# Bridging Stochastic Flow Maps and Boltzmann Generators with Normalizing Flows

arXiv: https://arxiv.org/abs/2609.31978 · HTML: https://mlsanity.com/paper/2609.31978/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 63 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 4 of 4)

- [Target Score Matching](https://mlsanity.com/paper/2402.08667/) (2024)
- [Control Variate Score Matching for Diffusion Models](https://mlsanity.com/paper/2512.20003/) (2025)
- [Diffusion Path Samplers via Sequential Monte Carlo](https://mlsanity.com/paper/2601.21951/) (2026)
- [Efficient and Unbiased Sampling of Boltzmann Distributions via Consistency Models](https://mlsanity.com/paper/2409.07323/) (2024)

## Abstract (arXiv)

Generating independent, equilibrium samples of molecular systems at scale remains a central obstacle in computational statistical mechanics. Boltzmann Generators address this by pairing a generative model with importance sampling to obtain consistent samples from the target distribution. We introduce Normalizing Flow Flow Maps (NF2M), which combines the strengths of recent stochastic flow maps with the tractability of classic normalizing flows to build a Boltzmann Generator. Unlike most methods, which correct the generative model only at the end, NF2M reweighs each denoising transition as generation proceeds, avoiding wasted compute on trajectories that are ultimately discarded. At each denoising step, a conditional normalizing flow proposes clean configurations given the current noisy state (a simpler task than sampling directly from the target) and its exact likelihood enables correcting each proposal toward the true denoising transition of the target Boltzmann distribution. This is in contrast to most existing methods, whose likelihoods are approximate or expensive to evaluate, undermining the statistical reliability of the correction. We establish consistency of the corrected transitions and bound how approximation errors propagate through the sampling chain. We evaluate NF2M on peptide systems, demonstrating improved sampling efficiency and sample quality.