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

Jul 16, 2026

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Many real-world systems are naturally represented as metric graphs, where spatio-temporal dynamics evolve along connected edges rather than only between discrete nodes. Existing physics-informed graph neural approaches struggle on metric graphs, especially for advection-dominated transport, due to directional antisymmetry, long-range temporal dependencies, and preservation of sharp gradients. MeGA-MP introduces a physics-informed message passing operator with a linear-advection inductive bias that recovers exact dynamics without training and extends to advection-reaction water-distribution settings with improved performance and graph-topology generalization.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages76

Reading Time3 h 12 min

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arXiv :2607 .05 167v 1 [ cs .LG] 6 Jul 2026 MeGA-MP: Metric Graph Advection Message Passing A Physics-Informed Message Passing Operator for Advection-Dominated Metric Graphs Janine Strotherm∗ [jstrotherm@techfak.uni-bielefeld. de](jstrotherm@techfak.uni-bielefeld. de) Bielefeld University Luca Hermes∗ [lhermes@techfak.uni-bielefeld. de](lhermes@techfak.uni-bielefeld. de) Bielefeld University André Artelt [aartelt@techfak.uni-bielefeld. de](aartelt@techfak.uni-bielefeld. de) Bielefeld University Barbara Hammer [bhammer@techfak.uni-bielefeld. de](bhammer@techfak.uni-bielefeld. de) Bielefeld University 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. 1 Introduction In the last years, physics-informed graph neural networks (GNNs) have shown impressive zero-shot generalization capabilities when addressing dynamical systems on conventional graphs (Thangamuthu et al. , 2022) . However, many real-world systems do not exhibit discrete node-to-node dynamics. Instead, dynamics unfold along connections between nodes. Examples include quantum dynamics, water distribution systems, gas networks, electrical grids, and traffic flow networks (Böttcher & Porter, 2024 ; Rossman et al. , 2020) . While a common approach is to discretize the domain and recover a conventional graph, metric graphs naturally capture the geometry of these systems. In a metric graph, each edge is associated with a one-dimensional domain – an interval of edge-dependent length – governed by edge dynamics, expressed as an ordinary differential equation (ODE) or, more commonly, a partial differential equation (PDE) . In combination with suitable coupling conditions at nodes, such as continuity or conservation of mass or flux, this defines a coupled system of PDEs (Böttcher & Porter, 2024 ; Blechschmidt et al. , 2025) . We identify advection, i.e., the transport of a substance through the presence of a flow field, as the main driver for common edge dynamics on a variety of application domains, such as utility networks like water distribution systems (Rossman et al. , 2020) and gas networks (Gugat & Herty, 2022), traffic flow networks ∗ Authors contributed equally. (Gugat et al. , 2005), telecommunication networks, and blood flow through vascular systems (Bressan et al. , 2014) . From the machine learning (ML) perspective, an intuitive approach is to model such dynamics with GNNs. However, spatio-temporal advection-dominated dynamics on metric graphs pose well-known challenges in the context of GNNs. First, the directional nature of transport requires antisymmetric message passing. Second, spatial dependencies can become global over time, requiring long-range spatial integration. Furthermore, sharp gradients and shocks are

FAQs

What are metric graphs and why do they matter for modeling real-world systems?

Metric graphs represent networks where dynamics unfold along edges connected at vertices, independent of an external global Euclidean space. They capture geometry through edge lengths and coupled PDE dynamics via node conditions like continuity or conservation.

What makes advection-dominated dynamics difficult for typical GNN approaches on metric graphs?

Advection requires antisymmetric, direction-aware message passing, and transport induces long-range dependencies over time. Sharp gradients and shocks persist instead of being smoothed, which conflicts with common over-smoothing behavior in GNNs.

How does MeGA-MP address linear advection on metric graphs?

MeGA-MP builds a physics-informed message passing operator derived from physical priors. It solves the linear advection PDE using the Method of Characteristics and integrates it into an iterative message passing architecture, yielding exact dynamics up to a discretization error without training.

Jul 16, 2026

Upload by Liam