, PSO-SVR, and CNN only method) and the average model performance of different prediction methods are presented in Tab.3 and Tab.4, respectively. It can clearly find from Tab.4 that the HMD method outperforms the traditional model-driven method as well as the commonly used GA-BPNN and PSO-SVR methods in all four evaluation indexes. Additionally, the calculation cost of the HMD method is also significantly lower compared to the GA-BPNN and PSO-SVR methods. This indicates that the proposed HMD method can significantly improve both the accuracy and time efficiency of concrete creep predictions. To more intuitively illustrate the performance advantages of the HMD method over the traditional model-driven method, the relative errors between the predicted and measured values of all test samples in one run are displayed in Fig.4. As depicted in this figure, the relative error of the test samples predicted by the HMD method are mainly concentrated around the zero axis. The average MAPE calculated based on these relative errors is only 0.0253, which is far lower than the corresponding value of 0.2711 for the model-driven method. In other words, the prediction results of the HMD closely align with the measured values overall.

Although both the HMD and CNN only methods have high accuracy from the perspective of the four evaluation indexes in Tab.4, the overall prediction accuracy of the HMD method is superior. Among these four indexes, the MAE is improved most significantly. The MAE value decreases from 2.3627 × 10⁻⁶ to 2.1644 × 10⁻⁶ MPa⁻¹, with a reduction of 8.39%. In addition, the distribution of the estimation errors for the test samples in one run is also provided in Fig.5. This figure illustrates that the estimation errors of concrete creep compliance using the HMD method are mostly concentrated between −4.5 × 10⁻⁶ and 4.5 × 10⁻⁶ MPa⁻¹, whereas the corresponding values of the CNN only method are mostly concentrated between 5.5 × 10⁻⁶ and 5.5 × 10⁻⁶ MPa⁻¹. This further confirms that the HMD method exhibits slightly better accuracy than the CNN only method. Moreover, compared with the CNN only method, the total calculation time of the HMD method is also reduced by 6.32%. Therefore, the HMD method is superior to the CNN only method in both accuracy and time efficiency. This phenomenon can be attributed to the HMD method’s capability to provide target values of creep compliance with high-entropy features as input, thereby reducing the computational complexity of the CNN model’s solution process and improving its accuracy and efficiency of the CNN only method.

4.3 Prediction results of uncertainty prediction model

It can be seen from Eqs. (15) and (16) that the sampling times M has a significant impact on the prediction accuracy of the HMD model when implementing the MC dropout approach. Here, the above case is denoted as the Monte–Carlo dropout-based HMD (MCD-HMD) model, and it is a crucial component of the UPM. For this reason, the effect of sampling times M on model performance is first analyzed to determine its optimal value. As indicated in Tab.5, increasing M will enhance the PICP of the MCD-HMD model. However, this improvement is accompanied by a slightly larger interval width. From the perspective of CWC, increasing M can improve the model prediction performance, but the calculation time will also increase almost linearly. When M increases from 100 to 200 and 400, the variations of PICP, PINAW and CWC increase very slightly, while the calculation cost increases exponentially. Therefore, the value of M is set to 100 in the subsequent analysis.

For the proposed UPM, some influencing factors are not considered in each calculation due to the implementation of the MC dropout approach. This results in a partial degradation of the performance indexes, including MAE, RMSE, MAPE, and R². Nevertheless, these indexes are still significantly better than the calculation results obtained from traditional methods such as GA-BPNN and PSO-SVR, and the value of R² is still as high as 0.9402. On the other hand, the obtained PICP, PINAW and CWC are 0.9541, 0.1094, and 0.1094, respectively. That is, the PICP of the proposed method is slightly larger than the ideal 95% PI. Additionally, the PIs obtained by UPM also have a relatively moderate interval width and a small CWC value. These results indicate that the proposed uncertainty prediction method has a good prediction performance.

The comparison between the mean prediction results and measured values of all testing samples in one run is presented in Fig.6. It can be seen from Fig.6 that the mean prediction results closely align with the measured value overall. The average MAE of all testing samples is only 3.7826 × 10⁻⁶ MPa⁻¹. This means that the proposed method still has high accuracy. In addition, to illustrate the rationality of the proposed method, two sample sets with a large measurement time span from the NU database, as provided in Refs. [70,71], have been selected, as shown in Fig.7. As indicated, all measured data of the two sample sets are concentrated near the mean prediction results and fall within the 95% PI. This also demonstrates the effectiveness of the proposed method.

