Learning Coarse-Grained Dynamics on Graph

TL;DR

This paper develops a GNN framework for modeling coarse-grained dynamics on graphs, linking architecture design to the Mori-Zwanzig memory term, and demonstrates its effectiveness on oscillator and power system examples.

Contribution

It systematically derives GNN architecture requirements from the Mori-Zwanzig formalism for coarse-grained graph dynamics.

Findings

GNN with at least 2K message passing steps captures K-hop interactions.

Memory length decreases with interaction strength under power-law decay.

Proposed GNN accurately predicts dynamics on heterogeneous systems.

Abstract

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.