The SEM is employed to simulate the pavement surface deflection response under FWD loading. Compared with conventional finite element or finite difference schemes, the SEM achieves high accuracy by interpolating the field variables with high-order spectral shape functions within each element and by describing the distributed mass inertia exactly. In this study, a one-dimensional axisymmetric SEM formulation is adopted following Zhao et al. (2015) and Cao et al. (2020). The layered pavement structure is modeled as a stack of homogeneous, isotropic layers characterized by thickness, elastic modulus, Poisson’s ratio, and density, resting on a semi-infinite subgrade.

The governing equations of motion for the axisymmetric elastic medium are given in Equation 1: λ t + μ t ∇ ∇ · u + μ ∇ 2 u = ρ u ¨ (1) where u represents the displacement vector composed of the radial component u and the vertical component w , u ¨ refers to the acceleration vector, λ t presents Lame’s constant for the material, while μ t represents the shear modulus. ∇ denotes the gradient differential operator, ∇ · u represents the divergence of u , ∇ 2 u represents the Laplacian of u , and ρ signifies the material density.

In the vertical direction, each pavement layer is discretized by a 2-node axisymmetric spectral layer element, while the semi-infinite subgrade is represented by a 1-node throw-off element that conducts energy out of the system. Each node carries two degrees of freedom (radial and vertical displacements). Within a spectral element, the displacement field is interpolated by high-order spectral shape functions constructed from Lagrange polynomials passing through Gauss–Lobatto–Legendre points, so that one spectral element per physical layer is sufficient and no further mesh refinement is required through the thickness.

In the radial direction, the domain is discretized by a graded mesh that is refined beneath and near the FWD loading area and gradually coarsened toward an outer truncation radius. This radius is chosen sufficiently large such that the computed surface vibration decays to negligible levels, which avoids spurious reflections from the lateral boundary. Axisymmetry is enforced at the centerline r = 0 , and the pavement surface is traction-free outside the circular loading area where the FWD pressure is applied. At the bottom of the truncated domain, vertical displacement is fixed while radial displacement continuity is maintained through the throw-off spectral element to mimic the semi-infinite half-space.

The spatial discretization leads to a semi-discrete system of second-order ordinary differential equations in time. This system is advanced using an explicit central-difference time integration scheme, with the time step selected according to the standard SEM stability criterion based on the smallest element size and the maximum wave speed. Since all layers are modeled as linear elastic materials and no additional material or Rayleigh damping is introduced, the computed response corresponds to the undamped elastic wave propagation problem. The resulting SEM formulation has been validated in previous studies (Cao et al., 2020; Zhao et al., 2015), confirming its accuracy and stability for simulating pavement surface deflection histories. The peak values of the computed surface deflection basins at the sensor locations are then extracted and used as input features for the Transformer-based learning model described in Section 2.2.

It should be clarified that the term physics-informed in this study refers to the use of SEM-based forward simulations to generate physically consistent training and testing datasets, rather than to the explicit enforcement of physical laws or inequality constraints within the neural network architecture itself. The Transformer model is trained as a data-driven regression mapping from deflection basins to layer elastic moduli and does not impose hard constraints such as modulus ordering or monotonicity during learning. In this study, an intelligent back-calculation framework based on the Transformer architecture is established to predict the elastic modulus of pavement layers from measured deflection data. The Transformer model, originally proposed by Vaswani et al. (2017), has demonstrated exceptional performance in capturing long-range dependencies through its self-attention mechanism, making it well-suited for modeling complex nonlinear relationships in pavement structural systems.

The model’s predictive performance is evaluated using five standard statistical metrics: MAE, Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Percentage Error (MAPE), and the Coefficient of Determination (R²). These metrics assess both the magnitude and distribution of prediction errors, providing a comprehensive evaluation of model accuracy.

To train and evaluate the Transformer-based intelligent back-calculation model, a large-scale synthetic dataset was established through numerical simulations using the validated SEM model. The SEM approach provides high computational efficiency and accuracy for solving dynamic response problems of layered pavement systems, making it particularly suitable for simulating FWD tests.

