A Novel Neural Computing Model Applied to Estimate the Dynamic Modulus (DM) of Asphalt Mixtures by the Improved Beetle Antennae Search

Abstract

To accurately estimate the dynamic properties of the asphalt mixtures to be used in the Mechanistic-Empirical Pavement Design Guide (MEPDG), a novel neural computing model using the improved beetle antennae search was developed. Asphalt mixtures were designed conventionally by eight types of aggregate gradations and two types of asphalt binders. The dynamic modulus (DM) tests were conducted under 3 temperatures and 3 loading frequencies to construct 144 datasets for the machine learning process. A novel neural network model was developed by using an improved beetle antennae search (BAS) algorithm to adjust the hyperparameters more efficiently. The predictive results of the proposed model were determined by R and RMSE and the importance score of the input parameters was assessed as well. The prediction performance showed that the improved BAS algorithm can effectively adjust the hyperparameters of the neural network calculation model, and built the asphalt mixture DM prediction model has higher reliability and effectiveness than the random hyperparameter selection. The mixture model can accurately evaluate and predict the DM of the asphalt mixture to be used in MEPDG. The dynamic shear modulus of the asphalt binder is the most important parameter that affects the DM of the asphalt mixtures because of its high correlation with the adhesive effect in the composition. The phase angle of the binder showed the highest influence on the DM of the asphalt mixtures in the remaining variables. The importance of these influences can provide a reference for the future design of asphalt mixtures.

1. Introduction

To overcome the shortcomings of the empirical pavement design method applied by the American Association of State Highway and Transportation Officials (AASHTO), the AASHTO collaborated with the National Collaborative Highway Research Program (NCHRP) to launch the Mechanistic-Empirical Pavement Design Guide (MEPDG) in 2004 [1,2,3,4,5]. In July of the same year, the available version of the MEPDG design software became available for practical use. Since then, after constant updates and improvements, the latest version of the design program was published in April 2011 [6,7,8,9,10,11]. The MEPDG has been adopted and promoted in more than 40 states in the United States and has attracted widespread attention worldwide [8,11,12,13]. The new design guidelines use mechanical methods to calculate the critical responses (i.e., stress, strain, and deformation) of the pavement structure based on the properties of the layers and climatic conditions [14,15,16,17,18,19,20,21,22,23]. The empirical approach is designed to bridge the gap between laboratory tests and field performance, which in turn is used to reflect the actual level of local construction and other factors of variability [8,24,25,26,27,28,29,30,31,32]. MEPDG can consider uniformly the design of the flexible, rigid, and composite pavement and use common traffic, subgrade, environmental, and reliability design parameters to predict a wide range of pavement performance and to link materials, pavement structure design, construction, climate, traffic, and pavement management systems [10,11,12]. Compared with the traditional empirical method, the mechanistic-empirical method is another innovation of pavement design theory [6,10]. The dynamic properties of the asphalt mixtures are indispensable parameters employed in the MEPDG and it is typically characterized by the dynamic modulus determined by varying loading conditions. To estimate the dynamic properties of the asphalt mixtures, various mathematical or empirical models have been adopted in the past years to reduce the expensive and time-consuming laboratory tests [5,9,33,34,35]. The so-called Witczak model has been adopted in the MEPDG model as the empirical part. Nevertheless, the accuracy of this model has been doubted in some of the early studies, especially when the loading temperatures are lower [5,10]. To overcome the drawbacks of mathematical or empirical regression models in the MEPDG, some more complex models using machine learning techniques should be taken into consideration to establish more advanced models to accomplish the “empirical” part of the MEPDG pavement design [36,37,38,39,40,41,42,43,44,45]. Artificial intelligence (AI) models have been adopted in recent years to construct the relationship between the input parameters of the construction and building materials (e.g., asphaltic mixtures and cementitious materials) and the mechanical performances [45,46,47,48,49,50]. Varying machine learning models have been employed to predict the properties and performances of the reinforced concrete (RC) slab [51]. Compared to the prediction model based on mathematical regression, the prediction model based on the artificial neural network has higher accuracy and smaller deviation [52,53,54,55,56,57]. Various research has been conducted to develop other AI models to estimate dynamic properties [45,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73].

