Causal graph convolution neural differential equation for spatio-temporal time series prediction

Abstract

Multivariate time series prediction has attracted wide research interest in recent decades. However, implicit spatial topology information and rich temporal evolution information bring many challenges to multivariate time series prediction. In this paper, a novel graph convolution module based on Granger causality is introduced to adaptively learn the causality between nodes. In detail, the ordinary differential equation (ODE) of a graph is used to model the propagation of spatial information between its nodes, and a temporal neural differential equation (NDE) is used to model the temporal evolution of the given nonlinear system. The Granger causality between multivariate time series is revealed by applying a multilayer perceptron (MLP) while imposing the \(L \)2 regularization constraint on the weights. A long short-term memory (LSTM)-based network is used as the nonlinear operator to reveal the underlying evolution mechanism of the input spatio-temporal time series. Furthermore, the forward Euler integration method is used to solve the graph ODE, which aims to enhance the representation ability of the proposed model while solving over-smoothing when the graph convolutional network (GCN) becomes too deep. The Euler trapezoidal integration method is used to simulate the evolution processes of dynamical systems and obtain the high-dimensional states of the medium and long-term prediction by solving the temporal NDE. The proposed model can explicitly discover the spatial correlations through its GCN-based causality module. We also combine the graph ODE module and the temporal NDE module to model the spatial information aggregation and temporal dynamic evolution processes, respectively, thus making the proposed model more interpretable. The experimental results demonstrate the effectiveness of our method in terms of spatio-temporal dynamic discovery and prediction performance.