High-resolution highway traffic state information is essential for Intelligent Transportation Systems, but typical traffic data acquired from loop detectors and probe vehicles are often too sparse and noisy to capture the detailed dynamics of traffic flow. We propose PMA-Diffusion, a physics-guided mask-aware diffusion framework that reconstructs unobserved highway speed fields from sparse, incomplete observations. Our approach trains a diffusion prior directly on sparsely observed speed fields using two mask-aware training strategies: Single-Mask and Double-Mask. At the inference phase, the physics-guided posterior sampler alternates reverse-diffusion updates, observation projection, and physics-guided projection based on adaptive anisotropic smoothing to reconstruct the missing speed fields. The proposed framework is tested on the I-24 MOTION dataset with varying visibility ratios. Even under severe sparsity, with only 5% visibility, PMA-Diffusion outperforms other baselines across three reconstruction error metrics. Furthermore, PMA-diffusion trained with sparse observation nearly matches the performance of the baseline model trained on fully observed speed fields. The results indicate that combining mask-aware diffusion priors with a physics-guided posterior sampler provides a reliable and flexible solution for traffic state estimation under realistic sensing sparsity.

Traffic state estimation (TSE) becomes challenging when probe-vehicle penetration is low and observations are spatially sparse. Pure data-driven methods lack physical explanations and have poor generalization when observed data is sparse. In contrast, physical models have difficulty integrating uncertainties and capturing the real complexity of traffic. To bridge this gap, recent studies have explored combining them by embedding physical structure into Gaussian process. These approaches typically introduce the governing equations as soft constraints through pseudo-observations, enabling the integration of model structure within a variational framework. However, these methods rely heavily on penalty tuning and lack principled uncertainty calibration, which makes them sensitive to model mis-specification. In this work, we address these limitations by presenting a novel Physics-Embedded Gaussian Process (PEGP), designed to integrate domain knowledge with data-driven methods in traffic state estimation. Specifically, we design two multi-output kernels informed by classic traffic flow models, constructed via the explicit application of the linearized differential operator. Experiments on HighD, NGSIM show consistent improvements over non-physics baselines. PEGP-ARZ proves more reliable under sparse observation, while PEGP-LWR achieves lower errors with denser observation. Ablation study further reveals that PEGP-ARZ residuals align closely with physics and yield calibrated, interpretable uncertainty, whereas PEGP-LWR residuals are more orthogonal and produce nearly constant variance fields. This PEGP framework combines physical priors, uncertainty quantification, which can provide reliable support for TSE.

Traffic State Estimation and Prediction (TSEP) has been extensively studied to reconstruct traffic state variables (e.g., flow, density, speed, travel time, etc.) using (partial) observed traffic data (Antoniou et al., 2013; Ban et al., 2011; Shi et al., 2021; Li et al., 2020). In recent years, advancements in data collection technologies have enabled TSEP methods to integrate traffic data from diverse sources for more accurate and robust estimation and prediction (Wang et al., 2016; Makridis and Kouvelas, 2023). These data sources can be broadly categorized into infrastructure-collected data and user-contributed data. Infrastructure-collected data typically includes information collected from loop detectors, traffic cameras, and radars installed on roadways or at intersections. In contrast, user-contributed data is derived from individuals, often through vehicles or personal devices, such as GPS traces, vehicle trajectories, and probe data collected via mobile apps or in-vehicle systems.