Regarding those artificial intelligence technologies, the backpropagation neural network (BPNN) is widely used to better predict the mechanical properties of cement concrete (or DM of asphalt mixture) [74,75]. The main reason for this feature is that the BPNN can estimate arbitrary functions stably and efficiently because a large number of parallel operations provide a high computation rate [48,65,76,77,78,79,80]. Although it has been widely used, the BPNN model design is still carried out through a laborious trial and error method. To find the optimal structure of the BPNN, it needs to spend a lot of time and energy to solve the problem [81]. The more complex the problem, the more time and effort will be needed. Therefore, this study proposed for the first time that the improved beetle antenna search (BAS) algorithm be adopted to adjust the BPNN architecture for the DM prediction of asphalt mixture, which has the advantages of simple implementation, fast convergence speed, and little possibility of falling into local optimization by changing the step size strategy [82,83]. The number of hidden layers, the number of neurons in each hidden layer, and the connection weights are adjusted through this meta-heuristic approach to minimize the time and effort required [83]. Due to the importance of dynamic characteristics in the prediction of pavement performance in MEPDG, the main research objective of this study is to predict the DM of asphalt mixture through the critical influence variable. Based on the improved BAS algorithm and combined with the influence variables, a new DM prediction neural network calculation model for asphalt mixture was proposed. This pioneering work provides a new method for the prediction of DM of asphalt mixtures.

2. Materials and Methods

The asphalt mixtures of 8 different types selected in this study were applied directly from those conventional ones used in industrial production. The information on the nominal maximum aggregate sizes, passing or retained percents for different sieves (including the percent passing #200, retained on #4, retained on #3/8), etc. can be found in the previous study.

2.1. Raw Materials

A base and a modified asphalt binder were used for the mixture design. Granite stones were employed as aggregates in the composition of the asphalt mixtures. Table 1 presents the basic properties of the raw materials employed in this study. The detailed physical and index properties of the raw materials can be found in the early studies [84].

2.2. Mix Design

The Superpave (SUperior PERforming Asphalt PAVEments) design method for the asphalt mixtures was adopted for the combinations of two binders and eight aggregate gradations to determine the optimal asphalt binder contents, considering the volumetrical and physical properties of the asphalt mixtures [85,86,87,88]. The asphalt mixture was compacted with a design gyration number (which was 75 times) to the standard sample with a diameter of 100 mm. Table 2 summarizes the asphalt mixtures employed in this research (in which Va is the air voids in the asphalt mixture).

2.3. Experimental Tests

Two types of dynamic tests (one is for the asphalt binders and the other is for the asphalt mixtures) were conducted in this study. The detailed procedures are introduced as follows.

2.3.1. Determination of the Dynamic Shear Modulus of the Asphalt Binders

The dynamic shear rheometer (DSR) was employed in the present research to determine the dynamic properties of the asphalt binders. The test was conducted under 80 combinations of testing conditions (including 6 temperatures and 16 frequencies). The master curves of the dynamic shear modulus and phase angle are constructed by using these 80 dynamic loading conditions. Then, the dynamic shear modulus and phase angle at the specific loading frequency and temperature were determined to be used for the prediction in the machine learning process.

2.3.2. Determination of the DM of the Asphalt Mixtures

In this study, the asphalt mixture performance tester was employed to determine the DM of the asphalt mixtures, and it was conducted according to the standard AASHTO TP62 [89]. For one type of asphalt mixture, three specimens should be produced to arrive at the target void. Then, the sample was cut with a core diameter and height of 100 mm and 150 mm, respectively, to make sure that the sawed surface was smooth, the end was uniform, and the surface unevenness was minimal. The sampled instrument is composed of the called standard glue-point system. In this study, 6 measurement points were selected on one sample.

2.4. Methods

In the present study, a new DM prediction neural network calculation model for asphalt mixture was proposed based on the improved BAS algorithm and combined with the influence variables. The detailed description of the methods is as follows.

