To model the spatiotemporal evolution of road distress, the PED model integrates discrete cellular transitions, stochastic diffusion dynamics, and physics-embedded learning into a unified framework, as illustrated in Fig. 1. PED builds a continuous latent distress field {\mathrm{\Phi }}_t\in {ℝ}^{m\times n}, which couples discrete local updates and global stochastic propagation through shared latent variables. This allows simultaneous representation of local transitions, global diffusion, and intervention-driven disturbance within a single mathematical structure.

The overall workflow of PED consists of 3 hierarchical layers:

To identify the optimal hyperparameter configuration for the proposed PED model, we conducted a comprehensive grid search across 5 key parameters:

A total of 3⁵ = 243 experiments were carried out to explore all possible hyperparameter combinations. The experiments were conducted on the China Fine-grained Road Distress Dataset (FRD-CN) (see Methods).

Prior to simulation, all other parameters and constants of the PED model were carefully initialized. The physical road domain considered in the FRD-CN case study is 15 m wide and 160 km long. It was represented using a 2-dimensional computational grid comprising 150 cells across the road width and 1,600 cells along the longitudinal direction. Accordingly, one grid row corresponds to 0.1 m in the transverse direction, while one grid column corresponds to 0.1 km (100 m) in the longitudinal direction. The full road was further divided into 32 spatial sectors, each covering 5 km and corresponding to 50 longitudinal grid columns. Considering the computational cost of training, we selected 25% of the dataset under initial setup to conduct the hyperparameter evaluation experiments. A total of 265 iterations were performed per simulation to ensure temporal convergence, with each iteration representing 1 d. The degradation probabilities were initialized as p_{\mathrm{e}1}=0.1 for cracks and p_{\mathrm{e}2}=0.075 for potholes based on study of typical cases in the data. Longitudinal cracks spread along the road’s length, transverse cracks across its width, and alligator cracks diagonally through interconnected patterns, and potholes expand outward in all directions. Precipitation, traffic volume, and temperature were incorporated into the model through a relative environmental influence factor rather than as fixed absolute inputs. Following normalization and temporal aggregation, the 3 factors were combined and scaled against their global mean, producing a sector-specific coefficient that dynamically adjusted road distress propagation and repair probabilities. In addition, min–max normalization was applied to ensure that the magnitudes of loss components are comparable during training. Model performance was assessed using the area MWPE index.

The parallel-coordinates visualization in Fig. 2 reveals that the lower learning rate \eta (0.0001) slightly improves stability, while a medium discount factor \gamma (0.99) enhances long-term consistency. Similarly, medium epsilon decay ({ϵ}_d=0.99) maintains exploration longer and achieves lower errors. The physical-loss weight ℯ shows an optimal range around 0.3; too small weakens physical constraints, and too large limits flexibility. The moderate diffusion rate \pi (0.02) improves spatial smoothness and predictive accuracy. In summary, the best configuration features a lower learning rate (0.0001); moderate discount factor (0.99), epsilon decay (0.99), and physical-loss weight (0.3); and relatively strong diffusion (0.02), underscoring the need to balance exploration, long-term reward guidance, physical consistency, and diffusion-driven stabilization for optimal performance.

To determine the optimal environmental history window length, we conducted comparative experiments under same initialization settings and fixed optimal hyperparameters from the prior evaluation. Model performance was assessed using area MWPE, PCI root mean square error (PCI RMSE), PCI mean absolute error (PCI MAE), and PCI mean absolute percentage error (PCI MAPE). The results are summarized in Table 1.

The results showed that shorter windows (30) achieved reasonable accuracy in area MWPE (=4.391%) but exhibited higher variability in PCI metrics (PCI MAPE = 9.298%). A length of 50 consistently yielded the best results, with the lowest PCI MAPE (9.262%), indicating strong agreement with observed deterioration and modest spatial consistency (area MWPE = 4.484%). Longer windows (55 to 70) offered no further gains and occasionally increased uncertainty, suggesting redundant information.

