Application of the ANN with swarm optimization algorithm in layer property backcalculation for asphalt pavement

State-of-the-art techniques for pavement performance evaluation have attracted considerable attention in recent years. Artificial Neural Networks (ANNs) can simulate the human brain to discover hidden patterns within datasets, thereby enhancing the accuracy of performance evaluations in civil engineering. Routine backcalculation of layer properties from Falling Weight Deflectometer (FWD) deflection time histories provide critical information for asphalt pavement performance assessment. This study proposes a general framework that integrates the ANN with swarm optimization algorithms for asphalt pavement layer property backcalculation at the project level. The proposed procedure employed the spectral element method (SEM) to calculate the deflection time histories, explicitly accounting for the dynamic effect of FWD loading and the viscoelastic property of asphalt concrete (AC). The geometrical characteristic of the load time history and the load-deflection hysteresis curve were extracted to construct the training dataset for the ANN model. Three swarm optimization algorithms were utilized to determine the initial weights and biases of the ANN. Subsequently, the parameters of the Williams-Landel-Ferry (WLF) function were optimized using the field temperatures and the backcalculated layer property to characterize the temperature-dependent behaviour of AC. A well agreement between the backcalculated and measured deflection time histories, together with the consistency of the backcalculated layer properties within reasonable ranges, demonstrated the feasibility of the proposed procedure. In contrast to conventional backpropagation neural networks (BPNNs), whose built-in optimization schemes tend to become trapped in local optima and may yield unreasonable layer properties, swarm optimization algorithms expand the candidate solution space and facilitate global optimization. The standard deviation (STD) and coefficient of variation (COV) obtained from ANNs enhanced with swarm optimization are significantly lower than those from BPNNs; the maximum reduction in COV for ANN + HPO (Hunter–Prey optimization) relative to BPNN exceeds 50%. The ANN + MA (Mayfly Algorithm) and ANN + HPO exhibit superior performance and greater computational efficiency for layer property backcalculation compared with a conventional ANN model.

Introduction

Asphalt pavement is a major infrastructure component. The structural performance evaluation and routine rehabilitation play an important role in extending the service life and enabling rational budget planning. The property of the paving material is critical for structural assessment. Various non-destructive techniques have been developed for monitoring the quality of paving materials under different conditions. Wang et al. (2020) developed various gauge length-based optical fiber Bragg grating sensing techniques for monitoring of large-span pavement through laboratory and field tests, which can serve as accurate tools for condition assessment. Zhang et al. (2025a) and Zhang et al. (2025b) proposed a quasi-distributed fiber optics system to collect the strain distribution curves of small asphalt beams subjected to fatigue three-point bend test and freeze–thaw cycling tests, providing precise identification of crack initiation and propagation. The falling weight deflectometer (FWD) applies an impulse load by dropping a specific mass from a given height to simulate high-speed traffic loading. It is widely accepted as a reliable apparatus for assessing pavement structural conditions (Abd El-Raof et al., 2018). The measured deflection was used to evaluate the structural bearing capacity or backcalculation of the layer property through backcalculation algorithms.

The layer property backcalculation procedure was composed of the forward analysis engine and nonlinear optimization. The theoretical deflection was computed based on the pavement mechanics model with the forward analysis engine. Considering the geometric property of the FWD load and the pavement structure, an axisymmetric model is generally adopted for the forward analysis. Various analytical models and numerical procedures have been employed in the pavement analysis under FWD loading. The multilayer elastic theory (MLET), derived from Boussinesq’s equation, has been implemented in forward and backcalculation programs, such as BISAR and EVERSTRESS. However, these multilayer analysis programs assume the static load and linear elastic behavior of all layers (Zhao et al., 2015). The dynamic nature of the FWD load and the viscoelastic property of asphalt concrete (AC) result in a significant discrepancy between theoretical simulations and field performance. The layered elastic solution was extended to the viscoelastic medium by considering the material’s viscoelastic behavior, dynamic loading effects, or both. Al-Khoury et al. (2001a) and Al-Khoury et al. (2002) developed the layered media dynamics analysis (LAMDA) procedure based on the spectral element method (SEM), where the governing equation was solved in the frequency–wavenumber domain via Fourier transform. The major advantage of the SEM is that the mass distribution can be accurately simulated using a single layer in the frequency domain, thus significantly enhancing the computational efficiency. Lee (2014) proposed a dynamic viscoelastic analysis procedure based on the Laplace–Hankel transforms, where the Prony series was used to simulate the viscoelastic properties of AC. Zhao et al. (2015) and Zhao et al. (2013) introduced the modified Havriliak–Negami (MHN) model that characterizes the linear viscoelastic behavior of AC and performed the dynamic viscoelastic analysis of asphalt pavement under FWD impulse loading. Varma and Kutay (2016) proposed a method for asphalt pavement based on the elastic–viscoelastic correspondence principle. Quan et al. (2022) used the fractional viscoelastic model for parameter identification and then implemented the model into the finite-element (FE) analysis through a user-defined material subroutine. The accuracy and efficiency of the forward analysis are major concerns since it must be run numerous times during layer property backcalculation.

The backcalculation process is a nonlinear optimization problem, where the difference between the measured and theoretical deflections is the objective function (Egan and Hall, 2015; Cao et al., 2020). The iteration-based algorithm is a widely used method in traditional procedures. The layer properties are initially predetermined and then updated iteratively until the difference meets the termination criterion. The major concern in the iteration is that the seed value and the optimization strategy have a marked influence on the performance of the procedure. The famous backcalculation software EVERCAL (Lee et al., 1988) used the MLET as a forward analysis engine and the “search and expand” approach for optimizing layer moduli. Al-Khoury et al. (2001b) documented that the Powell hybrid algorithm in addition to the SEM forward analysis exhibits a better performance in layer property backcalculation than the factored Secant update method and the modified Levenberg–Marquardt method. Zaabar et al. (2014) used Lee’s method in forward analysis and a hybrid approach that combined the genetic algorithm (GA) and the Levenberg–Marquardt algorithm in the optimization process. Varma and Kutay (2016) utilized the GA along with the Nelder–Mead simplex algorithm for backcalculation of the viscoelastic properties of AC and the nonlinear property of unbound materials. The mechanic behavior of the AC is temperature-dependent, and a temperature correction of the backcalculated layer property is necessary to accurately characterize the material behavior under specified temperatures. Various models have been developed for correcting pavement deflections and backcalculated moduli for temperature. Kim et al. (1995) presented a temperature correction procedure for deflections and backcalculated the AC moduli for flexible pavements based on tests conducted in North Carolina. Chou et al. (2017) developed a temperature adjustment model using only the surface temperature through a regression routine. Zheng et al. (2019) proposed a quadratic factor for deflection adjustment. The time–temperature superposition principle or a logarithmic function was used for the temperature correction of the backcalculated modulus for asphalt layers (Park et al., 2001; Le et al., 2023). Le et al. (2023) proposed a temperature correction factor based on the field backcalculated elastic modulus and the laboratory test dynamic modulus. Ashtiani et al. (2025) developed a temperature correction method for adjusting the backcalculated asphalt modulus based on the field airfield pavements. The temperature correction models mentioned above primarily address deflection peaks or backcalculated elastic moduli; temperature correction for backcalculated complex moduli, however, remains unresolved.