In summary, compared with the traditional data-driven methods, the proposed prediction method not only ensures relatively good prediction accuracy, but also quantifies the model and measurement uncertainties during the prediction process. This can greatly reduce the contingency of prediction in actual engineering applications and is very beneficial for the timely adoption of maintenance or reinforcement strategies.

4.4 Effect of uncertainty

For the entire filtered NU database of concrete creep, the obtained PICP, PINAW and CWC are respectively 0.9541, 0.1094, and 0.1094 when both the model and measurement uncertainties are considered (i.e. UPM model). These values are 3.43%, 2.82%, and −79.87% higher, respectively, compared to the corresponding values when only the model uncertainty is considered (i.e. MCD-HMD model). Although the interval width slightly increases, the coverage of the prediction interval has been improved to some extent and the CWC has decreased significantly. This indicates that incorporating the measurement uncertainty into concrete creep predictions can improve the prediction performance of UPM.

To further illustrate the effect of measurement uncertainty on concrete creep, a typical data set [72] from the NU database is also used. This data set contains 35 measured data of concrete creep with calculation ages ranging from 0 to 238.9 d. Since these data are all obtained from the same group of test specimens, the input parameter values of each group of data in this data set are consistent except for the calculation age. The detailed information can be found in Ref. [72]. By using the proposed UPM, the uncertainty prediction results of concrete compliance at different calculation ages are obtained, as shown in Fig.8(a). As indicated, when the loading time exceeds 176.83 d, the 95% PI of the UPM model is significantly wider than the 95% PI of the MCD-HMD model, while the difference between the two cases is not obvious in other loading times. Accordingly, the PICP of UPM is up to 100%, which is 8.57% higher than that of the MCD-HMD model. In addition, the probability distribution of the prediction creep compliance at 57.1 and 139.5 d are presented in Fig.8(b). From this figure, one can clearly see that the mean prediction values are close to the measured values, and the probability distribution curves obtained by UPM are more flatter than that obtained from the MCD-HMD model. This is caused by the larger variance of the former when the measurement uncertainty is considered. The above analysis not only verifies the effectiveness of the proposed method, but also demonstrates the necessity of considering the effect of measurement uncertainty.

To further compare the importance degree of the model and measurement uncertainties, the variance composition of the UPM is analyzed by using the data set provided in Ref. [72] and all testing data (as described in Subsection 4.1). As shown in Fig.9(a), the mean contribution of variance attributed to model uncertainty to the total variance are about 90% for the above two data sets. This percentage is significantly higher than that of the measurement uncertainty. This conclusion is also applicable to the analysis of the proportion of dominant effect in Fig.9(b). Therefore, the prediction interval of concrete creep compliance is more sensitive to the model uncertainty.

5 Conclusions

An HMD method for predicting concrete creep is proposed in this paper, and then a novel UPM of concrete creep is also developed. The effectiveness of the proposed method is validated by using the NU database of creep, and the performance evaluation of HMD, the prediction result of UPM and the effect of uncertainty on prediction results are also presented. The following conclusions can be drawn.

1) Compared with the model-driven method and three data-driven methods, including the GA-BPNN, PSO-SVR and CNN only method, the proposed HMD method exhibits the highest accuracy. Furthermore, it demonstrates better time efficiency than the above three data-driven methods.

2) The test results of NU database of concrete creep show that the proposed UPM has a good prediction performance. In this case study, the PICP of UPM is up to 0.9541, which is slightly larger than the ideal 95% PI.

3) Compared with the traditional data-driven method, the proposed concrete creep prediction method not only ensures relatively good prediction accuracy, but also quantifies the model and measurement uncertainties during the prediction process. This can greatly reduce the contingency of predictions in real engineering applications.

4) Incorporating the measurement uncertainty into concrete creep prediction improves the prediction performance of UPM. Nevertheless, the prediction interval of concrete creep compliance is more sensitive to the model uncertainty than measurement uncertainty, and the mean contribution of variance due to the model uncertainty to the total variance is about 90%.

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