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As deep learning gains popularity in modelling dynamical systems, we exposean underappreciated misunderstanding relevant to modelling dynamics onnetworks. Strongly influenced by graph neural networks, latent vertexembeddings are naturally adopted in many neural dynamical network models.However, we show that embeddings tend to induce a model that fits observationswell but simultaneously has incorrect dynamical behaviours. Recognising thatprevious studies narrowly focus on short-term predictions during the transientphase of a flow, we propose three tests for correct long-term behaviour, andillustrate how an embedding-based dynamical model fails these tests, andanalyse the causes, particularly through the lens of topological conjugacy. Indoing so, we show that the difficulties can be avoided by not using embedding.We propose a simple embedding-free alternative based on parametrising twoadditive vector-field components. Through extensive experiments, we verify thatthe proposed model can reliably recover a broad class of dynamics on differentnetwork topologies from time series data.

Do We Need an Encoder-Decoder to Model Dynamical Systems on Networks?