With the advancement of artificial intelligence (AI), artificial neural networks (ANNs) have garnered considerable attention in recent years. The superiority of the ANN lies in their ability to discover knowledge and hidden underlying patterns, especially in large-scale datasets. Various ANN models and the deep learning techniques have successfully been implemented in infrastructure assessment. The ANN architecture consists of input layers, hidden layers, and an output layer. The primary strength of ANNs is their ability to establish the mapping between input and output variables. The trained mapping relationship can be used to simultaneously predict the objectives, which is more appropriate for large-scale engineering problems. Saltan et al. (2013) first utilized an ANN framework in pavement assessment to simultaneously determine layer parameters including elastic modulus, Poisson’s ratio, and thickness of each layer. Zhang et al. (2024) introduced a hybrid intelligent model to predict the central point’s response based on peripheral gratings, which could significantly improve the precision of health monitoring for high-speed railroads. The combinations of various neural networks, such as the Kolmogorov–Arnold Network (KAN) fused with the transformer fusion model, informer model, and long short-term memory (LSTM) networks, have been developed to enhance the robustness and adaptability in large-scale data analysis (Zhang et al., 2025c; Zhang et al., 2025d). Two strategies have emerged from the principles underlying the application of ANNs in layer property backcalculation. The first strategy is to use the ANN as a surrogate forward analysis engine to simulate structural behavior, and the layer property or pavement response was predicted following an iteration-based method (Li and Wang, 2018; Fathi et al., 2023). The more widespread strategy involves taking the measured deflections as input variables to construct an inverse mapping between deflections and structural parameters. The core challenge in application of the ANN framework is selecting appropriate hyperparameters. The gradient descent algorithm is commonly used to optimize these hyperparameters by moving them in the direction opposite to the gradient of the objective function, which is the most efficient direction to update parameters. Although gradient descent offers superior optimization efficiency, it is prone to converging to local optima rather than global optima. Li and Wang (2019) utilized the GA to optimize ANN parameters and performed backcalculation of the layer property based on surface deflection, where only peak deflection was used. Wang and Zhao (2022) implemented particle swarm optimization (PSO) in the ANN framework to predict the layer property and bedrock depth. Cao et al. (2025) combined the mayfly algorithm (MA) and ANN to perform layer property backcalculation at the project level. Project-level pavement assessment requires backcalculation of layer properties at multiple test stations having similar structural information; therefore, the ANN-based procedure is particularly suitable for large-scale layer-property backcalculation.

To overcome the local optima of the ANN and validate the performance of multiple swarm intelligence optimization methods in layer property backcalculation, we propose a universal ANN-based framework for layer property backcalculation at the project level. The main components of the proposed framework include deflection geometric characteristic database generation, ANN framework construction, hyperparameter optimization, layer property backcalculation, and temperature correction. The architecture of the framework is determined based on the structural information and the test configuration. Population-based optimization algorithms, namely, PSO, MA, and hunter–prey Optimization (HPO), are incorporated into the neural network to determine the initial weights and biases for each neuron in the ANN structure. The optimal weights and biases are then obtained through ANN training using the built-in backpropagation algorithm. The temperature-related parameter for the viscoelastic material is determined through an optimization process instead of temperature modification on the measured deflection. A theoretical three-layer flexible pavement is employed to investigate the feasibility and the robustness of the ANN procedure. A field flexible pavement with 50 FWD test stations is further used to evaluate the performance of the population-based optimization algorithms in the layer property backcalculation.

Methodology

Dynamic analysis of asphalt pavement

The asphalt pavement behavior under the FWD was modelled as an axisymmetric problem. The FWD loading was assumed to be uniformly distributed in a circular area with the radius of 0.15 m. The multilayered pavement consists of layers with a finite thickness above the infinite half-space. All layers are homogeneous, isotropic, and horizontally infinite. The asphalt layer is modelled as linear viscoelastic and other layers as linear elastic. Considering the linear viscoelastic property of AC and the dynamic effect of FWD loading, the spectral element method (SEM) was utilized as the forward analysis engine to simulate the structural behavior of asphalt pavement (Zhao et al., 2015). The governing equation for SEM in terms of displacement is shown in Equation 1.

$\left(\lambda +\mu \right)\nabla \left(\nabla ·\mathbf{U}\right)+2\mu {\nabla }^2\mathbf{U}=\rho \ddot{\mathbf{U},}(1)$

where $\lambda =\nu E/\left(\left(1+\nu \right)\left(1-2\nu \right)\right)$ and $\mu =E/\left(2\left(1+\nu \right)\right)$ are Lamé constants for materials related to the modulus $E$ and the Poisson’s ratio $\nu$; $\mathbf{U}$ represents the displacement vector of the materials; $\nabla$ is the differential operator; $\nabla ·\mathbf{U}$ and ${\nabla }^2\mathbf{U}$ are the divergence and the Laplacian of the displacement vector $\mathbf{U}$, respectively; dots above the symbol represent the second differentiation of the vector with respect to time; and $\rho$ is the density of materials.

In SEM, the linear viscoelastic property of AC was characterized by the modified MHN model (Zhao et al., 2013), as shown in Equation 2. Five parameters of the MHN in addition to the temperature-related factor could accurately simulate the linear viscoelastic behavior of AC.

$E^{*}\left(\omega \right)=E_0^{*}+\left(E_{\infty }^{*}-E_0^{*}\right)/{\left[1+{\left(1+\frac{{\omega }_0}{\mathrm{i}\omega }\right)}^{\alpha }\right]}^{\beta },(2)$

where $E^{*}\left(\omega \right)$ is the complex modulus related to load frequency $\omega$; $E_0^{*}$ and $E_{\infty }^{*}$ are complex moduli as $\omega$ approaches 0 and $\infty$, respectively; ${\omega }_0$, $\alpha$, and $\beta$ are model coefficients. The parameters $E_{\infty }^{*}$ and $E_0^{*}$ characterize the equilibrium and instantaneous behavior of the linear viscoelastic property. The coefficients $\alpha$ and $\beta$ control the width and the skewness of the master curve, which are related to the relaxation mechanism behavior. The parameter ${\omega }_0$ controls the horizontal position of the master curve along the frequency axis and comes into effect in the horizontal shifting between different test temperatures together with the time–temperature shift factor. It should be noted that the MHN model is essentially a complex modulus model, which means that the dynamic modulus and phase angle could be derived through the linear viscoelastic theory using the same coefficients. Only five coefficients of the asphalt layer are required to be determined, which can significantly enhance the accuracy of the layer property backcalculation compared to the Prony series, which requires more than 13 coefficients.

The governing equation was transferred into the frequency–wavenumber domain via Fourier transform. The dynamic effect and the inertia distribution of mass can be described accurately without using sublayers. One spectral element is used to simulate the whole layer of the pavement, and the total number of SEM elements is equal to the number of layers. The spectral element stiffness matrix for each layer is constructed, and the global stiffness matrix is assembled in each frequency–wavenumber domain following the same procedure as the finite-element method. The displacement is obtained based on the boundary conditions and the equilibrium equation. The solution is summed over the wavenumber domain to capture the frequency-dependent behavior of the system, and the inverse Fourier transform is used to obtain the solution in the time domain. The superiority of the SEM is owing to the fact that it integrates the analysis efficiency of the spectral analysis and the organization of the finite element method. The formulated spectral element can precisely characterize the wave propagation dynamics, and solving is carried out in the frequency domain without model discretization. The SEM significantly reduces the total number of degrees-of-freedom and enhances the computational efficiency. Considering the dynamic effect of the FWD impulse load and the viscoelastic properties of AC, the analysis model could objectively simulate the structural behavior under the FWD load. The detailed information and validation of SEM are not introduced herein and can be found elsewhere (Zhao et al., 2015; Cao et al., 2025; Cao et al., 2019).

