AI summaryⓘ
The authors study systems like water pipes and power grids, which can be thought of as networks made of edges and points, where things flow along the edges rather than just hopping from point to point. They point out that common machine learning tools don't work well for these because they don't handle the flow direction and distance properly. To solve this, they create a new method that uses the idea of message passing to model how stuff moves through these network edges accurately, especially for flow processes like advection. Their method can exactly replicate simple flow dynamics without needing training, and when combined with learning components, it performs better than previous methods on more complex systems like water distribution networks, even when tested on new network shapes.
metric graphsadvectionmessage passingphysics-informed neural networks (PINNs)operator learningtransport dynamicsinductive biaswater distribution systemsgraph topologiesmachine learning
Authors
Janine Strotherm, Luca Hermes, André Artelt, Barbara Hammer
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.