Research paper · arXiv:2605.16126 · 2026

Entropy Across the Bridge

A conditional–marginal entropy-rate objective for allocating a small number of function evaluations along flow and Schrödinger-bridge paths.

Bruno Trentini et al.

Problem

When a trained flow-based generative model has a small inference budget, sample quality depends on where the sampler spends its few function evaluations. Common grids are heuristic or inherited from one-endpoint diffusion, even when the model follows a probability bridge with two endpoints.

Method

The paper separates endpoint-conditioned bridge geometry from marginal flow evolution and derives a conditional–marginal entropy-rate objective. This produces a training-free inference-time scheduler. For Gaussian Brownian bridges the rate is available in closed form and has a U-shape, which motivates denser evaluation near the boundaries.

Conceptual scheduler: more evaluations are allocated near the bridge boundaries.

Role and collaboration

I am the first author. The work was developed with Dejan Stancevic, Michael M. Bronstein, Alexander Tong and Luca Ambrogioni.

Results

On trained two-dimensional bridge and flow models, the estimated profile recovers the predicted shape. The same scheduling principle was also evaluated on EDM/CIFAR-10 and AlphaFlow protein generation.

18.1%better 10-step ODE-Heun MMD than linear
22.7%paired SDE-Heun improvement
186.3best tested five-step CIFAR-10 FID

The CIFAR-10 comparison reported 186.3 ± 4.0 for the entropic grid, 200.5 ± 2.9 for linear and 238.0 ± 5.3 for cosine. On CAMEO22 and ATLAS protein benchmarks, the conditional–marginal schedule showed an advantage in low-NFE regimes.