Modelling Networked Dynamical System by Temporal Graph Neural ODE with Irregularly Partial Observed Time-series Data

TL;DR

The paper tackles modeling networked dynamical systems from irregular, partially observed time-series data. It introduces a GNODE-GGRU framework that combines a Spatial Graph Neural ODE for continuous latent dynamics with a Graph Gate Recurrent Unit-based imputation, augmented by a reliability and time-aware mechanism to account for variable observation quality and time gaps. A loss weighting scheme $w_{t_{ij}} = \beta e^{-\zeta (t_i-t_{ij})}$ prioritizes higher-quality imputations during training, and the imputed states $\widetilde{\bm{X}}_{t_i} = \bm{M}_{t_i} \odot \bm{X}_{t_i} + \overline{\bm{M}}_{t_i} \odot \widehat{\bm{X}}_{t_i}$ feed into a Temporal Graph Neural Network for prediction. Experiments on a simulated 8-node network with irregular sampling show the proposed method outperforming RNN-based and Neural ODE baselines in terms of MSE across varying levels of observed data, demonstrating the model’s ability to leverage spatial-temporal dependencies for accurate imputation and forecasting. The approach offers a principled way to handle missing data and irregular sampling in graph-structured dynamical systems, with potential applications across traffic, energy, and communications networks.

Abstract

Modeling the evolution of system with time-series data is a challenging and critical task in a wide range of fields, especially when the time-series data is regularly sampled and partially observable. Some methods have been proposed to estimate the hidden dynamics between intervals like Neural ODE or Exponential decay dynamic function and combine with RNN to estimate the evolution. However, it is difficult for these methods to capture the spatial and temporal dependencies existing within graph-structured time-series data and take full advantage of the available relational information to impute missing data and predict the future states. Besides, traditional RNN-based methods leverage shared RNN cell to update the hidden state which does not capture the impact of various intervals and missing state information on the reliability of estimating the hidden state. To solve this problem, in this paper, we propose a method embedding Graph Neural ODE with reliability and time-aware mechanism which can capture the spatial and temporal dependencies in irregularly sampled and partially observable time-series data to reconstruct the dynamics. Also, a loss function is designed considering the reliability of the augment data from the above proposed method to make further prediction. The proposed method has been validated in experiments of different networked dynamical systems.

Figures (4)

• Figure 1: The structure of GNODE-GGRU to impute data. The hidden state between observations is estimated by Graph neural ODE and then updated by Graph GRU. The impute data can be used to make prediction in predict network.
• Figure 2: Irregular sampling time-series graph-structured data with unobserved states. The blue one represents the observed state and the white one is unobserved.
• Figure 3: Blue one is the observable state and the red one is the missing information. Green one is the impute state by GNODE-GGRU. Weight of each term in loss function is determined by $w_{t_{ij}} = \beta e^{-\zeta (t_i-t_{ij})}$.
• Figure 4: Modelling the trajectories of the system according to the temporal irregular and spatial missing data by the proposed method in this paper. The system contains 8 nodes and each node has a 2-dimensional dynamic function. The left figure shows trajectories of node 1 on $x$ plane and $y$ plane. The right figure is the phase portrait of node 1.