Fitting Spatial-Temporal Data via a Physics Regularized Multi-Output Grid Gaussian Process: Case Studies of a Bike-Sharing System

Fitting and modeling spatial-temporal processes are essential research topics in transportation studies. Recently, due to the analytically tractable formulation and good fitting accuracy, the Gaussian Process (GP) is becoming increasingly preferable in fitting transportation processes. However, conventional GPs are inapplicable for large-scale problems due to computational issues. Moreover, they dismiss physics laws when conducting multi-output fitting tasks. This paper proposes a physics regularized multi-output grid Gaussian Process Model (PRMGGP) model for fast and multi-output fitting of large-scale spatial-temporal processes in transportation systems. The PRMGGP model adopts a grid input structure to capture inherent spatial-temporal correlations in the fitting process, takes advantage of the Kronecker algebra to notably accelerate the computation speed, and utilizes a shadow GP to incorporate physics laws of the process. Model training and predictive algorithms are developed coordinately and are tested via synthetic datasets. Furthermore, we apply the proposed model and other widely used machine learning models to fit the numbers of pickups, returns, and idle bikes of a large-scale bike-sharing system based on Citi Bike data from New York City. The results demonstrate the computational efficiency, interpretable results, and the prediction accuracy of the PRMGGP model, which can be a promising methodology for modeling multi-output processes in transportation systems.