2.4.1. Backpropagation Neural Network (BPNN)

The neural network can simulate the human brain neurons, and forms a neural network by connecting them according to man-made rules, to realize the processing and storage of complex information. Stimulated by the input of data samples, it constantly iterates and updates, thus changing and adjusting the connection weights and thresholds of the network, so that the output results are closer to the expected value. The backpropagation neural network (BPNN) is one of the most widely used neural network models. It is a multi-layer feed-forward network with error backpropagation [90].

A BPNN is divided into the input layer, hidden layer (one layer or multiple layers), and output layer, as shown in Figure 1 [76,77]. In this study, 7 groups of samples including Vbeff, Va, ρ200, ρ4, ρ3/8, G*, and δ were included in the input layer and the expected output was set to DM to achieve the nonlinear mapping between input variables and output variables. In this determination process, the relationship between the error and the weighting coefficient was obtained to find the change law of the weighting coefficient. It should be noted that each sample consists of input information (Vbeff, Va, ρ200, ρ4, ρ3/8, G*, and δ) and expected output (DM).

The main function of the input layer is to accept the external characteristic parameters and pass the parameter information to the neurons of the hidden layer after processing according to the prescribed rules. The neurons in the hidden layer also process the information and transmit it to the next layer and finally to the output layer, which is the process of forwarding propagation. When there is a large error between the output and the actual value, the weight of each layer is corrected from the output layer according to the method of gradient descent, and then propagated to the hidden layer and the input layer, which is the backpropagation process of the neural network [45,48,65,79]. In the continuous alternating process of forwarding propagation and backpropagation, the weights of each layer of the neural network are constantly revised until the error of the output result is less than the expected error, and the learning and training of the neural network are finished [48,78].

The number of neural network layers has a direct relationship with the training duration, training accuracy, and efficiency. With the increase of neural network layers, the error can be reduced and the accuracy of prediction can be improved, but it also leads to a more complex network and increased training time. However, the error reduction can be achieved by increasing the number of neurons, which is easier than increasing the number of layers. When the increase in the number of neurons still fails to meet the accuracy requirements, the increase in the number of layers should be considered. There is no relevant theory to determine the number of hidden layers. The common method is to first establish a neural network of hidden layers and increase the number of neurons to improve accuracy. If it is not enough to add another hidden layer since blindly increasing the hidden layer may greatly increase the training time. A neural network containing a hidden layer of the transfer function and an output layer of a linear function can be used to simulate arbitrary nonlinear functions. Therefore, in most current studies, a BPNN with one or two hidden layers is mostly used. In the present study, 1 to 4 hidden layers were determined according to the trail tuning results.

2.4.2. Improved BAS Algorithm

To improve the computing efficiency and the reliability of the computing model, an improved BAS model was proposed in the study, which is detailed as follows.

Traditional BAS Algorithm

Unlike colony insects such as ants and bees, which rely on external information and survival information obtained from the communication between large groups, beetles can find enough food to survive only by themselves [82]. The beetle antennae search (BAS) algorithm is based on biological and foraging characteristics. During the foraging process, the beetle mainly senses the outside world according to its two antennae, which is equivalent to having two information receivers in different directions. The two antennae have unique nerve cells, and whenever the beetles feel that the odor signal is strong or weak, they will move in the direction of the tentacles with greater odor intensity, and finally achieve fast foraging. Based on this, the BAS simplified the beetle individuals and regarded them as particles capable of sensing left and right directions. In each search process, the direction of the head of the beetle is random, so the right antennae will generate a direction vector pointing to the left one. The length of the antennae is closely related to the size of the longhorn beetle, and the distance between the two antennae can be used as a parameter to measure the size of the individual. Regarding the longhorn beetle as a feasible solution for the function to be optimized, the position of the longhorn beetle is the code of the feasible solution [82].

