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

TL;DR

FlowSymm introduces a physics-aware, symmetry-preserving graph attention network that effectively recovers missing network flows while respecting local conservation laws, outperforming existing methods on real-world benchmarks.

Contribution

The paper presents a novel architecture combining symmetry group actions, graph attention, and Tikhonov refinement for flow completion respecting physical laws.

Findings

Outperforms state-of-the-art baselines in RMSE, MAE, and correlation metrics.

Successfully applied to traffic, power, and bike flow datasets.

Demonstrates the effectiveness of symmetry-preserving and physics-aware modeling.

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.