Higher-order Spatio-temporal Physics-incorporated Graph Neural Network for Multivariate Time Series Imputation

Exploring the missing values is an essential but challenging issue due to the complex latent spatio-temporal correlation and dynamic nature of time series. Owing to the outstanding performance in dealing with structure learning potentials, Graph Neural Networks (GNNs) and Recurrent Neural Networks (RNNs) are often used to capture such complex spatio-temporal features in multivariate time series. However, these data-driven models often fail to capture the essential spatio-temporal relationships when significant signal corruption occurs. Additionally, calculating the high-order neighbor nodes in these models is of high computational complexity. To address these problems, we propose a novel higher-order spatio-temporal physics-incorporated GNN (HSPGNN). Firstly, the dynamic Laplacian matrix can be obtained by the spatial attention mechanism. Then, the generic inhomogeneous partial differential equation (PDE) of physical dynamic systems is used to construct the dynamic higher-order spatio-temporal GNN to obtain the missing time series values. Moreover, we estimate the missing impact by Normalizing Flows (NF) to evaluate the importance of each node in the graph for better explainability. Experimental results on four benchmark datasets demonstrate the effectiveness of HSPGNN and the superior performance when combining various order neighbor nodes. Also, graph-like optical flow, dynamic graphs, and missing impact can be obtained naturally by HSPGNN, which provides better dynamic analysis and explanation than traditional data-driven models. Our code is available at https://github.com/gorgen2020/HSPGNN.