To assess the influence of historical feature decay, additional experiments were performed with the window fixed at 50 and decay rates ranging from 0.3 to 0.7. The performance metrics are summarized in Table 2.

Area MWPE demonstrated moderate sensitivity to variations in the decay rate. A decay rate of 0.5 achieved the lowest area MWPE (4.484%), reflecting balanced accuracy across scales. Higher rates (0.6 to 0.7) produced higher errors, indicating that excessive retention or rapid forgetting of historical effects reduces accuracy. Lower rates (0.3) caused higher area MWPE as well. Therefore, a decay rate of 0.5 provides the optimal trade-off, capturing meaningful environmental history without overemphasizing outdated information.

To evaluate the effectiveness of each component in our proposed framework, ablation studies were also conducted on both FRD-CN and the US Long-Term Pavement Performance Dataset (LTPP-US). The FRD-CN ablation study adopted the same road-grid configuration described above. For the LTPP-US ablation study, a 150 × 1,800-cell grid was used, with the road sections grouped into 36 spatial regions according to their geographic coordinates.

The experiments used the same initialization settings and fixed optimal hyperparameters, testing 5 model variants with different core module combinations: (a) full model, (b) no transitional diffusion, (c) no physics-embedded multifidelity constraint, (d) no disturbance-aware module, and (e) no external environmental inputs.

The FRD-CN and LTPP-US ablation models were trained and evaluated for 265 and 33 time steps, respectively. Prediction accuracy was measured using area MWPE and PCI MAPE. Figure 3 illustrates the error distributions across time steps using boxplots and scatter points, with the horizontal axis denoting the 5 model variants and the vertical axis showing error values, including mean, maximum, and minimum statistics.

As shown in Fig. 3A, the full model yielded the lowest mean area MWPE error (4.484%), whereas excluding disturbance-aware (5.383%) or environmental inputs (6.666%) caused the largest degradation. The diffusion-removed variant has a slightly higher mean area MWPE (4.778%), suggesting that diffusion improves area-level estimation. The physical-constraint-removed model also yields higher error (5.042%) and variance, underscoring the importance of physical constraints.

A similar pattern appears in PCI MAPE (Fig. 3B): The full model achieves the lowest prediction error, with a PCI MAPE of 9.262%. Removing the diffusion module slightly increases the error to 9.285%, while excluding the physics-informed constraint results in a further increase to 9.329%. Although these differences are modest, the consistently lower error of the full model indicates that both mechanisms contribute to improved predictive accuracy. The diffusion module helps represent the spatial propagation and neighborhood-dependent evolution of road distress, whereas the physics-informed constraint regularizes the deterioration process by encouraging predictions to remain consistent with the underlying physical evolution pattern. Their combined use therefore improves the model’s ability to capture both local distress interactions and the overall temporal deterioration trend. The variants without the disturbance-aware mechanism and without environmental inputs yield the highest PCI MAPE values, at 9.470% and 9.346%, respectively. This result highlights the importance of incorporating both maintenance-related disturbances and environmental conditions to improve the model’s predictive performance.

Consistent trends were found on the LTPP-US dataset (Fig. 3C and D). The full model again achieves the best PCI prediction (mean MAPE = 28.054%) with removing diffusion (28.431%), while removing the physical constraint (29.554%), disturbance (30.030%), or environmental input (28.373%) modules significantly degrades performance. In area MWPE, the full model (8.957%) outperforms all variants, with the environmental-input-removed model showing the largest deterioration (10.839%). The stronger sensitivity to environmental information removal in the LTPP-US dataset suggests that environmental factors become more influential in long-term forecasting.

Overall, both datasets indicate that the full model achieves an optimal balance between global PCI prediction accuracy and fine-grained distress-level prediction performance. The cross-dataset comparison also demonstrates strong generalization across regions and time scales (short- and long-term), although performance on LTPP-US is lower due to its coarser temporal granularity and greater heterogeneity. These findings also highlight that fine-grained data substantially enhance model accuracy and reliability.

