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Beckmann Transport Models: From Autonomous Flows to One-Step Maps
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Proposes Beckmann Transport Models: time-independent (autonomous) flow matching drifts that exactly transport a regular base to a singular (lower-dimensional manifold) target, plus a stop-gradient one-step map learning rule, validated on 2D toy targets and ImageNet 256x256.
Problem
Equilibrium Matching proposes time-independent drifts for generative modeling, but it is unclear whether the autonomous flow transports the base to the target, and its loss may bias endpoint weights; learning one-step maps directly is also not well grounded.
Approach
Minimize the standard FM loss over time-independent fields so the drift is the time-averaged FM current divided by its occupation density; under codimension-two singular targets the autonomous flow transports the base to the target, and the endpoint map satisfies a conservation equation b·∇T=0 that motivates a stop-gradient one-step map objective.
Key results
- BTM-corrected EqM-XL/2 achieves FID 1.87 vs 1.90 for uncorrected EqM-XL/2 on ImageNet 256x256.
- One-step BTM map achieves FID 17.58 on ImageNet 256x256 without guidance, vs iCT 30.10 and Shortcut Models 10.60 (with CFG).
- On a 5-atom target with unequal weights, BTM basin weights match target with MAE 0.005 vs 0.102 for EqM.
Surprising findings
- The original EqM loss yields a 20x larger weight error on a 5-atom target, with the smallest basin nearly disappearing.
- Iterating a partially trained one-step map sharpens samples toward the flow's quality.
Limitations
- Exact transport results are population-level; finite-data neural approximations are evaluated empirically.
- Theory requires target codimension at least two; codimension-one counterexamples exist.
- One-step ImageNet FID (17.58) remains modest and relies on extra empirical modifications without CFG; the stop-gradient rule lacks proof of uniqueness or optimization convergence.
Experimental setup
Datasets: ImageNet 256x256 2D synthetic atomic targets 2D spiral embedded in 6D
Benchmarks: ImageNet 256x256
Metrics: FID MAE of basin weights
Baselines: Equilibrium Matching (EqM-XL/2) iCT Shortcut Models SiT MeanFlow
Builds on: flow matching Equilibrium Matching Poisson Flow Beckmann transportation problem Drifting
Closest papers by meaning
- On the Relation between Rectified Flows and Optimal Transport (NeurIPS 2025 · similarity 0.90)
- Optimal Flow Matching: Learning Straight Trajectories in Just One Step (NeurIPS 2024 · similarity 0.90)
- Categorical Flow Matching on Statistical Manifolds (NeurIPS 2024 · similarity 0.90)
- Metric Flow Matching for Smooth Interpolations on the Data Manifold (NeurIPS 2024 · similarity 0.89)
- Flow Matching from Viewpoint of Proximal Operators (NeurIPS 2026 · similarity 0.88)
- Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once (NeurIPS 2026 · similarity 0.88)
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