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.

本文说明了在使用图神经网络时易忽视的一种在网络动力学建模方面的误解，指出利用嵌入在神经动力网络模型中往往会导致模型拟合观测数据的表现良好但动力行为不正确，因此提出了一个不使用嵌入的替代方法，通过实验证明了该模型可以可靠地从时间序列数据中恢复不同网络拓扑下的广泛动态。

我们是否需要一个编码器-解码器来建模网络上的动态系统？