Do We Need an Encoder-Decoder to Model Dynamical Systems on Networks?
Bing Liu Wei Luo Gang Li Jing Huang Bo Yang
As deep learning gains popularity in modelling dynamical systems, we expose
an underappreciated misunderstanding relevant to modelling dynamics on
networks. Strongly influenced by graph neural networks, latent vertex
embeddings are naturally adopted in many neural dynamical network models.
However, we show that embeddings tend to induce a model that fits observations
well but simultaneously has incorrect dynamical behaviours. Recognising that
previous studies narrowly focus on short-term predictions during the transient
phase of a flow, we propose three tests for correct long-term behaviour, and
illustrate how an embedding-based dynamical model fails these tests, and
analyse the causes, particularly through the lens of topological conjugacy. In
doing so, we show that the difficulties can be avoided by not using embedding.
We propose a simple embedding-free alternative based on parametrising two
additive vector-field components. Through extensive experiments, we verify that
the proposed model can reliably recover a broad class of dynamics on different
network topologies from time series data.