We consider a Graph Neural Network (GNN) non-Markovian modeling framework to identify coarse-grained dynamical systems on graphs. Our main idea is to systematically determine the GNN architecture by inspecting how the leading term of the Mori-Zwanzig memory term depends on the coarse-grained interaction coefficients that encode the graph topology. Based on this analysis, we found that the appropriate GNN architecture that will account for $K$-hop dynamical interactions has to employ a Message Passing (MP) mechanism with at least $2K$ steps. We also deduce that the memory length required for an accurate closure model decreases as a function of the interaction strength under the assumption that the interaction strength exhibits a power law that decays as a function of the hop distance. Supporting numerical demonstrations on two examples, a heterogeneous Kuramoto oscillator model and a power system, suggest that the proposed GNN architecture can predict the coarse-grained dynamics under fixed and time-varying graph topologies.

翻译：我们提出了一种基于图神经网络（GNN）的非马尔可夫建模框架，用于识别图上的粗粒度动力系统。核心思想是通过分析Mori-Zwanzig记忆项的首项如何依赖于编码图拓扑结构的粗粒度相互作用系数，从而系统性地确定GNN架构。基于此分析，我们发现，能够描述$K$跳动力学相互作用的合适GNN架构必须采用至少$2K$步的消息传递（MP）机制。同时，我们推导出，在相互作用强度服从随跳距衰减的幂律假设下，精确闭合模型所需的记忆长度随相互作用强度的增加而减小。基于两个算例（异质Kuramoto振子模型和电力系统）的数值验证表明，所提出的GNN架构能够在固定和时变图拓扑下预测粗粒度动力学。