MeGA-MP: Metric Graph Advection Message Passing -- A Physics-Informed Message Passing Operator for Advection-Dominated Metric Graphs

2026-07-06

AI Summary

Existing graph neural networks struggle to effectively model convection-dominated spatiotemporal dynamics on metric graphs, particularly in capturing asymmetric propagation and long-range dependencies. This work proposes a physics-informed message-passing operator that embeds linear convection dynamics as an inductive bias into graph neural networks, enabling exact recovery of pure convection dynamics on metric graphs without any training and supporting zero-shot generalization across graph topologies. By integrating a physics-guided convection operator with a learnable MLP component, the proposed framework significantly outperforms existing methods in modeling convection-reaction dynamics in water distribution systems.

Abstract

Many real-world systems are organized as networks where spatio-temporal dynamics unfold along connections and not discretely between nodes. Examples include utility networks such as water distribution systems or gas networks, electrical grids, and traffic flow networks. Such systems are naturally modeled as metric graphs, where edges correspond to one-dimensional Euclidean subspaces connected at vertices. Metric graphs are independent of an underlying global Euclidean space, limiting direct application of typical PINNs and operator-learning methods. Especially transport dynamics like advection require a methodology able to capture antisymmetric and long-range dependencies on graphs, which is itself a challenge. We propose a novel physics-informed message passing operator that encodes linear advection on metric graphs as an inductive bias. In the purely advective setting, the operator provably recovers the exact dynamics up to a theoretically derived discretization error without any training. Combined with trainable components like MLPs, our message passing operator extends to realistic advection-reaction dynamics in water distribution systems, where we achieve superior performance compared to baselines and zero-shot generalization across different graph topologies.

Problem

• metric graphs
• advection-dominated dynamics
• physics-informed learning
• long-range dependencies
• non-Euclidean networks

Innovation

• metric graphs
• advection-dominated dynamics
• physics-informed message passing
• zero-shot generalization
• inductive bias