  Revealing the continuous dynamics on the networks is essential for
understanding, predicting, and even controlling complex systems, but it is hard
to learn and model the continuous network dynamics because of complex and
unknown governing equations, high dimensions of complex systems, and
unsatisfactory observations. Moreover, in real cases, observed time-series data
are usually non-uniform and sparse, which also causes serious challenges. In
this paper, we propose an Autoregressive GNN-ODE GRU Model (AGOG) to learn and
capture the continuous network dynamics and realize predictions of node states
at an arbitrary time in a data-driven manner. The GNN module is used to model
complicated and nonlinear network dynamics. The hidden state of node states is
specified by the ODE system, and the augmented ODE system is utilized to map
the GNN into the continuous time domain. The hidden state is updated through
GRUCell by observations. As prior knowledge, the true observations at the same
timestamp are combined with the hidden states for the next prediction. We use
the autoregressive model to make a one-step ahead prediction based on
observation history. The prediction is achieved by solving an initial-value
problem for ODE. To verify the performance of our model, we visualize the
learned dynamics and test them in three tasks: interpolation reconstruction,
extrapolation prediction, and regular sequences prediction. The results
demonstrate that our model can capture the continuous dynamic process of
complex systems accurately and make precise predictions of node states with
minimal error. Our model can consistently outperform other baselines or achieve
comparable performance.
