  Despite the success of classical traffic flow (e.g., second-order
macroscopic) models and data-driven (e.g., Machine Learning - ML) approaches in
traffic state estimation, those approaches either require great efforts for
parameter calibrations or lack theoretical interpretation. To fill this
research gap, this study presents a new modeling framework, named physics
regularized Gaussian process (PRGP). This novel approach can encode physics
models, i.e., classical traffic flow models, into the Gaussian process
architecture and so as to regularize the ML training process. Particularly,
this study aims to discuss how to develop a PRGP model when the original
physics model is with discrete formulations. Then based on the posterior
regularization inference framework, an efficient stochastic optimization
algorithm is developed to maximize the evidence lowerbound of the system
likelihood. To prove the effectiveness of the proposed model, this paper
conducts empirical studies on a real-world dataset that is collected from a
stretch of I-15 freeway, Utah. Results show the new PRGP model can outperform
the previous compatible methods, such as calibrated physics models and pure
machine learning methods, in estimation precision and input robustness.