The modeled pavement structure consists of three primary layers: a surface course, a base course, and a subgrade. Each layer is characterized by its elastic modulus, Poisson’s ratio, and thickness, as summarized in Table 2. In the SEM simulations, pavement layers are modeled as linear elastic materials, and light viscous damping is introduced at the dynamic response level, following standard practice in SEM-based dynamic analysis of pavements (Al-Khoury et al., 2001; 2002). The mechanical parameters were randomly combined within reasonable engineering ranges to ensure adequate representation of various pavement conditions, resulting in a total of 20592 combinations of pavement structures.

During the simulation, the FWD test applies an impulsive load to the pavement surface in the form of a half-sine pulse with a peak pressure of 0.7 MPa and a duration of 25 ms. The loading plate radius is set to 15 cm, following standard FWD testing procedures. The pavement response was monitored at nine measurement points located at 0, 20, 30, 50, 80, 110, 140, 170, and 200 cm from the load center, corresponding to the typical sensor arrangement used in field testing.

For each simulation case, the SEM model outputs the deflection time history at all nine sensors. The peak deflection values were extracted from these time histories using an automated peak detection algorithm, representing the maximum surface displacement under the dynamic load. These peak values form the input features for the back-calculation model, while the corresponding elastic moduli of the three pavement layers serve as the output targets. Consequently, a comprehensive dataset with nine input variables (deflection peaks) and three output variables (layer moduli) was constructed, containing 20592 samples in total. This dataset was subsequently normalized and divided into training and testing subsets for the Transformer model development and performance evaluation.

The benchmark case, without any added noise, represents the ideal data condition for model evaluation. As shown in Figure 5; Table 4, the Transformer model demonstrates excellent agreement between predicted and true elastic moduli for all pavement layers. The predicted points in Figure 5 closely follow the 1:1 reference line, indicating strong consistency across the entire modulus range.

Quantitatively, the average MAE reaches 710.95 MPa, and the MAPE remains as low as 5.93%, signifying a high prediction accuracy. The average R² of 0.96 further confirms that the model captures over 96% of the variance in the true modulus values.

Among individual layers, the subgrade modulus exhibits a very high statistical correlation with the reference values (R² close to 1.00) and relatively small absolute errors (MAE = 0.79 MPa), reflecting its dominant influence on the overall deflection basin under the considered parameter ranges and sensor configuration. Conversely, the surface course exhibits slightly larger deviations due to its higher stiffness and greater sensitivity to small perturbations in deflection measurements.

These results highlight the Transformer model’s powerful feature extraction ability and its capacity to establish a robust nonlinear mapping between deflection patterns and pavement layer moduli under ideal conditions.

To emulate random fluctuations in field measurements, Gaussian noise with zero mean and specified variance was introduced into the input data. The corresponding results are illustrated in Figure 6; Table 5. Remarkably, even in the presence of random noise, the model maintains a high level of predictive accuracy. The average MAE (674.37 MPa) and RMSE (979.83 MPa) are slightly lower than those in the noise-free case, and R² remains above 0.95, suggesting that the model benefits from minor data perturbations, which can enhance generalization by reducing overfitting.

The R² values remain consistently high (≥0.95 for all layers), indicating that the random disturbances do not significantly affect the model’s regression capability. This robustness can be attributed to the self-attention mechanism in the Transformer architecture, which effectively identifies key spatial dependencies among deflection features and suppresses the influence of random noise.

In particular, the base course achieves an R² of 0.95 with MAPE below 8.1%, demonstrating the model’s adaptability to intermediate stiffness layers. The scatter distribution in Figure 6 remains tightly clustered around the reference line, further confirming the model’s insensitivity to random fluctuations.

Systematic errors, such as sensor calibration bias or consistent drift in FWD equipment, were next introduced to evaluate the model’s resilience to directional deviations. The outcomes are summarized in Figure 7; Table 6. Compared with the previous cases, the performance metrics show a moderate decline. The average MAE increases to 856.30 MPa, RMSE to 1205.76 MPa, and MAPE to 8.19%, while the average R² decreases slightly to 0.94.