In the space with a multidimensional BAS algorithm, the global optimum equation can be described as follows:

(1)$\frac{\mathrm{Minimize}}{\mathrm{Maximize}}f\left(x\right),x={\left[x_1,x_2,\cdots ,x_N\right]}^{\mathrm{T}}$

where $x_1,x_2,\cdots ,x_N$ represent the position of the beetle’s center of mass. To characterize the searching behavior of the beetle, the following equations should be defined:

(2)$b=\frac{rnd\left(k,1\right)}{\left|\right|rnd\left(k,1\right)\left|\right|}$

(3)$x_r=x^i+d^{i}b$

(4)$x_l=x^i-d^{i}b$

where $rnd$ is the random function. $k$ is the dimension of the problem to be optimized. $x_r$ and $x_l$ represent the calculation of the left and right whiskers. The detecting characterization of the beetle can be determined as follows:

(5)$x^{i+1}=x^i+{\delta }^{i}b·\mathrm{sign}\left(f\left(x_r\right)-f\left(x_l\right)\right)$

where the $\mathrm{sign}$ is the sign function and δⁱb is the step of the beetle at time t. The equation used to update the step size can be determined as follows:

(6)${\delta }^{i+1}=\eta {\delta }^i$

Improved BAS with Higher Searching Efficiency

For traditional BAS, the beetle step size is constant or decreases with each iteration, leading to problems in the step size adjustment strategy. When the given step size is too small, the BAS algorithm may converge slowly or be in the local optimal state. If the step size is very large, global optimality may be missed and the results may oscillate. Therefore, in this study, Levy flight and autoinertial weight were used to adjust the BAS step size and improve it into an improved BAS algorithm (called the MBAS algorithm). The MBAS algorithm can achieve the following objectives: first, the step size can be adjusted quickly according to the current fitness value, and the adaptive weight can be used to reduce the oscillation; secondly, when the BAS is locally optimal, Levy flight is used to increase step size randomly. In the traditional BAS algorithm, the step size of the beetle is constant or decreases in each iteration, which makes the BAS algorithm easily fall into the local optimal state. To solve this problem, Levy flight is adopted in the BAS algorithm to adjust step size.

In the present study, when the local optimal solution of the BAS algorithm is obtained, the step size in the calculation can be increased by the following equation:

(7)${\delta }^{\left(i\right)}=\alpha \left|Levy\right|\otimes {\delta }^{\left(i-1\right)}$

where $\alpha$ can be used to describe the randomization; $\otimes$ can be used to represent the entrywise multiplications; $\left|Levy\right|$ is used to describe the so-called Levy distribution with infinite ($Levy~u=t^{-\lambda },(1<\mathsf{\lambda }\le 3)$) variance.

In the present study, the adaptive inertia weight is described by the monotone reduction equation as follows:

(8)${\delta }^{i+1}={\eta }^i\times {\delta }^i$

where ${\delta }^i$ can be used to describe the step size at the present position. ${\eta }^i$ can be employed to describe the inertial weight adaptively, which is determined as:

(9)${\eta }^i=\left(1-\alpha \right)0.95+\alpha \frac{f_w^i-f^i}{f_w^i-f_b^i}$

where $f^i$ is used to describe the fitting equation at the present position; $f_b^i$ is used to characterize the optimized fitting values. In this study, α was determined as 0.2 after the preliminary trial calculations. The pseudo-code of the MBAS algorithm (which is modified by the Levy flight and the adaptive inertia weight) is given in Algorithm 1. As shown in the figure, for an n-dimensional space optimization problem, $x_l$ was used to represent the left whisker coordinate, $x_r$ was used to represent the right whisker coordinate, and $x_{ }$was used to represent the centroid coordinate. The head is arbitrary, so the vector from the right whisker to the left whisker is arbitrary, so a random vector can be generated to represent it. The expression of the center of mass can be obtained by the normalization of the random vector. For the function to be optimized, the values of the left and right whiskers are obtained, and then the cycle is iterated.

Algorithm 1. Pseudo-code of the MBAS algorithm [84]

Input Fitness function f(x^i), initial position of the beetle x^0, initial step-size δ^0, maximum iterations n, ratio of antennae length to step-size c, attenuation coefficient of step-size η

Output: Optimal position x_b, optimal fitness function value f_b.