Architecture of the ANN-based backcalculation frame

Intelligent optimization has become a valuable tool to solve problems across various research fields, especially those related to engineering optimization. The ANN is the typical fundamental technique that enables simulating human behavior to discover the knowledge or the underlying patterns within the datasets. The unit of the ANN, known as a neuron, receives the information from various excitation sources and outputs the result through the active function in addition to a bias value, as shown in Equation 3. The primary tasks in constructing an ANN include designing the network architecture and determining the model’s hyperparameters (Li et al., 2022). The commonly utilized ANN structure consists of input layers, hidden layers, and output layers. The input layer is used to receive information from input variables, and the number of neurons corresponds to the number of input parameters. The output layer is the final layer of the ANN to return the prediction, and the number of neurons is related to the number of optimized parameters. The hidden layers connect the input layer and the output layer and perform the computation as a black box. The number of the hidden layer and the neurons in each layer varies and was related to the complexity of the problem.

$output=f\left(input\right)+Biasis.(3)$

The layer property backcalculation that identifies a reasonable combination of layer properties that accurately simulates the structural behavior under the FWD load can be formulated as a nonlinear optimization problem. The traditional method treated the difference between the measured and calculated peak deflections as the objective function, and the seed layer property was iteratively updated until the difference fell within a specified tolerance. Important characteristics related to the dynamic FWD load and damping materials, such as the time lag between the load peak and the peak deflection at various sensor locations, along with the deflection rate, are considered. However, only the peak value of the deflection time history is used in the static procedure, which limits the utilization of the full information contained in the measured data. Since the forward analysis engine must be executed numerous times, both the accuracy and computational efficiency are critical concerns for the iterative approach.

We used the ANN framework to build the mapping between the measured deflection and the layer property. In order to capture the influence of dynamic loading and damping materials, the measured time history of the FWD load and the deflection at various locations were considered as input variables in the ANN architecture. The length and complexity of the data posed a significant consideration for constructing the ANN input layer. The geometric characteristics were carefully extracted from the measured data to form the input variables. For the load time-history, four features were selected to represent the load properties, namely, peak load, peak time when the peak load occurs, load area, and load duration. For the deflection time history, the hysteresis curve was introduced to characterize the geometric property. Since the head and tail portions of the data are prone to environmental noise, the measured deflection greater than 30% of the magnitude and the corresponding loading were utilized to form the hysteresis curves, as shown in Figure 1. Six characteristics were extracted from each hysteresis curve. The peak deflection, the area under the hysteresis curve, and the ratio of the long axis to the short axis were considered the fundamental descriptors of each deflection curve. The peak deflection represents the deflection basin, while the area under the hysteresis curve and the ratio of the long axis to the short axis describe the relationship between deflection and loading. The enclosure degree was represented by the vertical distance between the initial and final deflection points, which were denoted as the length of the segment AB. The curve density was defined as the coordinate of the intersection point between the long and short axes, labelled as point O, which represented the center of the deflection peaks and the corresponding load. The inclination degree was defined as the tangent of the angle between the long axis and the tangent to segment AB. These characteristics were considered geometric indices to characterize the properties of the load–deflection hysteresis curve. The input variable characteristics are summarized in Table 1. It should be noted that the hysteresis curve was formed using the deflection and load time histories. The extracted geometric indices for deflections capture time domain characteristics, which allows the dynamic effects and damping influences to be reasonably represented. The number of neurons in the input layer was determined based on the features of the load time history and the load–deflection hysteresis curves. The output variable contains the mechanical property of each layer, including the linear viscoelastic coefficient of the asphalt layer and the elastic moduli for other layers.

Figure 1

Figure 1. Schematic of the hysteresis curve characteristic.

Table 1

Table 1. Evaluation indicator of the applied load and deflection.

This study employed a four-layer ANN model to establish the mapping between the measured deflection and layer properties (Wang and Zhao, 2022). The number of neurons in the input and the output layers was determined based on the structural information and the FWD configuration, while the number of neurons in the hidden layer was based on the relationships between adjacent layers. The weights and the biases of each neuron were considered as hyperparameters to be optimized. The weight connects the neurons and adjusts the importance of the contribution of each input excitation. The bias provides the flexibility of the neural network by acting as an offset or threshold. The ANN hyperparameters were initially predetermined and subsequently optimized during the training process (Marcelino et al., 2021; Mohammadi and Sheikholeslam, 2023). The built-in algorithm for the determination of hyperparameters of the ANN is the gradient-based method, which is prone to premature convergence and local optima (Abdolrasol et al., 2021). Various techniques have been applied to optimize the hyperparameters of the ANN. The widely used swarm intelligent optimization algorithms, namely, PSO, MA, and HPO, were utilized in this study to optimize the hyperparameters. Thus, a brief introduction of the algorithms mentioned above is provided in the following section.

Particle swarm optimization

The PSO is a well-known swarm-based optimization algorithm inspired by the social behavior observed in bird and fish flocks (Kennedy and Eberhart, 1995). The number of particles and the dimension of the PSO are related to the complexity of the optimization problem. The position and the velocity of the individual particles are important characteristics for the performance of the PSO algorithm. The position of each particle represents a candidate solution to the optimization objective, while the velocity is the critical parameter related to optimization efficiency. Initially, the position of each particle is uniformly distributed within the constrained search space and is updated by adding the current velocity vector. The position with the best fitness performance is recorded as the personal best ($pbest$) of the current particle and the global best ($gbest$) across all particles and generations. The exploration direction and velocity are updated based on the distance from the current position to the $pbest$ and $gbest$ positions, respectively, as shown in the following Equation 4.

$v_i^{t+1}=w{v}_i^t+c_1{r}_1\left(pbes{t}_i-x_i^t\right)+c_2{r}_2\left(gbest-x_i^t\right)(4)$

where, $v_i^t$ and $v_i^{t+1}$ are the velocity of the i^th particle at the current iteration $t$ and the next iteration $t+1$, respectively. $pbes{t}_i$ and $gbest$ are the optimal solutions of the current and the historical generations, respectively. $w$ is the inertia factor governing the contribution of the current speed; $c_1$ and $c_2$ are the acceleration factors related to the contribution of the $pbest$ and $gbest$, respectively. $r_1$ and $r_2$ are random numbers distributed between [0,1]. The kernel point of the PSO is that the global optimal solution is updated via shared information among the generations. The individual particle adjusts its exploration direction by updating the velocity using Equation 4 in each iteration until the objective error meets the requirement or the maximum iteration is reached.

Mayfly algorithm

Inspired by the social behavior of mayfly insects, the Mayfly algorithm was proposed for single-objective or multi-objective optimization problems (Zervoudakis and Tsafarakis, 2020). The position of each individual mayfly is considered the potential candidate for the optimization objective, while the speed is the optimization direction related to computational efficiency and precision, which is the same as that in PSO. The advantage of MA is that the mayfly in each generation is grouped into female and male sets, and the offspring are generated through crossover and mutation operations, similar to that in the GA. The MA combined the advantages of the PSO and the GA to enhance the optimization performance. The male MA will gather in swarms and dance up and down around the water level. The velocity of the position is updated following the Equation 5, considering the distance between from individual position to the p_best and g_best, respectively.