To further illustrate the simulation results, we visualize the spatiotemporal evolution of road conditions corresponding to the FRD-CN dataset. In Fig. 4, the horizontal axis represents the longitudinal direction of the road, and the vertical axis corresponds to the lane width. Four lanes are delineated by white dashed lines. For each cell, light green represents a healthy state; shades of blue (from light to dark) denote longitudinal, transverse, and alligator cracks and potholes as shown in the legend; and dark green indicates repaired areas. The figures present 4 representative snapshots: the initial condition and the simulated states after 70, 140, 210, and 230 time steps.

It can be observed that crack patterns progressively intensified over time. Alligator cracking developed severely compared with other crack types, with noticeable deterioration emerging around stake number −5. Sudden pothole initiations are observed near stake −20 around iteration 70 (approximately 2022 August 3). This rapid initiation appears to be associated with short-term dynamics, as a localized rainfall peak can be identified around the same period (see the “Road distress data source” section in Methods), which may have accelerated the deterioration process. From a long-term perspective, this event coincides with a period of rising temperatures beginning in July and reaching a peak around August, along with sustained rainfall during July and early August. Although these rainfall events are not the most intense, their persistence may have contributed to cumulative moisture-related damage, facilitating the observed distress evolution. Similarly, around iteration 230 (approximately 2023 January 10), new potholes are observed near stake +25. This event also appears to be associated with a short-term rainfall peak around that date. In addition, longer-term conditions, including gradually decreasing temperatures during winter and relatively low precipitation in the surrounding period, may have contributed to material stiffening and increased vulnerability to traffic-induced damage. Pothole regions have undergone rapid repair. Nevertheless, even after these repairs, areas of previously distressed areas might still exhibit renewed signs of distress (e.g., around stake numbers −45 and +72), which was highly consistent with real-world deterioration pattern.

At the section level, several consistent patterns can also be observed. First, transverse cracking within the region tends to exhibit more severe deterioration (into potholes) during winter periods (around stake −80), which is likely associated with temperature drops that induce material contraction and increase tensile stress. Second, cracks tend to extend more frequently during both hot-humid periods (summer, around iteration 70) and cold-wet periods (winter, around iteration 230). In summer, higher humidity and sustained rainfall may promote moisture infiltration and cumulative distress, while in winter, lower temperatures can lead to increased stiffness and brittleness of the asphalt binder, making it more susceptible to aggregate loss under traffic loading. In addition, larger diurnal temperature variations in both seasons may induce repeated thermal expansion–contraction cycles, contributing to fatigue accumulation and crack propagation.

In addition, Fig. 5 illustrates the temporal evolution of different distress types throughout the simulation. The horizontal axis represents distress categories, while the vertical axis shows their corresponding proportions. Blue curves, deepening in color with increasing iteration steps, depict the progression over time. Initially, various types of distress were distributed across the road section. In the FRD-CN dataset (Fig. 5A), the proportions of longitudinal and transverse cracks gradually decreased, and alligator cracking also diminished. Potholes had a relatively small proportion. The LTPP-US dataset (Fig. 5B) exhibited a similar pattern. As observation years progressed, longitudinal, transverse and alligator cracking repeatedly exhibited a pattern of decrease. The proportion of potholes remained consistently low, likely due to effective preventive maintenance, routine resurfacing, and strict safety standards in US highway networks. The extent of repairs showed a rise that stabilized at a high level, further evidencing the sustained effectiveness of maintenance practices.