Traffic state estimation (TSE) fundamentally involves solving high-dimensional spatiotemporal partial differential equations (PDEs) governing traffic flow dynamics from limited, noisy measurements. While Physics-Informed Neural Networks (PINNs) enforce PDE constraints point-wise, this paper adopts a physics-informed deep operator network (PI-DeepONet) framework that reformulates TSE as an operator learning problem. Our approach trains a parameterized neural operator that maps sparse input data to the full spatiotemporal traffic state field, governed by the traffic flow conservation law. Crucially, unlike PINNs that enforce PDE constraints point-wise, PI-DeepONet integrates traffic flow conservation model and the fundamental diagram directly into the operator learning process, ensuring physical consistency while capturing congestion propagation, spatial correlations, and temporal evolution. Experiments on the NGSIM dataset demonstrate superior performance over state-of-the-art baselines. Further analysis reveals insights into optimal function generation strategies and branch network complexity. Additionally, the impact of input function generation methods and the number of functions on model performance is explored, highlighting the robustness and efficacy of proposed framework.

Network-wide Traffic State Estimation (TSE), which aims to infer a complete image of network traffic states with sparsely deployed sensors, plays a vital role in intelligent transportation systems. With the development of data-driven methods, traffic dynamics modeling has advanced significantly. However, TSE poses fundamental challenges for data-driven approaches, since historical patterns cannot be learned locally at sensor-free segments. Although inductive graph learning shows promise in estimating states at locations without sensor, existing methods typically handle unobserved locations by filling them with zeros, introducing bias to the sensitive graph message propagation. The recently proposed Dirichlet Energy-based Feature Propagation (DEFP) method achieves State-Of-The-Art (SOTA) performance in unobserved node classification by eliminating the need for zero-filling. However, applying it to TSE faces three key challenges: inability to handle directed traffic networks, strong assumptions in traffic spatial correlation modeling, and overlooks distinct propagation rules of different patterns (e.g., congestion and free flow). We propose DGAE, a novel inductive graph representation model that addresses these challenges through theoretically derived DEFP for Directed graph (DEFP4D), enhanced spatial representation learning via DEFP4D-guided latent space encoding, and physics-guided propagation mechanisms that separately handles congested and free-flow patterns. Experiments on three traffic datasets demonstrate that DGAE outperforms existing SOTA methods and exhibits strong cross-city transferability. Furthermore, DEFP4D can serve as a standalone lightweight solution, showing superior performance under extremely sparse sensor conditions.

Accurately estimating spatiotemporal traffic states on freeways is a significant challenge due to limited sensor deployment and potential data corruption. In this study, we propose an efficient and robust low-rank model for precise spatiotemporal traffic speed state estimation (TSE) using lowpenetration vehicle trajectory data. Leveraging traffic wave priors, an oblique grid-based matrix is first designed to transform the inherent dependencies of spatiotemporal traffic states into the algebraic low-rankness of a matrix. Then, with the enhanced traffic state low-rankness in the oblique matrix, a low-rank matrix completion method is tailored to explicitly capture spatiotemporal traffic propagation characteristics and precisely reconstruct traffic states. In addition, an anomaly-tolerant module based on a sparse matrix is developed to accommodate corrupted data input and thereby improve the TSE model robustness. Notably, driven by the understanding of traffic waves, the computational complexity of the proposed efficient method is only correlated with the problem size itself, not with dataset size and hyperparameter selection prevalent in existing studies. Extensive experiments demonstrate the effectiveness, robustness, and efficiency of the proposed model. The performance of the proposed method achieves up to a 12% improvement in Root Mean Squared Error (RMSE) in the TSE scenarios and an 18% improvement in RMSE in the robust TSE scenarios, and it runs more than 20 times faster than the state-of-the-art (SOTA) methods.