Visual inspection of Figure 7 reveals that the predicted moduli tend to deviate systematically from the 1:1 line, producing a slight offset pattern. This shift reflects the influence of persistent bias in the input data, which cannot be fully corrected by the model’s internal learning process. The Transformer architecture, while capable of capturing complex nonlinear relationships, inherently inherits a portion of the systematic bias embedded in the training data distribution.

Nevertheless, even under such challenging conditions, the model’s prediction accuracy remains acceptable for engineering applications. The R² values for all layers remain above 0.90, demonstrating that the model retains substantial predictive capability. These findings suggest that moderate systematic measurement errors do not critically impair the Transformer’s inference reliability, making it feasible for use with field FWD data where small calibration biases are common.

The most realistic testing condition involves the coexistence of both random and systematic errors. Figure 8; Table 7 show that under this comprehensive noise environment, the Transformer model continues to perform robustly. The average MAE increases modestly to 732.56 MPa, while the average R² remains high at 0.95. The MAPE of 7.48% indicates that overall prediction deviations remain within an acceptable engineering range.

The subgrade layer once again demonstrates the highest stability, with an R² of 0.96, reflecting its lower sensitivity to noise due to smaller deflection amplitude variability. The surface and base layers experience minor performance degradation; however, the overall trend remains consistent, confirming that the Transformer effectively generalizes the underlying input–output relationship even when measurement uncertainty increases.

The results collectively demonstrate that the Transformer-based back-calculation model is not only accurate under ideal conditions but also robust and reliable under realistic noise perturbations.

For a fair and consistent comparison, all baseline models (BPNN, SVR, and XGBoost) were trained and evaluated under the same experimental conditions as the proposed Transformer model. Specifically, all models used identical input features (nine FWD deflection peaks), output targets (layer elastic moduli), train–test split (70%/30%), and data preprocessing procedures, including Z-score normalization. The comparative evaluation was conducted on the same synthetic dataset with combined random and systematic errors, and model performance was assessed on an identical test set using the same evaluation metrics (MAE, MSE, RMSE, MAPE, and R²) for the surface course, base course, and subgrade. The hyperparameters of all baseline models were selected using standard tuning strategies within commonly accepted ranges. For the BPNN, the number of hidden neurons and the learning rate were adjusted empirically based on validation performance. For SVR, key hyperparameters including the kernel type, penalty parameter, and kernel width were optimized using grid search. For XGBoost, the tree depth, learning rate, and number of estimators were tuned through empirical validation. Default parameter settings were avoided when they resulted in clear underfitting or overfitting. Consequently, the adopted configurations represent reasonable and competitive baselines rather than minimally tuned models. The corresponding test results are summarized in Table 8.

As shown in Table 8, the Transformer model achieves the lowest overall error levels and the most consistent performance across all three layers. In terms of MAPE, the average values across the three layers are approximately 7.48% for the Transformer, compared with 8.44% for BPNN, 12.83% for SVR, and 18.62% for XGBoost. The corresponding average R² values are about 0.95 for both the Transformer and BPNN, but decrease to roughly 0.89 and 0.77 for SVR and XGBoost, respectively. These results indicate that although BPNN can reach a comparable average R², it still yields larger errors than the Transformer, whereas SVR and XGBoost suffer from a clear degradation in predictive accuracy under the random and systematic noisy condition.

The advantage of the Transformer is particularly evident for the base course, which is generally the most difficult layer to identify due to its intermediate position and strong interaction with both the surface course and the subgrade. For this layer, the Transformer attains MAE = 882.85 MPa, RMSE = 1172.39 MPa, MAPE = 8.83%, and R² = 0.94, outperforming BPNN (MAE = 1073.63 MPa, MAPE = 10.63%, R² = 0.91) and substantially surpassing SVR and XGBoost (e.g., XGBoost yields MAPE = 28.04% and R² = 0.56). For the surface course and subgrade, the Transformer also provides competitive MAE/RMSE and high R² values, remaining at least as accurate as, and in several cases more accurate than, the baseline models.