We conducted a performance comparison between the proposed model and state-of-the-art models. The state-of-the-art models include the PED model enhanced with a neural PDE evolver, and 4 advanced physics-informed neural network (PINN) variants: (a) finite-difference PINN (fPINN), (b) residual-based PINN, (c) Fourier PINN (FNN), and (d) adaptive multidomain PINN (XPINN). fPINN is designed to incorporate finite-difference discretization into the physics-informed learning framework, making it particularly suitable for approximating localized spatial variations and sharp state transitions in road deterioration. Since road surface degradation often exhibits strong local dependency and neighborhood-driven propagation behavior, the finite-difference formulation in fPINN provides a natural mechanism for modeling discrete spatial gradients and localized distress diffusion. Residual-based PINN emphasizes enforcing governing equations through residual minimization, thereby improving the physical consistency of the predicted deterioration process by constraining the solution space according to predefined evolution dynamics. FNN leverages Fourier feature representations to capture periodic and globally smooth patterns. This design is theoretically motivated by the fact that road deterioration often contains low-frequency global trends driven by long-term environmental loading, traffic accumulation, and seasonal cyclic effects. Fourier mappings enhance the network’s ability to represent multiscale spatial-frequency components and alleviate the spectral bias problem commonly observed in standard neural networks when approximating high-frequency variations. XPINN further decomposes the computational domain into multiple subregions, enabling localized physics constraints and independent subnet optimization. This mechanism is particularly suitable for road systems because road deterioration processes are highly heterogeneous across different spatial regions due to varying traffic loads, environmental exposure, and maintenance histories. Domain decomposition, therefore, improves the model’s capability to capture spatial heterogeneity and multiscale interactions within large-scale road networks. Overall, these model variants were selected because their underlying mathematical structures correspond to different physical characteristics of road deterioration, including localized diffusion behavior, global temporal evolution, spatial heterogeneity, and multiscale deterioration interactions.

All models were implemented within the original training framework and interface definitions, allowing seamless switching under the same setup to ensure experimental comparability. Evaluation was conducted using area MWPE, PCI RMSE, PCI MAE, PCI MAPE, and computing time, with the results summarized in Table 3. All computational experiments were conducted on a laptop equipped with an 8-core ARM-based processor comprising 4 performance cores and 4 efficiency cores. All models were executed in CPU-only mode without discrete GPU acceleration. The implementation was developed using Python 3.9.18 and PyTorch 2.0.0. The computation times reported in Table 3 were measured under the same hardware and software environment for all models to ensure a consistent comparison.

The neural PDE model exhibited a modest improvement over the PED model in macroscopic PCI prediction, reducing the error from 9.262% to 9.041%, indicating that incorporating more complex fatigue mechanisms at the macroscopic scale enhances the physical representation of deterioration processes. However, at the individual-distress level, the error increased markedly from 4.484% to 45.581%. This discrepancy suggests that although the neural PDE formulation improves the overall smoothness and global consistency of deterioration dynamics, it may simultaneously weaken the model’s capability to preserve fine-grained local distress patterns. In particular, the diffusion-driven propagation mechanism tends to spatially redistribute distress states, which can amplify local deviations in regions with sparse or highly heterogeneous distress distributions.

Most PINN variants performed worse in both macroscopic and microscopic tasks, except residual-based PINN variant. The others’ macroscopic prediction errors (9.276%, 9.286%, and 9.317%) all exceeded that of the PED model (9.262%), while all individual-distress errors of the 4 variants (4.749%, 4.700%, 4.874%, and 4.516%) were substantially higher than the PED model’s 4.484%. The inferior performance of neural PDE and PINN variants may be partly due to their reliance on predefined or stationary governing equations, which are difficult to maintain under highly nonstationary real-world conditions (e.g., changing traffic loads and environmental stressors). In contrast, the proposed model better accommodates such nonstationarity through data-driven learning guided by mechanism-informed constraints. The residual-based PINN variant improves the predictive accuracy of the macroscopic PCI metric, but at the cost of substantially increased computational time and complexity, with a computing time of 6,423.39 s.

It is worth noting that extreme values can be observed in the area MWPE results for the PED model (45.581%). These extreme values are primarily attributed to the inherent instability of physics-informed models when applied to highly heterogeneous and nonstationary real-world deterioration data. When the model fails to accurately capture localized distress evolution, even relatively small absolute prediction errors in regions with low true values can result in disproportionately large percentage errors, thereby amplifying the area MWPE metric. Moreover, PINN-based approaches rely on predefined governing equations and boundary conditions, which are often difficult to specify consistently under complex and evolving real-world conditions. This mismatch can lead to unstable solutions in localized regions. In addition, the presence of sparse observations and measurement noise may further exacerbate sensitivity to input perturbations, resulting in large fluctuations in percentage-based error metrics.