$v_{ij}^{t+1}=g{v}_{ij}^t+a_1{e}^{-\beta r_p^2}\left(pbes{t}_{ij}-x_{ij}^t\right)+a_2{e}^{-\beta r_g^2}\left(gbes{t}_{ij}-x_{ij}^t\right),(5)$

where $v_{ij}^t$ and $v_{ij}^{t+1}$ are the velocities of the j^th element in the i^th male mayfly at iteration t and t+1, respectively; $a_1$ and $a_2$ are the positive attraction coefficients of social function; $r_p$ is the Cartesian distance between $x_{ij}^t$ and $pbes{t}_{ij}$; $r_g$ is the Cartesian distance between $x_{ij}^t$ and $gbes{t}_{ij}$; and g is the gravity parameter.

Both the female and male mayfliesare ranked based on the fitness error. The female mayfly flies toward the male mayfly with the same rank order following the attraction function. The position of the female mayfly is updated using the following Equation 6.

$v{f}_{ij}^{t+1}=\left\{\begin{array}{ll}gv{f}_{ij}^t+a_2{e}^{-\beta r_{rmf}^2}\left(x_{ij}^t-x{f}_{ij}^t\right),& iff\left(x{f}_i\right)>f\left(x_i\right)\\ gv{f}_{ij}^t+f{l}_t\times r,& iff\left(x{f}_i\right)<f\left(x_i\right)\end{array}\right.,(6)$

where $v{f}_{ij}^t$ and $v{f}_{ij}^{t+1}$ are the velocities of the j^th element of the i^th female mayfly at time step $t$ , and $t+1$, $x_{ij}^t$ and $x{f}_{ij}^t$ are the positions of the male and female mayflies with the same fitness rank. $fl$ is a random walk coefficient, and $r$ is a random value in the range [−1, 1]. Subsequently, the position of the female mayfly is updated by adding the velocity of the female at the current step, similar to the male mayfly.

After the optimized male and female mayflies are obtained at the end of the generation, the female will be paired with the male mayfly based on the attraction rules in MA. The crossover and the mutation operations are conducted to generate the offspring. The random selection strategy is utilized to extract the information from the parent generation. The inherited element from the parent generation is combined to form two offspring using the following Equation 7.

$\left\{\begin{array}{l}offsprin{g}_1=L\times male+\left(1-L\right)\times female\\ offsprin{g}_2=\left(1-L\right)\times male+L\times female,\end{array}\right.(7)$

where L is a random value within a certain range. It is noted that the offspring inherits both the advantage and disadvantage from the parents; a premature convergence will occur during the optimization process. A Gaussian distribution with the average of 0 and standard deviation of 1 is added into the offspring to lead the offspring to visit around the search space to obtain the global optimization, as shown in Equation 8.

$offsprin{g}_n=offsprin{g}_n+\sigma N_n\left(0,1\right),(8)$

where $\sigma$ is the standard deviation of the normal distribution and $N_n\left(0,1\right)$ represents the normal Gaussian distribution. Thus, the generated offspring will be treated as initial members of the next generation. The optimization process will be performed following the routine until the accuracy meets the requirements.

Hunter–prey optimization method

The HPO method (Naruei et al., 2022) is another population-based method to simulate the dynamic interaction behavior of the animal hunting process. The core principle of the HPO is that the hunter will dynamically adjust the position based on the average position of the prey to capture the marginal prey, while the prey moves toward designated safety zones. The position search procedure is classified into two steps: exploration and exploitation. The exploration step contains a highly random behavior to discover the promising areas by updating the position significantly. The exploitation step contains a less random behavior to update the position around the promising regions. The position of the hunter will update following Equation 9.

$H_{ij}\left(t+1\right)=H_{ij}\left(t\right)+0.5*\left\{\left[2CZ{P}_{pos}-H_{ij}\left(t\right)\right]+\left[2\left(1-C\right)Z{\mu }_j-H_{ij}\left(t\right)\right]\right\},(9)$

where $H_{ij}\left(t+1\right)$ and $H_{ij}\left(t\right)$ are the positions of the hunter of the current and next iteration, respectively; $P_{pos}$ is the location of the prey; ${\mu }_j=1/N{\displaystyle \sum _{i=1}^N}{H}_{ij}$ is the average position of the hunter; $C$ is the optimization parameter that controls the balance between exploration and exploitation; $Z$ is the adaptive parameter. Once the prey is attacked, the prey will update its position to reach the safe zone, and the hunter may choose another prey. The global optimization is considered a safe place for the prey with a better chance of survival. The prey updates the position using the following Equation 10.

$P_{i,j}\left(t+1\right)=T_{pos\left(j\right)}+CZ\cos \left(2\pi R\right)\times \left(T_{pos\left(j\right)}-P_{i,j}\left(t\right)\right),(10)$

where $P_{i,j}\left(t\right)$ and $P_{i,j}\left(t+1\right)$ are the positions of the prey of the current and next iteration, respectively. $T_{pos\left(j\right)}$ is the global optimization position. $R$ is a random number in the range [−1, 1], and the COS function is to improve the prey exploitation performance by changing the radials and angles from the optimum position. According to the natural behavior, the trapped prey will die, and the hunter will move to the remaining prey swarm. A decreasing mechanism is adopted to simulate the variation in the swarm number following the Equation 11.

$kbest=round\left(C\times n\right),(11)$

where $n$ is the number of search agents. The number of the search agent will decrease during the optimization process, in accordance with the nature that the hunter attacks the prey at the farthest distance from the average position. The optimization will stop when the termination criteria are met and $kbest$ is equal to the requirements of the search agent.

Implementation of the intelligent optimization in layer property backcalculation

The key point of ANN-based optimization methods is to establish the mapping between input and output variables, of which the preliminary task is to determine the hyperparameter. Since the hyperparameters have a crucial influence on the accuracy and robustness of the method, various swarm optimization algorithms have been applied to perform hyperparameter optimization. Due to the complexity and variability of engineering problems, a fixed framework cannot be appropriate for all field problems. We, thus, propose a universal backcalculation framework and evaluate the performance of intelligent optimizations at a project level. Thus, the procedure of the intelligent optimization combined with a swarm optimization algorithm should be a universal framework, and the implementation of the intelligent optimization in layer property backcalculation was briefly introduced.

Step 1

Step 1. Generating the synthetic dataset using forward analysis

Step 2

Step 2. Constructing the ANN structure and determining the hyperparameters of the ANN

Step 3

Step 3. Optimizing the temperature-related parameter of the AC

Result and analysis

Feasibility analysis using a three-layer pavement

A typical three-layer asphalt pavement was used in theoretical evaluation of the effectiveness of intelligent optimizations. The pavement information is listed in Table 2. The layer thickness was determined according to typical asphalt pavement structures. The Poisson’s ratio, damping ratio, and density were assigned according to the material type (Cao et al., 2025). During the forward analysis, the range for layer properties, including the MHN model coefficient of the AC layer and the elastic modulus of other layers, was determined according to the material type. The dynamic modulus range for viscoelastic materials was set between 5,000 MPa and 45,000 MPa, the elastic modulus for cement-treated aggregate ranged from 2,000 MPa to 25,000 MPa, and the elastic modulus for soil ranged from 30 MPa to 400 MPa. In order to improve the accuracy of the proposed method by extending the cover content, the layer property was carefully selected within the recommended range and randomly combined to create analysis scenarios. In total, 10,000 scenarios were generated to perform forward analysis.

