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

ICLR 2026 Conference Submission14572 Authors

Keywords: graphs, networks, flow graphs, graph attention networks, group action, bilevel-optimization, physics-aware graph neural networks

TL;DR: FlowSymm is an end-to-end graph neural model that completes missing flows by enforcing a divergence-free group-action prior, scoring corrections with attention, and refining with feature-conditioned Tikhonov regularization

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 endow the flow manifold with 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 substantially outperforms state-of-the-art baselines in RMSE, MAE and correlation metrics.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 14572