Physics-informed deep learning (PIDL)-based models have recently garnered remarkable success in traffic state estimation (TSE). However, the prior knowledge used to guide regularization training in current mainstream architectures is based on deterministic physical models. The drawback is that a solely deterministic model fails to capture the universally observed traffic flow dynamic scattering effect, thereby yielding unreliable outcomes for traffic control. This study, for the first time, proposes stochastic physics-informed deep learning (SPIDL) for traffic state estimation. The idea behind such SPIDL is simple and is based on the fact that a stochastic fundamental diagram provides the entire range of possible speeds for any given density with associated probabilities. Specifically, we select percentile-based fundamental diagram and distribution-based fundamental diagram as stochastic physics knowledge, and design corresponding physics-uninformed neural networks for effective fusion, thereby realizing two specific SPIDL models, namely \text{$\alpha$}-SPIDL and \text{$\cal B$}-SPIDL. The main contribution of SPIDL lies in addressing the "overly centralized guidance" caused by the one-to-one speed-density relationship in deterministic models during neural network training, enabling the network to digest more reliable knowledge-based constraints.Experiments on the real-world dataset indicate that proposed SPIDL models achieve accurate traffic state estimation in sparse data scenarios. More importantly, as expected, SPIDL models reproduce well the scattering effect of field observations, demonstrating the effectiveness of fusing stochastic physics model knowledge with deep learning frameworks.

The growing number of connected vehicles offers an opportunity to leverage internet of vehicles (IoV) data for traffic state estimation (TSE) which plays a crucial role in intelligent transportation systems (ITS). By utilizing only a portion of IoV data instead of the entire dataset, the significant overheads associated with collecting and processing large amounts of data can be avoided. In this paper, we introduce a novel framework that utilizes sparse IoV data to achieve cost-effective TSE. Particularly, we propose a novel spatial-temporal attention model called the convolutional retentive network (CRNet) to improve the TSE accuracy by mining spatial-temporal traffic state correlations. The model employs the convolutional neural network (CNN) for spatial correlation aggregation and the retentive network (RetNet) based on the attention mechanism to extract temporal correlations. Extensive simulations on a real-world IoV dataset validate the advantage of the proposed TSE approach in achieving accurate TSE using sparse IoV data, demonstrating its cost effectiveness and practicality for real-world applications.

The expanding role of Artificial Intelligence (AI) in diverse engineering domains highlights the challenges associated with deploying AI models in new operational environments, involving substantial investments in data collection and model training. Rapid application of AI necessitates evaluating the feasibility of utilizing pre-trained models in unobserved operational settings with minimal or no additional data. However, interpreting the opaque nature of AI's black-box models remains a persistent challenge. Addressing this issue, this paper proposes a science-based certification methodology to assess the viability of employing pre-trained data-driven models in new operational environments. The methodology advocates a profound integration of domain knowledge, leveraging theoretical and analytical models from physics and related disciplines, with data-driven AI models. This novel approach introduces tools to facilitate the development of secure engineering systems, providing decision-makers with confidence in the trustworthiness and safety of AI-based models across diverse environments characterized by limited training data and dynamic, uncertain conditions. The paper demonstrates the efficacy of this methodology in real-world safety-critical scenarios, particularly in the context of traffic state estimation. Through simulation results, the study illustrates how the proposed methodology efficiently quantifies physical inconsistencies exhibited by pre-trained AI models. By utilizing analytical models, the methodology offers a means to gauge the applicability of pre-trained AI models in new operational environments. This research contributes to advancing the understanding and deployment of AI models, offering a robust certification framework that enhances confidence in their reliability and safety across a spectrum of operational conditions.

Traffic big data can be generally divided into two categories. The first type is Eulerian observations measured with fixed roadside sensors (e.g., loop detectors, traffic cameras, radars, etc.) installed on a small set of road links. This type of data is typically owned by municipal authorities (MAs). The second type is Lagrangian observations obtained from vehicle trajectory data (e.g., real-time vehicle locations and speeds), typically possessed by mobility providers (MPs) with large fleets, such as ride-hailing companies and public transport operators. Although these data can be useful for ITS applications, they provide only an incomplete picture of transportation systems due to the partial spatial coverage of the road network (in terms of Eulerian observations) and partial penetration rate over the vehicle population (in terms of Lagrangian observations). Traffic state estimation (TSE) is an important research topic that leverages these partially observed traffic data to infer key traffic state variables (e.g., flow, density, speed) on road segments.