Table 2

Table 2. Structural information regarding the theoretical three-layer pavement.

Typical loading time history from various test equipment was collected from the long-term pavement performance (LTPP) program dataset, with the amplitude of 0.7 MPa. The surface deflection at the fixed offsets from the load center of 0, 210, 330, 500, 800, 1,100, 1,400, and 1,700 mm was computed to perform the layer property backcalculation. Four characteristics of the load and six characteristics of the deflection–load hysteresis curve of each offset were utilized as input variables, and the MHN model coefficient of the AC layer and the elastic moduli of other layers were used as output variables. A four-layer ANN (52-26-13-7) structure was established to build the mapping between input and output parameters. The weights and biases were optimized using the procedure mentioned above and the built-in algorithm.

A pure backpropagation neural network (BPNN) was used to validate the feasibility of the ANN in layer property backcalculation. The pure BPNN utilizes the gradient-based algorithm to optimize the hyperparameter. Figure 2 plots the comparison between the calculated and the backcalculated deflection time history, showing a good agreement between the measured and backcalculated deflection time history, thus indicating that the backcalculated layer property could be considered an appropriate solution for pavement behavior characterization. The backcalculated layer property and the corresponding actual values are listed in Table 3. Once MHN was available, the complex modulus could be derived based on the linear viscoelastic theory. The comparison between the actual and back-calculated dynamic modulus and phase angle master curves is plotted in Figure 3. The phase angle shows a discrepancy at lower frequencies, and a slight difference in the dynamic modulus could be observed at higher frequencies. Both the backcalculated dynamic modulus and phase angle master curves exhibit a good agreement during the effective frequency range (Fu et al., 2020). The normalized difference between the backcalculated elastic moduli of the base and subgrade was less than 10%. The negligible difference demonstrates the feasibility of the ANN-based layer backcalculation.

Figure 2

Figure 2. Backcalculated deflection time history result from the BPNN.

Table 3

Table 3. Backcalculated layer properties of the BPNN and actual values.

Figure 3

Figure 3. Comparison between the actual and backcalculated complex modulus.

However, a crucial concern is the robustness of the ANN-based method, the performance of which depends on the optimization strategy. The results of the two analysis scenarios under duplicated trials are listed in Table 4. Both the backcalculated deflection histories matched well with the calculated data, which is not shown here for brevity. The dynamic modulus at the frequency of 10 Hz was calculated for comparison. The difference between the actual and the backcalculated layer modulus is listed in parentheses. The positive value denotes that the backcalculated modulus was larger than the actual one, and vice versa. A significant difference between the layer properties from different trials could be observed in the results. For scenario #1, the backcalculated surface layer property from trial 1 was larger than the actual one, while the backcalculated modulus of the base layer was lower than the actual one, which could be interpreted using the concept of layer compensation (Cao et al., 2019; Stubstad et al., 2006; Nakhaei and Timm, 2023). However, the backcalculated layer properties of both the surface and base layer from trial 2 were less than the actual ones. The normalized difference (ND) between the results obtained from the different trials was calculated and is listed in the last row. The ND for the surface layer between the two trials was 14.72%. For scenario #2, the difference between the backcalculated and actual base layer property could be more than 3,000 MPa, with an ND of approximately 20%. The result indicated that the robustness of the ANN-based method may produce an unreasonable result for pavement assessment.

Table 4

Table 4. Backcalculated result of the BPNN for two duplicated operations.

In order to compare the performance of the backcalculation method with that of various intelligent optimization algorithms, the layer properties of scenarios #1 and #2 were backcalculated using the ANN model with different swarm optimization algorithms. The result was marked by the algorithms used in the hyperparameter optimization. The ND between the actual and the backcalculated parameter was also calculated and is listed in Table 5. The backcalculated layer properties from different models were reasonable, indicating the effectiveness of the backcalculation methods. The ND of the backcalculated layer property from the ANN model with swarm optimization algorithms was lower than that from the BPNN model. For the asphalt layers, the layer properties obtained from ANN + MA and ANN + HPO were closer to the actual values than those from BPNN and ANN + PSO. Especially for the base layer of scenario 2, the ND from the BPNN was 15.89%, while those from ANN + PSO, ANN + MA, and ANN + HPO were 12.04%, 4.34%, and 7.87%, respectively. The ND from the ANN + MA and ANN + HPO was almost 60% lower than that of the BPNN. The result indicated that the ANN + MA and ANN + HPO models exhibit a better overall performance for layer property backcalculation compared to BPNN and ANN + PSO.

Table 5

Table 5. Comparison between backcalculation results from various algorithms.

Performance evaluation using the field pavement at the project level

In order to evaluate the performance of the proposed ANN-based model at the project level, a four-layer field asphalt pavement was selected to perform layer property backcalculation. The four-layer highway is located in Liaoning Province, China. The layer material and the thickness were determined by the as-designed document and are shown in Table 6. Three asphalt surface lifts were combined into one AC layer to reduce the complexity of the optimization. The field FWD test was performed with an interval of 50 m along the vehicle path. The magnitude of the test was set as 0.7 MPa, and the deflection time-history at various offsets of 0, 200, 300, 450, 600, 900, 1,200, 1,500, 1,800, and 2,100 mm was collected.

Table 6

Table 6. Field pavement information.

The mechanical property of each layer varied within the reasonable range according to the material type. A total of 30,000 scenarios were generated, and the deflection time history was calculated using the SEM. The characteristics of the loading and the deflection–load hysteresis curve were calculated to generate the database for training the ANN model with various swarm optimization algorithms. The input variable consisted of four characteristics of the load and six characteristics of the deflection–load hysteresis curve of 10 sensors. The output variable was the mechanical property of each layer. Based on the structural information and the test configuration, a four-layer ANN (64-32-16-8) architecture was developed to map the input to the output parameters. The ANN structure was trained using the theoretical deflection database following the procedure mentioned above. The trained ANN model was used to backcalculate the layer property based on field-measured deflection. The typical comparison between the measured and the backcalculated deflection from the BPNN model is shown in Figure 4, yielding a good agreement, indicating that the backcalculated layer property can accurately simulate the pavement behavior under FWD load conditions. The backcalculation results from other models showed a similar performance, which is not plotted here for brevity. The backcalculated layer properties from various models are listed in Table 7.

Figure 4

Figure 4. Comparison between the measured and backcalculated deflection.

Table 7

Table 7. Typical backcalculated layer properties from various ANN-based frames.

It can be seen that the layer property from various models fell within reasonable ranges and matched well with each other, indicating that the trained ANN model can effectively capture the essential relationship between the layer property and pavement performance. The complex modulus of the AC layer could be derived from the backcalculated MHN model and is shown in Figure 5. Within the effective frequency range of 5 Hz–65 Hz (Li and Wang, 2019), the master curve from all the models matched well with each other. Notably, the master curve was calculated using the backcalculated MHN model, for which the reference temperature was the field-measured data. The backcalculated layer property should be primarily corrected into the same temperature condition.

Figure 5

Figure 5. Backcalculated (a) dynamic modulus and (b) phase angle master curve from various models.