From a practical deployment perspective, the computational overhead of the proposed PED model remains manageable. The computation time for processing about 1 year over a 160-km road network is approximately 467.17 s, demonstrating its potential for scalable urban applications, compared with 290.60 s for the neural PDE model and over 1,000 s for the 4 PINN variants. The model operates on a cell-based discretization framework, where state transitions are computed locally and aggregated across the network. This structure enables efficient scaling, as computations can be parallelized across spatial units and further benefit from ongoing improvements in computational resources. In addition, our analysis shows that a 50-d historical environmental window provides an effective balance between capturing sufficient temporal context and maintaining computational efficiency. In practical applications, this window can be extended when computational resources allow, while retaining a 50-d window remains sufficient to achieve reliable performance under constrained conditions.

These substantial computational costs of PINN variants hinder rapid iteration for city-scale DT applications. Although the neural PDE model demonstrated an advantage in computational efficiency, its prediction errors at the distress level were considerably larger. Therefore, when both predictive accuracy and computational efficiency are considered, the lightweight PED model appears more suitable for large-scale deployment and practical DT applications.

To further evaluate the robustness of the proposed model under realistic perturbations, a sensitivity comparison with state-of-the-art models was conducted. Four perturbation scenarios were considered: (a) overdetection bias (false positives), where nonexistent distress is incorrectly introduced into the model input (ratio: 0.1%); (b) underdetection bias (false negatives), where existing distress is missed in the input (ratio: 1%); (c) spatial missing, where 5% of road segments are randomly missed (masked); and (d) macro-measurement noise, where Gaussian noise with a standard deviation of 1.0 is added to the PCI observations.

The results are summarized in Table 4. Compared with the baseline results in Table 3, the PED model exhibits increased prediction errors in area MWPE under all 4 perturbation scenarios. Among them, the model demonstrates the strongest robustness to macro-measurement noise, with area MWPE increasing only slightly from 4.484% to 4.542%. In contrast, the largest performance degradation is observed under the overdetection bias scenario, where area MWPE increases to 5.540%, indicating that the PED model is relatively sensitive to falsely detected distress inputs. Under the overdetection and underdetection bias scenarios, the PCI MAPE of PED all slightly decreases, suggesting that the model may perform relatively better in macro-tasks with noise, although its overall generalization ability remains limited.

Although the neural PDE model consistently achieved the lowest PCI MAPE values across most disturbance scenarios, its area MWPE increased dramatically, particularly under overdetection bias. This suggests that continuous PDE diffusion tends to amplify local disturbance propagation after categorical discretization, resulting in severe instability at the microscopic distress level despite relatively accurate macroscopic PCI prediction. The extremely large maximum error values (e.g., 141.125%) further indicate that PDE-driven smoothing may excessively diffuse localized disturbance patterns into neighboring regions.

PINN-based variants exhibited relatively stable microscopic distress consistency under disturbance conditions, suggesting that physics-informed constraints help suppress excessive local error propagation. However, the improvements did not translate into superior PCI prediction accuracy.

Among all disturbance scenarios, overdetection bias produced the most severe degradation in microscopic distress consistency. This phenomenon is mainly attributed to the spatial propagation mechanism, where falsely detected distress states can diffuse to neighboring regions and accumulate during iterative evolution. On the other hand, all models show relatively low sensitivity to measurement noise, suggesting that fluctuations in macro-level statistical indicators (e.g., PCI) have a limited impact on model performance. In comparison, perturbations at the micro-level (e.g., distress state modifications) are more critical and can notably affect prediction stability.

In addition, a comparison between Fig. 4 and Fig. S3 further reveals that the PED model exhibits stable and physically consistent behavior in capturing the full life cycle of road distress, including initiation, propagation, maintenance intervention, and subsequent redevelopment. In contrast, the neural PDE variant tends to maintain macroscopic stability by underestimating the evolution of distress, which deviates from the inherently nonstationary nature of real-world spatiotemporal deterioration processes. The 4 PINN variants, as shown in Fig. S3b to e, on the other hand, tend to overestimate both the temporal progression of distress and its spatial interactions. This is primarily because such models enforce predefined physical constraints that encourage smooth and globally consistent evolution patterns, often at the expense of accurately capturing localized and heterogeneous dynamics.

