FlowSymm: Physics Aware, Symmetry Preserving Graph Attention for Network Flow Completion

2026-01-29

AI Summary

This study addresses the problem of recovering missing edge flows in networks under strict adherence to local conservation laws, such as flow conservation. To this end, the authors propose a physically consistent and data-driven completion method that uniquely integrates the Abelian group symmetry induced by conservation laws with graph attention mechanisms, yielding a symmetry-preserving parameterization of the solution space. The framework leverages a GATv2 encoder, orthogonal basis representations, an implicit bilevel optimization scheme incorporating Tikhonov regularization and Cholesky-based gradient propagation, and a convex solver to enable efficient training and inference. Evaluated on three real-world datasets—traffic, power grids, and bike-sharing—the method consistently outperforms state-of-the-art approaches across RMSE, MAE, and correlation metrics.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Computer Vision: Visual Reasoning & Symbolic Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Data transparency and provenance

Abstract

Recovering missing flows on the edges of a network, while exactly respecting local conservation laws, is a fundamental inverse problem that arises in many systems such as transportation, energy, and mobility. We introduce FlowSymm, a novel architecture that combines (i) a group-action on divergence-free flows, (ii) a graph-attention encoder to learn feature-conditioned weights over these symmetry-preserving actions, and (iii) a lightweight Tikhonov refinement solved via implicit bilevel optimization. The method first anchors the given observation on a minimum-norm divergence-free completion. We then compute an orthonormal basis for all admissible group actions that leave the observed flows invariant and parameterize the valid solution subspace, which shows an Abelian group structure under vector addition. A stack of GATv2 layers then encodes the graph and its edge features into per-edge embeddings, which are pooled over the missing edges and produce per-basis attention weights. This attention-guided process selects a set of physics-aware group actions that preserve the observed flows. Finally, a scalar Tikhonov penalty refines the missing entries via a convex least-squares solver, with gradients propagated implicitly through Cholesky factorization. Across three real-world flow benchmarks (traffic, power, bike), FlowSymm outperforms state-of-the-art baselines in RMSE, MAE and correlation metrics.

Problem

• network flow completion
• local conservation laws
• missing flow recovery
• divergence-free flows
• inverse problem

Innovation

• symmetry-preserving
• graph attention
• divergence-free flow
• implicit bilevel optimization
• physics-aware learning