The field-measured deflection was carefully selected based on the criterion that no macroscopic crack appeared at the test station. The backcalculated layer property and the measured temperature of 27.81 °C of station 1 were treated as the reference scenario. The backcalculated layer properties of other stations were used to obtain the optimal coefficient of the WLF function within the effective range. The optimized WLF coefficients were 10.6 and 145.25 for $C_1$ and $C_2$, respectively. Once the coefficient of the MHN model and the WLF function was available, the dynamic modulus at the artificial condition could be derived via the LVE theory. The dynamic modulus of the test stations could be obtained based on the layer property of station 1 and the WLF function coefficient. The comparison of the corrected dynamic modulus and the backcalculated dynamic modulus was plotted and is shown in Figure 6. A good agreement could be observed, indicating the feasibility of the temperature modification procedure.

Figure 6

Figure 6. Comparison of the dynamic modulus before and after temperature modification.

In order to evaluate the performance of the proposed backcalculation method, the dynamic modulus at the frequency of 10 Hz and the temperature of 20 °C was calculated using the backcalculated LVE parameter and the measured temperature of each station. The box-and-whisker plot was adopted to illustrate the statistical characteristics of the backcalculated layer properties obtained from different methods, as shown in Figure 7. The length of the box-and-whisker plot represents the distribution performance of the result. The maximum, minimum, and average values of the backcalculated layer property were also inserted in the figure.

Figure 7

Figure 7. Comparison of the backcalculated complex modulus for (a) asphalt layers; (b) base; (c) subbase; (d) subgrade.

The backcalculated layer property from various methods varied within a reasonable range, indicating the effectiveness of the trained ANN model. For the asphalt layer, the difference between the maximum and minimum of the dynamic modulus from the BPNN model was greater than that from the ANN + PSO, ANN + MA, and ANN + HPO models. The interquartile range of the box-and-whisker from BPNN, ANN + PSO, and ANN + MA was greater than that from the ANN + HPO, indicating the ANN + HPO exhibits a better performance in layer property backcalculation for viscoelastic materials. The phase angle showed a similar trend to the dynamic modulus; however, the phase angle obtained from ANN + MA exhibited a more scattered distribution. For the elastic material, all backcalculation models can obtain moduli with a reasonable range in accordance with the corresponding material types. There were only slight differences among the average moduli obtained from the different methods. In order to quantitatively analyze the performance, the average, standard deviation (STD), and the coefficient of variation (COV) of the layer property from various models were calculated and are listed in Table 8.

Table 8

Table 8. Statistical performance of the backcalculated layer property.

As shown in Table 8, the average moduli for all the layers from various ANN-based models had a reasonable range in accordance with the material type. The difference of the average backcalculated layer property from different models was less than 10%. The STD from the BPNN model for all the layers was significantly greater than that from the ANN model with a swarm optimization algorithm. For the asphalt layer, the STD from the ANN + HPO model was 140.33, which was obviously less than 341.24 observed from the BPNN model. The COV of the backcalculated layer property from ANN + HPO was only 50% of that from the BPNN model. For the elastic layers, both the STD and COV from the BPNN and ANN + PSO were greater than those from ANN + MA and ANN + HPO. The STD and COV for the base layer from the ANN + MA were 38% and 45% less than those from the BPNN, respectively. The statistical characteristic of the ANN + MA was slightly less than that of the ANN + HPO model. Among the swarm optimization algorithm models, the ANN + HPO shows a better performance in the backcalculation of the asphalt layer, while the ANN + MA has a better performance for the elastic layers. The computational time for layer property backcalculation is a critical factor for engineering applications. The time consumption of the backcalculation procedure included forward analysis for generating the synthetic dataset and the hyperparameter optimization process. The backcalculation was performed on a computer with Intel i7 CPU of 3.20 GHz. The number of generations was set to 100, and the number of iterations was set to 150 for all procedures. The computational time for the SEM was approximately 3 s per scenario, while the optimization processes required 3,830, 3,780, 3,215, and 2,797 s for BPNN, ANN + PSO, ANN + MA, and ANN + HPO, respectively. The time consumption of the ANN + HPO model was significantly less than that of the other models.

The superiority of intelligent models lies in the ability of the swarm population to extend the candidate solution range and utilize information from each generation to approach the optimal result. The MA algorithm combines the advantages of PSO and the inheritance mechanisms of the GA; crossover and mutation operations leverage individual information to escape the local optima and find the global optimum. The HPO algorithm divides particles into hunters and prey, both of which update their positions toward the best solution. The interaction between these two groups can enhance the performance of the ANN model. The number of the search agents decreases during the optimization process, which could reduce the optimization amount and significantly enhance the computer efficiency. Integrating the ANN framework with swarm optimization algorithms enables the construction of a mapping between the measured deflection and layer properties, thereby improving pavement assessment at the project level. The influence of environmental noise, pavement deterioration on the accuracy of layer property backcalculation, and the application of the backcalculated layer property in the maintenance decision-making will be analyzed in subsequent research to improve the applicability of the proposed procedure.

Conclusion

This study proposed an intelligent optimization procedure for project-level backcalculation of asphalt pavement layer properties. A four-layer ANN model was utilized to establish the underlying mapping between the measured deflection time history and the layer property, where the hyperparameter of the ANN was determined using the swarm optimization algorithm. The deflection dataset under FWD loading was generated using a dynamic viscoelastic analysis engine. The geometric characteristics of the measured time history, including four indices of the applied load and six indices of the deflection–load hysteresis curve, were extracted to determine the input layer of the ANN. The viscoelastic property of the AC layer and the elastic moduli of other layers were backcalculated and used as the output layer of the ANN. The PSO, MA, and HPO were selected to determine the initial weights and biases of each neuron. The trained ANN model was utilized to perform large-scale, parallel layer property backcalculation. Subsequently, an optimization was conducted to obtain the WLF coefficient to characterize the temperature-dependent behavior of the AC layer, rather than applying temperature correction directly to the deflection data. Theoretical three-layer and field four-layer asphalt pavements were utilized to evaluate the performance of the intelligent optimization procedure in layer property backcalculation. The good agreement between the backcalculated and measured deflection history and the reasonable layer property demonstrated the effectiveness of the proposed procedure. The performance of the ANN was highly dependent on the optimization strategy employed. The ND between the backcalculated and actual layer property obtained from the ANN with swarm optimization algorithms was less than that obtained from the BPNN. The swarm optimization algorithm could extend the candidate search space and guided the hyperparameters toward global optima. The temperature-dependent behavior of the asphalt layer could be precisely characterized using the backcalculated layer property and WLF coefficient. The backcalculated layer property of the four-layer asphalt pavement fell within reasonable ranges. The average moduli from various models exhibited a difference of less than 10%, denoting the effectiveness of the ANN-based framework for layer property backcalculation. The STD and COV from the ANN with swarm optimization algorithms were significantly less than those from the BPNN. The ANN + MA and the ANN + HPO exhibit a relatively better performance in layer property backcalculation compared to the BPNN and the ANN + PSO. The COV of the BPNN could be up to 50% higher than that of the ANN + MA and ANN + HPO. The ANN + HPO with a decreased mechanism in optimization process exhibited a significantly greater computational efficiency over all the intelligent models. The result indicated that the ANN with swarm optimization algorithms provided an efficient procedure for pavement condition assessment at a project level. The influence of environmental noise and pavement deterioration on backcalculated layer properties, along with a comparison to experimental data, will be investigated in subsequent research works to enhance their practical engineering applications.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

GW: Data curation, Methodology, Conceptualization, Investigation, Writing – review and editing, Formal Analysis, Writing – original draft. YZ: Visualization, Writing – review and editing, Validation, Resources, Writing – original draft, Project administration, Funding acquisition, Data curation, Conceptualization, Investigation, Supervision, Software. DC: Software, Investigation, Writing – original draft, Resources, Funding acquisition, Visualization, Conceptualization, Validation, Data curation, Supervision, Writing – review and editing, Project administration. MW: Funding acquisition, Investigation, Writing – review and editing, Validation, Supervision, Formal Analysis, Methodology, Software, Resources, Data curation, Project administration, Visualization, Conceptualization. HY: Writing – review and editing, Validation, Supervision, Visualization.

