Physics-Informed Deep Learning for Traffic State Estimation: Illustrations With LWR and CTM Models

We present a physics-informed deep learning (PIDL) approach to tackle the challenge of data sparsity and sensor noise in traffic state estimation (TSE). PIDL strengthens a deep learning (DL) neural network with the knowledge of traffic flow theory to accurately estimate traffic conditions. The 'physics'-a priori information of the system-acts as a regularization agent during training. We illustrate the implementation of the proposed approach with two commonly used models representing traffic physics: Lighthill-Whitham-Richards (LWR) model and the cell transmission model (CTM). The LWR implementation is illustrated with Greenshields' and inverse-lambda fundamental diagrams; whereas, CTM model implementation works with any fundamental diagram of choice. Two case studies validate the approach by reconstructing the velocity-field. Case study-I uses synthetic data generated to resemble the trajectory of connected and autonomous vehicles as captured by roadside units. Case study-II employs NGSIM data mimicking scant probe vehicle observations. We observe that the proposed PIDL approach is particularly better in state estimation with a lower amount of training data, illustrating the capability of PIDL in making precise and timely TSE even with sparse input. E.g., With 10% CAV penetration rate and a 15% added-noise, relative error for PIDL was at 22.9% compared to 30.8% for DL.