Learning Coarse-Grained Dynamics on Graph

TL;DR

This work develops a Mori–Zwanzig-guided, non-Markovian reduced-order modeling framework for coarse-grained graph dynamics and couples it with a tailored Graph Neural Network (GNN) architecture. The key insight is that the leading Mori–Zwanzig memory term depends quadratically on coarse-grained interaction coefficients, which dictates a minimum MP depth of at least $2K$ to capture $K$-hop interactions; memory length decreases as the hop-strength decays under a power-law assumption. The authors implement a Chebyshev-convolution-based GNN with an encoder–processor–decoder, using topology indices to manage time-varying graphs and ensuring the model can learn from past states and the current coarse-grained topology. Numerical experiments on a heterogeneous Kuramoto network and a time-varying power grid demonstrate that the proposed GNN ROM outperforms non-topology baselines and accurately predicts coarse-grained dynamics under topology changes, with clear guidance on selecting MP depth based on interaction strength. Overall, the approach offers a scalable, topology-aware method for predicting reduced-order dynamics in networked systems with potential broad impact across engineering and physics applications.

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.

Figures (12)

• Figure 1: An example topology before and after coarse-graining.
• Figure 2: Model prediction on test data.
• Figure 3: Mean and standard deviation of predictive NRMSE based on 5 randomly generated topologies with $d\epsilon=0.75$.
• Figure 4: Comparison of convergence trend with increasing $d\epsilon$ with 2-hop GNN model ($N_C=1$ and $K=2$).
• Figure 5: Comparison of model performance with sparse topology changes during simulations.

Theorems & Definitions (12)

• Remark 2.1
• Example 2.2: Kuramoto model
• Example 2.3: Power system
• Remark 3.1
• Example 3.2: Kuramoto example, continued
• Example 3.3: Power system example, continued
• Proposition 3.4
• proof
• Remark 3.5
• Example 4.1: Kuramoto example, continued