To analyze the model’s applicability under specific operational scenarios, several working conditions were selected.

To evaluate the effectiveness of the proposed method over a broader spatial domain, we conducted validation on the LTPP-US dataset. The average PCI values for the 36 regions are shown in Fig. 12. Compared with the section-level fluctuations, the PCI values at the region level exhibited more pronounced oscillations and wider upper and lower bounds. This was attributable to the dataset’s longer temporal span and larger geospatial coverage, which introduced richer and more representative variations.

Figures 13 and 14 illustrate the cumulative normalized rewards and physical loss during regional-level training under different volumes of simulated data. As shown in Fig. 13, the cumulative normalized reward increases progressively with the number of iteration steps for all configurations. However, the rate of reward accumulation generally decreases as the number of simulated training regions increases. Configurations involving fewer regions exhibit steeper reward trajectories and higher final cumulative values, whereas those involving larger spatial domains tend to produce flatter curves and lower terminal rewards. This pattern suggests that increasing the number of regions introduces greater spatial heterogeneity and a more complex deterioration environment, thereby making the learning and optimization process more challenging. As shown in Fig. 14, the physical loss exhibits noticeable fluctuations during the early training stages, with several configurations showing short-lived peaks. Nevertheless, the loss decreases rapidly and converges to relatively low values as training proceeds across nearly all regional settings. These results indicate that, although expanding the simulated spatial domain increases the difficulty of reward optimization, the physics-informed training process remains numerically stable and capable of converging under different regional scales. Overall, the model demonstrates strong potential for simulating road deterioration across broader spatial domains.

The proposed PED model integrates stochastic evolution, continuous diffusion, disturbance-aware transitions, and physics-informed corrections into a unified spatiotemporal architecture for modeling road distress evolution. As illustrated in Fig. 17, the model begins with an initial distress field S_{i,j}^t, which evolves under the combined influence of environment inputs E_t and real distress observations R_t. The evolution process is governed by 3 complementary mechanisms.

First, a discrete stochastic transition module models the short-term and neighborhood-dependent distress growth P\left(S_{i,j}^{t+1}=k|S_{i,j}^t,N_{i,j}^t,E_t\right), where the temporal weight \zeta regulates the contribution of historical states and environmental factors in nonstationary environments. Second, a continuous diffusion component performs localized smoothing through a physics-inspired operator f_{\mathit{\text{local}}}\left({\mathrm{\Phi }}^t,{\mathcal{N}}^t,P,E_t\right), where the diffusion rate \pi controls the spatial propagation strength. Third, disturbance-aware transitions \mathcal{A}\left(s,a,t\right) incorporate the impacts of maintenance actions and external events, with the exploration-decay factor {ϵ}_d determining the extent of adaptive perturbation introduced during learning-based action selection.

In parallel, physical constraints enforce consistency with governing diffusion principles using a PDE residual-based loss {\mathcal{L}}_{\mathit{\text{phys}}}. The physics-loss weight ℯ balances this constraint against data-driven objectives {\mathcal{L}}_{\mathit{obs}}. HF observations u^{\mathit{HF}} and LF simulated fields u^{\mathit{LF}} are fused in the loss function to ensure multifidelity consistency and robust extrapolation across varying spatiotemporal regimes.

The training loss adjusts model parameters under a learning rate \eta. The reward is controlled by the discount factor γ to balance future rewards in the model’s update rule. The overall updated parameters determine the formation of the next state S_{i,j}^{t+1}, completing a full cycle. Together, these components allow the PED model to capture multiscale distress propagation, incorporate environmental and physical influences, and maintain stability under heterogeneous and disturbance-prone real-world conditions.

PED embeds interpretable physical priors within a data-driven framework, including local spatial dependency, diffusion-based distress propagation, deterioration–repair transitions, environmental modulation, and consistency between simulated results and observed condition indicators.