Funding

The author(s) declared that financial support was received for this work and/or its publication. The work is funded by the National Natural Science Foundation of China (52008012) and the Shanxi Province Transportation Research Project (19-JKKJ-4 and 23-JKCF-12). The authors gratefully acknowledge their financial support.

Conflict of interest

Author GW was employed by Shanxi Provincial Transportation Construction Engineering Quality Inspection Center (Co., Ltd.).

The remaining author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author HY declared that they were an editorial board member of Frontiers at the time of submission. This had no impact on the peer review process and the final decision.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

References

Abd El-Raof, H. S., Abd El-Hakim, R. T., El-Badawy, S. M., and Afify, H. A. (2018). Simplified closed-form procedure for network-level determination of pavement layer moduli from falling weight deflectometer data. J. Transp. Eng. Part B Pavements 144 (4), 04018052. doi:10.1061/JPEODX.0000080

CrossRef Full Text | Google Scholar

Abdolrasol, M. G. M., Hussain, S. M. S., Ustun, T. S., Sarker, M. R., Hannan, M. A., Mohamed, R., et al. (2021). Artificial neural networks based optimization techniques: a review. Electronics 10 (21), 2689. doi:10.3390/electronics10212689

CrossRef Full Text | Google Scholar

Al-Khoury, R., Scarpas, A., Kasbergen, C., and Blaauwendraad, J. (2001a). Spectral element technique for efficient parameter identification of layered media. I. Forward calculation. Int. J. Solids Struct. 38 (9), 1605–1623. doi:10.1016/S0020-7683(00)00112-8

CrossRef Full Text | Google Scholar

Al-Khoury, R., Kasbergen, C., Scarpas, A., and Blaauwendraad, J. (2001b). Spectral element technique for efficient parameter identification of layered media: part II: inverse calculation. Int. J. Solids Struct. 38 (48-49), 8753–8772. doi:10.1016/S0020-7683(01)00109-3

CrossRef Full Text | Google Scholar

Al-Khoury, R., Scarpas, A., Kasbergen, C., and Blaauwendraad, J. (2002). Spectral element technique for efficient parameter identification of layered media. Part III: viscoelastic aspects. Int. J. Solids Struct. 39 (8), 2189–2201. doi:10.1016/S0020-7683(02)00079-3

CrossRef Full Text | Google Scholar

Asefzadeh, A., Hashemian, L., and Bayat, A. (2017). Development of statistical temperature prediction models for a test road in Edmonton, Alberta, Canada. Int. J. Pavement Res. Technol. 10 (5), 369–382. doi:10.1016/j.ijprt.2017.05.004

CrossRef Full Text | Google Scholar

Ashtiani, A. Z., Murrell, S., and Ji, R. (2025). “Temperature correction factors for backcalculated asphalt modulus and HWD deflections of airfield pavements,” in Airfield and highway pavements, 152–161. Glendale. doi:10.1061/9780784486238.014

CrossRef Full Text | Google Scholar

Cao, D., Zhao, Y., Liu, W., Li, Y., and Ouyang, J. (2019). Comparisons of asphalt pavement responses computed using layer properties backcalculated from dynamic and static approaches. Road Mater. Pavement Des. 20 (5), 1114–1130. doi:10.1080/14680629.2018.1436467

CrossRef Full Text | Google Scholar

Cao, D., Zhou, C., Zhao, Y., Fu, G., and Liu, W. (2020). Effectiveness of static and dynamic backcalculation approaches for asphalt pavement. Can. J. Civ. Eng. 47 (7), 846–855. doi:10.1139/cjce-2019-0052

CrossRef Full Text | Google Scholar

Cao, D., Zhu, M., Niu, F., Zeng, W., Zhang, J., and Zhao, Y. (2025). An efficient dynamic backcalculation procedure for flexible pavement at a network level. Road Mater. Pavement Des., 1–19. doi:10.1080/14680629.2025.2499234

CrossRef Full Text | Google Scholar

Chou, C. P., Lin, Y. C., and Chen, A. C. (2017). Temperature adjustment for light weight deflectometer application of evaluating asphalt pavement structural bearing capacity. Transp. Res. Rec. 2641 (1), 75–82. doi:10.3141/2641-10

CrossRef Full Text | Google Scholar

Egan, J. R., and Hall, I. M. (2015). A review of back-calculation techniques and their potential to inform mitigation strategies with application to non-transmissible acute infectious diseases. J. R. Soc. Interface 12 (106), 20150096. doi:10.1098/rsif.2015.0096

PubMed Abstract | CrossRef Full Text | Google Scholar

Fathi, S., Mehravar, M., and Rahman, M. (2023). Development of FWD based hybrid back-analysis technique for railway track condition assessment. Transp. Geotech. 38, 100894. doi:10.1016/j.trgeo.2022.100894

CrossRef Full Text | Google Scholar

Fu, G., Zhao, Y., Zhou, C., and Liu, W. (2020). Determination of effective frequency range excited by falling weight deflectometer loading history for asphalt pavement. Constr. Build. Mater. 235, 117792. doi:10.1016/j.conbuildmat.2019.117792

CrossRef Full Text | Google Scholar

Kennedy, J., and Eberhart, R. (1995). “Particle swarm optimization,” in Proceedings of ICNN’95-International conference on neural networks (IEEE), 1942–1948. doi:10.1109/ICNN.1995.488968

CrossRef Full Text | Google Scholar

Kim, Y. R., Hibbs, B. O., and Lee, &Y.-C. (1995). Temperature correction of deflections and backcalculated asphalt concrete moduli. Transp. Res. Rec. 1473 (1), 55–62. doi:10.3141/1570-13

CrossRef Full Text | Google Scholar

Le, V. P., Le, A. T., Nguyen, M. T., and Nguyen, Q. P. (2023). Development and validation of a temperature correction model for FWD backcalculated moduli. Aust. J. Civ. Eng. 21 (1), 89–97. doi:10.1080/14488353.2022.2083045

CrossRef Full Text | Google Scholar

Lee, H. S. (2014). Viscowave–a new solution for viscoelastic wave propagation of layered structures subjected to an impact load. Int. J. Pavement Eng. 15 (6), 542–557. doi:10.1080/10298436.2013.782401

CrossRef Full Text | Google Scholar

Lee, S. W., Mahoney, J. P., and Jackson, N. C. (1988). Verification of a backcalculation of pavement moduli. Transp. Res. Rec. 1196, 85–95.