FOR I = 1 to n

Generate random antennae direction b;

Calculate the antennae length d^i = c × δ^i;

Calculate the left-hand and right-hand positions x_l and x_r, respectively;

Calculate the fitness function value f(xl) and f(xr) at the left and right antennae position;

Calculate the next position x^i;

Calculate the fitness function value f(x^(i + 1) ) at next position x^(i + 1);

IF f(x^(i + 1)) < f_b

THEN Update x_b to x^(i + 1);⋯Update f_b to f(x^(i + 1) );

END

Update step-size δ^(i + 1) using Equations (8) and (9);

IF |f (x^(i + 1)) − f (x^i)| < μ (fw − fb)

THEN Update step-size δ^(i + 1) using Levy flight according to Equation (9);

ELSE Update step-size δ^(i + 1) according to Equation (8)

END

i = i + 1;

END

3. Results and Discussion

The testing results obtained from the proposed model are analyzed in this section. Those data for the machine learning were characterized by the correlation coefficients. The results of the predictive effect and hyperparameters tuning are presented as well. The detailed results and discussion are shown as follows.

3.1. Testing Results and Dataset Description

Figure 2 gives the results of the DM of the asphalt mixtures.

It can be observed from Figure 2 that nearly all the predicted DM of the asphalt mixtures were higher than the measured ones, especially when the testing temperatures were lower (see those points characterizing higher DM). It is evident that the previous Witczak 1-40D model can overestimate the DM of the asphalt mixtures and the degree of overestimation is more significant in the low-temperature region. This conclusion is similar to previous studies by other scholars, suggesting that an AI technique to predict DM is urgently needed instead of mathematical regression.

Using this dataset, a correlation study between different input variables was conducted. Figure 3 shows the correlation matrix of the varying input variables, including Vbeff, Va, ρ200, ρ4, ρ3/8, G*, and δ, which are also used as the input parameters in the Witczak model. The element in row i and column j of the correlation matrix is the correlation coefficient between column i and column j of the original matrix.

It can be seen from Figure 3 that the correlation coefficients of most input variables (except the phase angle and shear dynamic modulus of the binder) are lower than 0.5, demonstrating that these elements (including Vbeff, Va, ρ200, ρ4, ρ3/8, G*, and δ) will not raise the issue of multicollinearity for the future machine learning process. It should be noted that the strong relationship between the phase angle and shear dynamic modulus is due to the viscoelastic property of the asphalt binder. However, such a strong correlation of only two elements is acceptable in the AI prediction. Hence, Vbeff, Va, ρ200, ρ4, ρ3/8, G*, and δ can be used to predict the dynamic modulus by the AI techniques for the next step.

3.2. Hyperparameter Tuning Results

To determine the optimized structure of the BPNN model, the hyperparameters were tuned by the 10-fold cross-validation by evaluating the RMSE (root mean square error) values obtained from the testing dataset. After the 10 folds of cross-validation during the hyperparameter tuning process, the relationship between the iteration and RMSE values for varying numbers of hidden layers was obtained, as shown in Figure 4.

A significant decrease in RMSE can be observed through one iteration, indicating that the BAS algorithm can effectively adjust the BPNN model to predict the DM of the asphalt mixtures, using the selected input parameters. Afterward, the RMSE value gradually decreases until the 19th iteration to arrive at the optimal BPNN structure for the prediction. At the same time, regarding 70% of the training datasets, BAS was used to adjust the connection weights and deviations of BPNN with a fixed structure by the 10-fold CV. In each of the 10 rounds, the BAS algorithm iterates 100 times and the obtained minimum RMSE value corresponds to the optimal weight and deviation value of this round. It is worth noting that the predictive results of the BPNN model should be validated by evaluating the testing dataset of the DM of asphalt mixtures. The RMSE values obtained in each round of hyperparameter tuning processes are shown in Figure 5.

It can be seen that the minimum value of RMSE was obtained in the fourth round of the hyperparameter tunning process. Therefore, the connection weights and deviations of the fourth round were selected as the initial weights and deviations of the BPNN model to predict the DM of the asphalt mixtures using the Witczak input parameters. It is suggested to employ four hidden layers for the prediction. The number of neurons in each hidden layer is 6, 5, 10, and 6, respectively. The optimal stable structure of the BPNN model will be used for the further prediction of DM of the asphalt mixtures.