The local propagation operator {\mathcal{T}}_{\mathit{\text{prop}}} controls the stochastic spread of active distress states to neighboring cells. For a cell c=\left(x,y\right) and one of its neighboring cells n\in \mathcal{N}\left(c\right), the transition probability can be written in a general form aswhere l\in \left\{1,2,3,4\right\} represents an active distress state, p_l is the baseline propagation probability for distress type l, a_t is the action-dependent multiplier, E_t\left(y\right) is the environmental scaling factor for location y, G_t\left(y\right) is the sector-level area-consistency gate, and D_t\left(n\right) is the diffusion spatial dependency signal. The coefficient {\eta }_l controls the influence of diffusion on the corresponding distress type.

The spatial diffusion operator \mathcal{D} represents neighborhood-dependent distress expansion. The operator generates a spatial probability field that modulates the likelihood of additional distress formation in locally susceptible regions. This can be expressed aswhere F_{\text{diff}}\left(S_t,c\right) is the diffusion probability factor estimated from the current cellular distress configuration, \delta is a normalization constant, and \text{clip}\left(·\right) restricts the signal to the range \left[0,1\right]. This operator is used to increase the probability of distress occurrence in cells that are both spatially exposed and adjacent to existing distress clusters. Therefore, it acts as a physics-informed spatial regularization mechanism rather than a deterministic mechanical diffusion model.

The environmental scaling operator \mathcal{E} adjusts deterioration and repair probabilities according to external exposure. Temperature, rainfall, and traffic volume are first combined into a normalized environmental exposure term e_t\left(y\right) for each road column or sector. The scaling factor is then defined aswhere c_e is a constant scaling coefficient; E_{\min } and E_{\max } define the lower and upper bounds of the environmental effect, respectively; and \gamma is the environmental sensitivity exponent. This operator reflects the assumption that deterioration transition probabilities should vary with environmental and loading conditions while remaining bounded to avoid unrealistic amplification.

The repair transition operator \mathcal{R} controls transition from active distress states to the repaired state. For a cell c=\left(x,y\right) with an active distress state S_t\left(c\right)\in \left\{1,2,3,4\right\}, the repair probability is defined aswhere p_r is the baseline repair probability, I_t\left(y\right) is the repair adjustment factor, \epsilon is a small numerical stabilization term, and r_{\min } and r_{\max } define the lower and upper bounds of the repair probability, respectively. This operator represents the physical assumption that maintenance and repair activities convert active distress cells into repaired cells, while environmental exposure can influence the effective repair transition rate. Repaired cells are also allowed to reenter active distress states with a small probability, representing subsequent deterioration after treatment.

The physics residual operator introduces a regularization term for the latent deterioration field u\left(x,t\right). The residual is defined aswhere u\left(x,t\right) denotes the latent deterioration intensity at location x and time t and k is a diffusion coefficient. This term encourages the learned latent deterioration field to evolve smoothly in space and time according to a diffusion-type regularity assumption. It does not imply that the model is a full continuum-mechanics road model; rather, it provides a physically motivated constraint to reduce unrealistic discontinuities in the learned deterioration process.

Finally, the equivalent-area and PCI consistency operator \mathcal{C} links the cellular distress states to observed road condition indicators. For each sector r, the equivalent distress area of label l is calculated aswhere N_{r,t}^{\left(l\right)} is the number of cells with distress label l in sector r; {\alpha }_l^0 and {\alpha }_l^1 are the start and end equivalent-area coefficients for distress type l, respectively; and {\rho }_t=t/T is the normalized simulation progress. The estimated sector-level PCI is then derived from the equivalent areas using a distress-weighted transformation:where A_{r,t}^{\left(1,2\right)} denotes the combined equivalent area of longitudinal and transverse cracking, A_{r,t}^{\left(3,4\right)} denotes the combined equivalent area of alligator cracking and potholes, A_{r,t}^{\left(5\right)} denotes the repaired or patched area, and L_r is the sector length in grid columns. This consistency operator ensures that the cell-level deterioration simulation remains linked to both observed distress-area evolution and sector-level PCI measurements.

