Physics-Embedded Gaussian Process for Traffic State Estimation
Yanlin Chen, Kehua Chen, Yinhai Wang
TL;DR
This work tackles traffic state estimation under sparse observations by embedding physical ARZ and LWR dynamics into Gaussian process priors through operator-embedded kernels, enabling principled uncertainty and cross-variable coupling. It develops a two-output PEGP framework with closed-form cross-covariances and a residual component to capture nonlinearities, evaluated on HighD and NGSIM. Results show PEGP-ARZ provides robust, calibrated uncertainty under sparse data, while PEGP-LWR achieves lower errors with denser data; ablations reveal regime-dependent benefits and complementary physics-residual interactions. The approach demonstrates how physics-informed kernel design can improve interpretability and reliability of TSE and suggests avenues for extending to networks and complex boundary conditions.
Abstract
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.