These quantitative comparisons help clarify why a Transformer is preferred over simpler models in the proposed SEM + Transformer framework. The multi-head self-attention mechanism enables the Transformer to explicitly capture global dependencies among all FWD deflection measurements, allowing the network to focus on physically informative deflection patterns and to down-weight noisy or less relevant components. In contrast, BPNN relies on fixed fully connected mappings, SVR depends on pre-defined kernel functions, and XGBoost aggregates a series of decision trees, all of which have more limited capacity to represent the highly nonlinear and ill-posed mapping from surface deflections to multilayer elastic moduli under noisy conditions. Consequently, the Transformer not only achieves lower errors and higher R² on the random and systematic noisy dataset, but also exhibits stronger robustness and generalization, especially for the critical base course.

The above results presented above indicate that the proposed SEM–Transformer framework achieves high predictive accuracy and strong robustness across all considered noise scenarios. Beyond numerical accuracy, however, two fundamental issues deserve careful discussion: physical plausibility of the predicted moduli and identifiability of the inverse mapping from FWD deflections to multilayer elastic properties. It should be emphasized that the adopted Transformer configuration is not claimed to be universally optimal. Rather, it represents a compact and effective design choice tailored to the specific characteristics of the FWD-based back-calculation problem considered in this study.

From the perspective of physical plausibility, the predicted moduli in this study remain consistent with expected pavement mechanics within the predefined parameter space. Across all experiments and noise levels, no non-physical outcomes such as negative elastic moduli or severe modulus layering were observed in the reported test cases. In particular, the stiffness ordering between the surface course, base course, and subgrade is generally preserved. This behavior can be attributed primarily to the use of SEM-generated training data, which inherently satisfy mechanical consistency and realistic stiffness hierarchies.

Nevertheless, it is important to emphasize that the Transformer model itself is unconstrained. No explicit monotonicity, ordering, or inequality constraints (e.g., E surface ≥ E base ≥ E subgrade ) are enforced during training or inference. As a result, the observed physical consistency arises implicitly from the data distribution rather than from hard constraints embedded in the learning model. In principle, when applied outside the training distribution or under substantially different field conditions, the model may produce physically inconsistent modulus combinations, such as a surface-layer modulus lower than that of the subgrade. This limitation is common to most purely data-driven back-calculation approaches and should be carefully considered in practical applications.

The issue of identifiability and uniqueness is intrinsic to FWD-based modulus back-calculation and is independent of the specific learning algorithm employed. The inverse mapping from surface deflection basins to multilayer elastic moduli is inherently ill-posed and non-unique: different combinations of layer properties may yield very similar surface deflection responses, particularly under limited sensor spacing and in the presence of measurement noise. Consequently, even under ideal noise-free conditions, a mathematically unique inverse solution does not generally exist.

In this context, the role of the proposed Transformer model is not to recover a unique physical solution, but rather to learn a statistically optimal inverse mapping conditioned on the assumed parameter ranges, pavement configurations, sensor layout, and noise characteristics represented in the training data. The predicted moduli should therefore be interpreted as the most probable estimates within this constrained statistical space, rather than as exact physical truths. This interpretation is consistent with both classical optimization-based back-calculation methods and recent data-driven approaches reported in the literature.

Finally, it should be recognized that a non-negligible domain shift exists between SEM-generated responses and real-world FWD measurements. Real pavements exhibit temperature-dependent and viscoelastic material behavior, layer heterogeneity, construction-induced variability, and non-ideal load–pavement contact conditions, whereas the present SEM model assumes linear elasticity, homogeneity, and axisymmetry. Field FWD data are also affected by sensor coupling effects and spatially correlated measurement errors that are difficult to reproduce numerically. While the noise models adopted in this study provide a first-order approximation of measurement uncertainty, they do not fully capture these complexities. Addressing both physical constraint enforcement and the simulation-to-field domain gap will be essential steps toward reliable deployment of the proposed framework in real-world pavement evaluation and digital twin–based management systems.

From the perspective of inverse problem theory, pavement modulus back-calculation based on FWD deflections is a fundamentally ill-posed problem. The surface deflection basin represents an aggregated structural response, and different combinations of layer moduli may produce very similar deflection profiles, particularly when sensor spacing is limited and measurement noise is present. The proposed Transformer model does not eliminate this ill-posedness, but rather provides a data-driven regularization by learning the most statistically probable inverse mapping under the assumed parameter ranges and noise conditions.

