Integrating PDE Observer with Deep Learning for Traffic State Estimation

Traffic state estimation (TSE) refers to the inference of traffic state information from partially observed traffic data and some prior knowledge of the traffic dynamics. TSE plays a key role in traffic management as traffic control relies on an accurate estimation of the traffic states. Macroscopic traffic models describe traffic dynamics with aggregated values such as traffic density, velocity, and flow and are often employed for TSE of a freeway road segment. This paper integrates Partial Differential Equation (PDE) observer design and deep learning paradigm to estimate spatial-temporal traffic states from boundary sensing. With the PDE observer providing an rigorous guarantee for state estimates, we propose Observer-Informed Deep Learning (OIDL) paradigm which is a data-driven solution to TSE that leverages the PDE observer design. An Observer-Informed Neural Network (OINN) is constructed by training NN to generate state estimates and use the boundary observer for regularization. The OINN forms a novel class of data-efficient function approximators that encode PDE observer as theoretical guarantee and improves the accuracy and convergence speed. Experiments using NGSIM data-set demonstrate that the proposed OIDL reduces the estimation error compared to either the model-based observer or either the data-driven neural network. We also compare OIDL with the existing Physics-informed Deep Learning (PIDL) approach for TSE.