Title: MeGA-MP: Metric Graph Advection Message Passing - Solving Dynamical Processes on Metric Graphs with Graph Neural Networks
Primary area: learning on graphs and other geometries & topologies
Keywords: metric graphs, dynamical systems, advection, message passing, graph neural networks, physics-informed machine learning, neural operators

Abstract: Many real-world systems are organized as networks, where spatio-temporal dynamics unfold not only at nodes, but also along the connections between them. 
Such networks are known as $\textit{metric graphs}$.
Examples include utility networks and the propagation of signals in physical or biological media.
The methods that approach such problems are mostly PINN-based with limited generalizability to PDE parametrization and boundary conditions. 
A recent work addresses the limitations of PINNs by proposing a neural operator tailored to drift-diffusion dynamics.
However, in many real-world settings, hyperbolic dynamics like advection dominate the spatial evolution of a system, which has not been addressed so far.
In this work, we propose a novel graph operator that solves linear advection on metric graphs via message-passing. 
We provide an error bound on the approximation of ground truth obtained through multiple MP-iterations without the necessity of training.
Empirically, we show that it solves advection competitive to numerical and neural solvers. 
Combined with trainable components like MLPs, we demonstrate how it can be applied to realistic advection-reaction dynamics in water distribution systems, where we achieve superior performance compared to baselines.

TL;DR: A message passing architecture that solves advection dynamical systems on metrics graphs.