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.

我们考虑使用图神经网络（GNN）非马尔科夫建模框架来识别图上的粗粒化动力系统。我们的主要思路是通过检查莫里 - 茨旺齐记忆项的主导项如何依赖编码图拓扑的粗粒化相互作用系数，系统地确定 GNN 的体系结构。基于这个分析，我们发现适当的 GNN 体系结构需要至少进行 2K 步的消息传递（MP）机制，以考虑 K - 跳动力学相互作用。我们还推断出，在相互作用强度以跳距离的幂律衰减的假设下，准确的闭包模型所需的记忆长度会随着相互作用强度的减小而减小。针对两个例子，异质库拉莫特振荡器模型和电力系统，我们进行了支持性的数值演示，表明所提出的 GNN 体系结构能够预测固定和时变图拓扑下的粗粒化动力学。