3.3. Predictive Performance of the Proposed Model

Figure 6 and Figure 7 give the predictive results of the training and testing datasets for the DM of the asphalt mixtures, respectively. Among the datasets completed in the laboratory, 70% (100 groups of datasets) were employed for the training process in machine learning, and 30% (44 groups of datasets) for the testing process. The histograms in Figure 6 and Figure 7 represent the difference between the measured and predicted dynamic moduli of the asphalt mixtures, while the horizontal lines indicate that the predicted results are in complete agreement with the measured results.

It can be seen from Figure 6 and Figure 7 that the predicted DM of asphalt mixture is close to the experimental results, which indicates that the improved BAS algorithm can effectively adjust the hyperparameters of the BPNN model and effectively learn the correlation between input variables (including Vbeff, Va, ρ200, ρ4, ρ3/8, G*, and δ) and output results for the prediction. To quantitatively evaluate the predictive performance of the BPNN model using the improved BAS algorithm, the comparison results were also shown in the form of a scatter chart (Figure 8) with a “1:1” curve.

It can be seen that the predicted DM of the asphalt mixtures is in good agreement with the measured DM, indicating that this method can well establish the nonlinear relationship between the input variables (including Vbeff, Va, ρ200, ρ4, ρ3/8, G*, and δ) and the DM of the asphalt mixtures. The statistical parameters for these comparisons between the training and testing datasets were obtained as well, as shown in Figure 8. The relatively low RMSE values (which were 1.2943 and 1.8148 for the training and testing datasets, respectively) and high correlation coefficient (R) values (which were 0.9725 and 0.9469 for the training and testing datasets, respectively) were obtained in the machine learning process. These results showed that the proposed MBAS-BPNN algorithm does not have the issue of overfitting when it was used to predict the DM of the asphalt mixtures. Additionally, the proposed model can be employed to replace those conventional mathematical regression models (e.g., the Witczak 1-40D model) to predict the DM of the asphalt mixtures.

3.4. Importance of the Input Variables

The sensitivity analysis was performed in the present study to determine the importance of the variables (including Vbeff, Va, ρ200, ρ4, ρ3/8, G*, and δ), referring to an uncertain determination to study the influence degree of a certain change of relevant factors on a certain or a group of key indicators from the perspective of quantitative analysis. The essence of this method is to explain the law of the influence of these factors on key indicators by changing the value of related variables one by one. Figure 9 gives the importance scores of these influential variables using the sensitivity analysis.

In Figure 9, the value corresponding to each influential variable indicates its importance score. For a parameter with a higher importance score, it means that its change has a greater impact on the DM of the asphalt mixtures. It can be seen from Figure 9 that the dynamic shear modulus of the asphalt binder is the most important parameter that affects the DM of the asphalt mixtures. This can be due to the adhesive effect of the asphalt binder in the composition of the mixtures correlates strongly to the shear modulus of the binder while such adhesive effect may significantly determine the DM of the asphalt mixtures. Regarding the remaining variables, the phase angle of the binder showed the highest influence on the DM of the asphalt mixtures. This may be determined by the close relationship between the shear dynamic modulus and the phase angle of the asphalt binder. For conventional asphalt mixtures, the effect of Va on the DM of the asphalt mixtures should be significant (e.g., one can compare the DM of the open-grade and dense-grade asphalt mixtures). However, it should be noted that the difference in Va of different asphalt mixtures in this study was not large. Such a small difference cannot have a great impact on the DM of asphalt mixture, resulting in a low importance score of Va. Similarly, this is why the importance score of Vbeff is not high.

4. Conclusions

The present study aims to evaluate and predict the DM of the asphalt mixtures, one of the most important parameters in the MEPDG, using a novel proposed neural computing technique. In the laboratory, the asphalt mixtures were designed with eight types of aggregate gradations and two types of asphalt binders, and then the DM tests were conducted under 3 temperatures and 3 loading frequencies to construct 144 datasets for the machine learning process. The prediction model was developed by using an improved beetle antennae search (BAS) algorithm to tune the hyperparameters of the neural computing model more efficiently. The predictive performance of the proposed model was evaluated by R and RMSE and the importance score of the input parameters was assessed as well. The conclusions obtained can be highlighted as follows.

