Learning Coarse-Grained Dynamics on Graph
This work addresses the challenge of predicting coarse-grained dynamics on graphs for applications like oscillator models and power systems, but it is incremental as it builds on existing GNN and Mori-Zwanzig methods.
The paper tackled the problem of modeling coarse-grained dynamics on graphs by proposing a Graph Neural Network (GNN) framework that determines architecture based on Mori-Zwanzig memory analysis, resulting in a requirement of at least 2K message-passing steps for K-hop interactions and reduced memory length with interaction strength decay.
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.