Google Scholar

Li, M., and Wang, H. (2018). Prediction of asphalt pavement responses from FWD surface deflections using soft computing methods. J. Transp. Eng. Part B Pavements 144 (2), 04018014. doi:10.1061/JPEODX.0000044

CrossRef Full Text | Google Scholar

Li, M., and Wang, H. (2019). Development of ANN-GA program for backcalculation of pavement moduli under FWD testing with viscoelastic and nonlinear parameters. Int. J. Pavement Eng. 20 (4), 490–498. doi:10.1080/10298436.2017.1309197

CrossRef Full Text | Google Scholar

Li, J., Yin, G., Wang, X., and Yan, W. (2022). Automated decision making in highway pavement preventive maintenance based on deep learning. Automation Constr. 135, 104111. doi:10.1016/j.autcon.2021.104111

CrossRef Full Text | Google Scholar

Marcelino, P., de Lurdes Antunes, M., Fortunato, E., and Gomes, M. C. (2021). Machine learning approach for pavement performance prediction. Int. J. Pavement Eng. 22 (3), 341–354. doi:10.1080/10298436.2019.1609673

CrossRef Full Text | Google Scholar

Mohammadi, A., and Sheikholeslam, F. (2023). Intelligent optimization: literature review and state-of-the-art algorithms (1965–2022). Eng. Appl. Artif. Intell. 126, 106959. doi:10.1016/j.engappai.2023.106959

CrossRef Full Text | Google Scholar

Nakhaei, M., and Timm, D. H. (2023). A new methodology to improve backcalculation of flexible pavements with stabilized foundations. Constr. Build. Mater. 368, 130405. doi:10.1016/j.conbuildmat.2023.130405

CrossRef Full Text | Google Scholar

Naruei, I., Keynia, F., and Sabbagh Molahosseini, A. (2022). Hunter–prey optimization: algorithm and applications. Soft Comput. 26 (3), 1279–1314. doi:10.1007/s00500-021-06401-0

CrossRef Full Text | Google Scholar

Park, D. Y., Buch, N., and Chatti, K. (2001). Effective layer temperature prediction model and temperature correction via falling weight deflectometer deflections. Transp. Res. Rec. 1764 (1), 97–111. doi:10.3141/1764-11

CrossRef Full Text | Google Scholar

Quan, W., Zhao, K., Ma, X., and Dong, Z. (2022). Fractional viscoelastic models for asphalt concrete: from parameter identification to pavement mechanics analysis. J. Eng. Mech. 148 (8), 04022036. doi:10.1061/(ASCE)EM.1943-7889.0002116

CrossRef Full Text | Google Scholar

Saltan, M., Uz, V., and Aktaş, B. (2013). Artificial neural networks based backcalculation of the structural properties of a typical flexible pavement. Neural Comput. Appl. 23 (6), 1703–1710. doi:10.1007/s00521-012-1131-y

CrossRef Full Text | Google Scholar

Stubstad, R., Jiang, Y., Clevenson, M., and Lukanen, E. (2006). Review of the long term pavement performance backcalculation results. Final Report. Washington, D.C.: Federal Highway Administration. Rep. No. FHWA-HRT-05-150.

Google Scholar

Varma, S., and Kutay, M. E. (2016). Viscoelastic nonlinear multilayered model for asphalt pavements. J. Eng. Mech. 142 (7), 04016044. doi:10.1061/(ASCE)EM.1943-7889.0001095

CrossRef Full Text | Google Scholar

Varol Altay, E., and Alatas, B. (2020). Bird swarm algorithms with chaotic mapping. Artif. Intell. Rev. 53 (2), 1373–1414. doi:10.1007/s10462-019-09704-9

CrossRef Full Text | Google Scholar

Wang, Y., and Zhao, Y. (2022). Predicting bedrock depth under asphalt pavement through a data-driven method based on particle swarm optimization-back propagation neural network. Constr. Build. Mater. 354, 129165. doi:10.1016/j.conbuildmat.2022.129165

CrossRef Full Text | Google Scholar

Wang, H. P., Xiang, P., and Jiang, L. Z. (2020). Optical fiber sensing technology for full-scale condition monitoring of pavement layers. Road Mater. Pavement Des. 21 (5), 1258–1273. doi:10.1080/14680629.2018.1547656

CrossRef Full Text | Google Scholar

Zaabar, I., Chatti, K., Lee, H. S., and Lajnef, N. (2014). Backcalculation of asphalt concrete modulus master curve from field-measured falling weight deflectometer data: using a new time domain viscoelastic dynamic solution and genetic algorithm. Transp. Res. Rec. 2457 (1), 80–92. doi:10.3141/2457-09

CrossRef Full Text | Google Scholar

Zervoudakis, K., and Tsafarakis, S. (2020). A mayfly optimization algorithm. Comput. and Industrial Eng. 145, 106559. doi:10.1016/j.cie.2020.106559

CrossRef Full Text | Google Scholar

Zhang, X., Xie, X., Tang, S., Zhao, H., Shi, X., Wang, L., et al. (2024). High-speed railway seismic response prediction using CNN-LSTM hybrid neural network. J. Civ. Struct. Health Monit. 14 (5), 1125–1139. doi:10.1007/s13349-023-00758-6

CrossRef Full Text | Google Scholar

Zhang, X., Liu, L., Zheng, Z., Quan, Y., Chen, Z., Cao, J., et al. (2025a). Bending strain progression and damage of asphalt beams based on distributed fibre optic sensors. Int. J. Pavement Eng. 26 (1), 2479645. doi:10.1080/10298436.2025.2479645

CrossRef Full Text | Google Scholar

Zhang, X., Wang, J., Chen, Z., Quan, Y., Zheng, Z., Zhang, T., et al. (2025b). A distributed fiber optic sensor-based approach for crack asphalt structure under freeze-thaw cycling tests. Constr. Build. Mater. 476, 141262. doi:10.1016/j.conbuildmat.2025.141262

CrossRef Full Text | Google Scholar

Zhang, X., Wang, J., Liu, L., Cao, J., Quan, Y., Xie, X., et al. (2025c). Prediction of freeze-thaw damage of asphalt concrete based on distributed fiber optic sensors and KAN-Transformer fusion model. Opt. Fiber Technol. 94, 104304. doi:10.1016/j.yofte.2025.104304

CrossRef Full Text | Google Scholar

Zhang, X., Wang, J., Cao, J., Quan, Y., Liu, L., Wen, K., et al. (2025d). Distributed fiber optic sensing and Informer-LSTM prediction for damage evolution of asphalt concrete under freeze–thaw cycling. Mater. Struct. 58 (7), 1–19. doi:10.1617/s11527-025-02758-y

CrossRef Full Text | Google Scholar

Zhao, Y., Liu, H., Bai, L., and Tan, Y. (2013). Characterization of linear viscoelastic behavior of asphalt concrete using complex modulus model. J. Materials Civil Engineering 25 (10), 1543–1548. doi:10.1061/(ASCE)MT.1943-5533.0000688

CrossRef Full Text | Google Scholar

Zhao, Y., Cao, D., and Chen, P. (2015). Dynamic backcalculation of asphalt pavement layer properties using spectral element method. Road Mater. Pavement Des. 16 (4), 870–888. doi:10.1080/14680629.2015.1056214

CrossRef Full Text | Google Scholar

Zheng, Y., Zhang, P., and Liu, H. (2019). Correlation between pavement temperature and deflection basin form factors of asphalt pavement. Int. J. Pavement Eng. 20 (8), 874–883. doi:10.1080/10298436.2017.1356172

CrossRef Full Text | Google Scholar