Network macroscopic fundamental diagram-informed graph learning for traffic state imputation

Traffic state imputation refers to the estimation of missing values of traffic variables, such as flow rate and traffic density, using available data. It furnishes comprehensive traffic context for various operation tasks such as vehicle routing, and enables us to augment existing datasets (e.g., PeMS, UTD19, Uber Movement) for diverse theoretical and practical investigations. Despite the superior performance achieved by purely data-driven methods, they are subject to two limitations. One limitation is the absence of a traffic engineering-level interpretation in the model architecture, as it fails to elucidate the methodology behind deriving imputation results from a traffic engineering standpoint. The other limitation is the possibility that imputation results may violate traffic flow theories, thereby yielding unreliable outcomes for transportation engineers. In this study, we introduce NMFD-GNN, a physics-informed machine learning method that fuses the network macroscopic fundamental diagram (NMFD) with the graph neural network (GNN), to perform traffic state imputation. Specifically, we construct the graph learning module that captures the spatio-temporal dependency of traffic congestion. Besides, we develop the physics-informed module based on the $\lambda$-trapezoidal MFD, which presents a functional form of NMFD and was formulated by transportation researchers in 2020. The primary contribution of NMFD-GNN lies in being the first physics-informed machine learning model specifically designed for real-world traffic networks with multiple roads, while existing studies have primarily focused on individual road corridors. We evaluate the performance of NMFD-GNN by conducting experiments on real-world traffic networks located in Zurich and London, utilizing the UTD19 dataset ¹. The results indicate that our NMFD-GNN outperforms six baseline models in terms of performance in traffic state imputation.