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

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.

Generative Modeling on Metric Graphs via Neural Optimal Transport

We introduce, to our knowledge, the first deep generative modeling framework for probability distributions continuously supported on compact metric graphs. Given source and target measures on a metric graph, our method embeds the graph into a smooth ambient space, solves an entropic Kantorovich problem via a neural semidual parameterization, and projects generated samples back onto the original graph. We study two embedded geometries: an extrinsic Euclidean realization and the intrinsic tropical Abel--Jacobi embedding into the Jacobian torus. In both cases, the resulting generator is graph-supported by construction. We prove that, in the joint limit of increasing neural expressivity, the learned generator converges weakly to a valid transport coupling between the original graph measures. Empirically, across a range of geometrically distinct graphs, our method matches or improves upon heuristic transport baselines based on discrete graph OT, while scaling more favorably. Finally, we demonstrate scalability on real-world urban mobility data by training our model on one million Uber pickup locations in Manhattan, New York City.