By using Levy flight weight and inertia weight, the BAS algorithm can be improved to avoid premature convergence to local optimum, thus improving the search efficiency of the traditional BAS algorithm. The convergence speed of the algorithm is fast and the RMSE is significantly reduced, which proves the effectiveness and accuracy of the improved method. The improved BAS algorithm can effectively tune the hyperparameters of the neural computing model to construct the predictive model for the DM of the asphalt mixtures, demonstrating higher reliability and validity compared to random hyperparameters selection.

The predicted results showed that the developed hybrid model can accurately be used to evaluate and predict the DM of the asphalt mixtures to be employed in the MEPDG. The relatively low RMSE and high R values were obtained for the learning and training process in the machine learning process, proving the good predictive performance of the developed model used in the future mix-design process.

The significance analysis showed the dynamic shear modulus of the asphalt binder is the most important parameter that affects the DM of the asphalt mixtures because of its high correlation with the adhesive effect in the composition. The phase angle of the binder showed the highest influence on the DM of the asphalt mixtures in the remaining variables. The importance of these influences can provide a reference for the future design of asphalt mixtures.

For future development, a comparative study of different algorithms should be carried out to verify the most effective and reasonable algorithm to predict the DM of asphalt mixtures. Additionally, due to the limitation of experimental tests carried out in the laboratory, 144 datasets in this study are used in the process of machine learning. Future research should try more data (including different aggregate gradations and different asphalt binders) to provide a more accurate and reliable DM prediction model for asphalt mixtures.

Author Contributions

Conceptualization, J.H. and M.Z.; methodology, J.H. and M.M.S.S.; writing—original draft preparation, J.H. and M.Z.; writing—review and editing, M.M.S.S. and M.Z.; visualization, M.Z.; supervision, J.H. and H.Y.; funding acquisition, M.M.S.S. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

MEPDG	Mechanistic-Empirical Pavement Design Guide
BAS	Beetle antennae search
DM	Dynamic modulus
AASHTO	American Association of State Highway and Transportation Officials
PI	Penetration index
AI	Artificial intelligence
RC	Reinforced concrete
BPNN	Backpropagation neural network
Superpave	SUperior PERforming Asphalt PAVEments
Va	Air voids
DSR	Dynamic shear rheometer
RMSE	Root mean square error
R	Correlation coefficient

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. BPNN model.

Figure 2. DM results of the asphalt mixtures as well as the comparison with the Witczak 1-40D model.

Figure 3. Correlation matrix of the varying input variables.

Figure 4. Relationship between the iteration and RMSE.

Figure 5. RMSE results of the hyperparameter tuning.

Figure 6. Predicted DM of the asphalt mixtures: training dataset.

Figure 7. Predicted DM of the asphalt mixtures: testing dataset.

Figure 8. Measured and predicted values of the DM of the asphalt mixtures.

Figure 9. Importance scores of the influential variables.

Table 1

Basic properties of the raw materials.

Basic Properties		Binder 1	Binder 2	Aggregates
Binder-index properties of asphalt	Penetration @ 25 °C (0.1 mm)	92.0	75.1	-
	Penetration index (PI)	−1.17	0.2	-
	Softening point (°C)	44.0	52.5	-
	Viscosity @ 135 °C (Pa∙s)	0.363	1.3	-
Physical properties of aggregate	LA abrasion value (%)	-	-	23.0
	Aggregate impact value (%)	-	-	22.0
	Water absorption (%)	-	-	0.14
	Combined elongation and flakiness indices (%)	-	-	28.0
	Soundness, magnesium sulfate solution (%)	-	-	0.4

Table 2

Summary of the asphalt mixtures used in the study.

Asphalt Mixtures	Nominal Aggregate Size (mm)	Binder Content (%)	Va (%)
Asphalt Mixture-1	19	4.5	4.0
Asphalt Mixture-2	19	4.4	4.0
Asphalt Mixture-3	19	4.8	4.0
Asphalt Mixture-4	19	5.0	4.0
Asphalt Mixture-5	12.5	5.0	4.0
Asphalt Mixture-6	12.5	5.3	4.0
Asphalt Mixture-7	12.5	5.5	4.0
Asphalt Mixture-8	12.5	5.4	4.0

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