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

TL;DR

FlowSymm tackles the challenge of imputing missing edge flows in networks while strictly respecting conservation laws modeled by $Bf=c$. It combines a physics-aware, symmetry-preserving approach with a learning-based attention mechanism over a divergence-free group-action basis, followed by a feature-conditioned Tikhonov refinement and end-to-end implicit bilevel training. The method demonstrates consistent improvements over nine baselines across traffic, power, and bike networks, reducing RMSE by about 8–10% and MAE by up to 16%, while maintaining low divergence residuals and high correlation with true flows. This approach provides a scalable, interpretable framework for physically grounded imputation in systems where exact balance is desired on observed components and where corrections must reside in a prescribed divergence-free subspace. The results highlight the practical impact of integrating algebraic symmetries with graph-attention and implicit optimization for reliable network-flow completion in transportation, energy, and mobility domains.

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.

Figures (2)

• Figure 1: Pipeline of FlowSymm: (1) construct a provisional balanced completion, (2) build a divergence‐free group‐action basis, (3) score and combine these actions via attention, (4) refine the missing flows with a feature‐conditioned Tikhonov solver, and (5) train all parameters end‐to‐end via implicit bilevel optimization.
• Figure 2: Distribution of learned attention weights $\alpha_{\theta,i}$ over $k{=}256$ basis actions on Traffic, Power, and Bike.