These operators collectively define the PED module as a physics-informed deterioration representation. The stochastic transition and repair operators describe the cellular evolution of distress states; the diffusion-like spatial operator introduces neighborhood-dependent propagation; the environmental scaling operator incorporates exposure effects from temperature, rainfall, and traffic volume; the physics residual operator imposes a smooth diffusion-type regularity on the latent deterioration field; and the equivalent-area and PCI consistency operator connects cell-level simulation outputs with observed road performance indicators.

Inspired by the principle of differentiated feature weighting in attention-based hybrid learning [55], an attention-inspired adaptive weighting mechanism was incorporated into the proposed framework to regulate the contributions of heterogeneous deterioration factors. Unlike conventional multihead self-attention, which learns query–key–value relationships from latent feature sequences, the proposed mechanism assigns physically interpretable, spatially varying weights to environmental exposure, physics-informed predictions, and local diffusion susceptibility. This design is more compatible with the cell-based deterioration process considered in this study because the model must regulate local state-transition probabilities rather than predict a single aggregated sequence-level variable.

For each longitudinal grid position y, the environmental influence factor e_y is first adjusted according to the normalized output of the physics-informed module:where e_y denotes the combined environmental influence derived from temperature, precipitation, and traffic volume; {\phi }_y is the normalized physics-informed influence at position y; and {\alpha }_p controls the contribution of the physics-informed signal. In the present implementation, {\alpha }_p=0.2, while e_{\min }=0.3 and e_{\max }=2.0. This formulation increases or decreases the local environmental contribution according to the spatial deterioration tendency estimated by the physics-informed component, rather than assigning an identical weight to all road cells.

The physics-informed signal is also used to adapt the local repair tendency when both the physics-informed and diffusion modules are activated:

In this expression, {\alpha }_r=0.2, a_{\min }=0.985, and a_{\max }=1.015. The resulting repair probability is calculated aswhere p_0^{\mathrm{rep}} is the baseline repair probability and \epsilon is a small constant introduced for numerical stability. In the current model, the probability is bounded between 0.008 and 0.045. This formulation gives relatively greater importance to cells exposed to adverse environmental and physical conditions, for which spontaneous repair transitions should occur less frequently.

To further represent local spatial dependence, the diffusion output is transformed into a normalized susceptibility weight:where f_{x,y} is the diffusion-derived probability factor at cell \left(x,y\right) and {\eta }_p controls its interaction with the physics-informed component. The diffusion factor is derived from the normalized local Laplacian of the binary distress field and is restricted to a narrow interval to prevent excessive modification of the original transition process. The effective propagation probability of distress type k is subsequently expressed as

In this formulation, p_k is the baseline propagation probability and g_k\left(t\right) is a stage-dependent diffusion weight. A larger d_{x,y} indicates stronger local spatial support for distress propagation and therefore increases the corresponding transition probability. The diffusion weights are larger during the early and intermediate stages and gradually decrease during later iterations, allowing the model to emphasize local spatial interactions during distress development while avoiding excessive diffusion after the deterioration pattern has become established.

Overall, this attention-inspired mechanism does not introduce an additional black-box attention network. Instead, it transfers the central principle of attention (differentiated weighting of information according to its relevance) to the physically interpretable transition rules of the cellular deterioration model. Environmental exposure, physics-informed estimates, and local spatial interactions are therefore weighted dynamically at each location and iteration, enabling the framework to capture heterogeneous deterioration conditions while preserving the interpretability and computational structure of the PED framework.

The Laplacian kernel is based on von Neumann neighborhood, using a convolution kernel to approximate the spatial Laplace operator to achieve neighborhood diffusion, as shown in Eq. (28):

The central negative weight ensures that the focal cell redistributes its value to neighbors, preserving the global mean. Larger weights for orthogonal neighbors capture primary directional diffusion, while smaller diagonal weights allow minor cross-directional smoothing. The sum-to-zero property guarantees numerical stability.