It should also be noted that different pavement layers exhibit markedly different sensitivities in FWD measurements. The subgrade modulus predominantly controls the overall curvature and far-field deflections of the basin, while the surface and base layers mainly affect near-load deflections. As a result, the inverse mapping is inherently more sensitive to subgrade stiffness variations than to variations in upper-layer moduli. This sensitivity imbalance explains the near-perfect R² values observed for the subgrade in the present study. Such high R² values reflect dominant sensitivity rather than guaranteed identifiability or uniqueness of the subgrade modulus.

The current study does not explicitly assess the model’s ability to distinguish between different modulus combinations that generate nearly indistinguishable deflection basins. A rigorous identifiability or sensitivity analysis—such as controlled perturbation studies or equivalence-class analysis—would be required to quantify this capability and is left for future work. The goal of this study is not to exhaustively benchmark all possible architectures, but to demonstrate feasibility and robustness of a Transformer-based inverse framework. The comparative analysis is intended to provide contextual performance references under consistent experimental conditions, rather than a statistically exhaustive or uncertainty-aware benchmark across all model families.

Across all four scenarios, the Transformer-based intelligent back-calculation framework demonstrates high accuracy, stability, and adaptability. The model achieves an average R² exceeding 0.94 under all noise conditions, confirming its robustness against both random and systematic measurement errors. The average MAPE values remain below 8%, well within acceptable limits for pavement engineering applications.

These findings verify that the Transformer model effectively learns the intrinsic relationships between FWD deflection responses and layer elastic moduli, even in the presence of complex noise patterns. Consequently, this method provides a reliable and data-driven solution for practical modulus back-calculation tasks, offering improved accuracy and interpretability compared with traditional approaches.

The Transformer-based model achieved excellent predictive performance, with average R 2 exceeding 0.94 and MAPE below 8% across all scenarios. Even under combined random and systematic noise, the model maintained stable accuracy, demonstrating strong generalization and robustness to measurement uncertainty.

By leveraging multi-head self-attention, the Transformer effectively captured global dependencies among deflection sensors, enabling precise mapping between surface deflection patterns and underlying layer moduli. The predicted trends were physically consistent with pavement structural behavior—subgrade moduli showed the highest stability due to smoother deformation responses, while the surface layer exhibited slightly higher variability owing to its stiffness contrast.

Once trained, the proposed model provided rapid, millisecond-level predictions, offering a computationally efficient and fully data-driven solution for modulus inversion. Its end-to-end design minimizes manual parameter tuning and avoids convergence issues common in traditional iterative back-calculation, supporting integration into real-time pavement condition evaluation and intelligent maintenance systems.

Overall, the developed SEM–Transformer framework demonstrates strong potential for intelligent, accurate, and efficient pavement structural evaluation, and provides a promising basis for data-driven digital twin systems in pavement management. However, the present study also has clear limitations. All training and testing data are synthetically generated using an SEM model, and no field FWD dataset is used for direct validation. As a result, the current findings should be interpreted as a numerical benchmark demonstrating feasibility and robustness, rather than as evidence of immediate field applicability. Future work will focus on validating the proposed framework using large-scale field FWD datasets, incorporating temperature-dependent and viscoelastic material behavior, and developing physics-guided or domain-adaptive learning strategies to mitigate the simulation-to-field gap. These efforts are essential before the proposed method can be reliably deployed in real-world pavement evaluation and digital twin–based infrastructure management systems. Future work will also explicitly address physical constraint enforcement by incorporating monotonicity or inequality constraints into the learning process, for example, through output reparameterization, physics-guided loss functions, or hybrid inversion frameworks. Such extensions are expected to further improve physical interpretability, reduce the risk of inconsistent predictions, and enhance robustness when applying the model to real-world FWD datasets. It should be emphasized that the reported prediction accuracy does not imply mathematical uniqueness of the inverse solution. The proposed framework provides statistically optimal estimates conditioned on the assumed data distribution, rather than resolving the intrinsic non-uniqueness of pavement modulus back-calculation. Future work will incorporate uncertainty quantification and statistical testing, such as repeated sampling, confidence interval estimation, or Bayesian approaches, to further strengthen the rigor of comparative performance evaluation.