# The Coupling Within: Flow Matching via Distilled Normalizing Flows

arXiv: https://arxiv.org/abs/2603.09014 · HTML: https://mlsanity.com/paper/2603.09014/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 32 of its 43 references that are in the map.
- First posted on arXiv 9 Mar 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 32)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Accelerating Large Language Model Decoding with Speculative Sampling](https://mlsanity.com/paper/2302.01318/) (2023)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Parallel WaveNet: Fast High-Fidelity Speech Synthesis](https://mlsanity.com/paper/1711.10433/) (2017)
- [Normalizing Flows are Capable Generative Models](https://mlsanity.com/paper/2412.06329/) (2024)
- [Minimizing Trajectory Curvature of ODE-based Generative Models](https://mlsanity.com/paper/2301.12003/) (2023)
- [Diffusion Normalizing Flow](https://mlsanity.com/paper/2110.07579/) (2021)
- [Non-Denoising Forward-Time Diffusions](https://mlsanity.com/paper/2312.14589/) (2023)
- [Accelerating Auto-regressive Text-to-Image Generation with Training-free Speculative Jacobi Decoding](https://mlsanity.com/paper/2410.01699/) (2024)
- [Accelerating Feedforward Computation via Parallel Nonlinear Equation Solving](https://mlsanity.com/paper/2002.03629/) (2020)
- [PaSS: Parallel Speculative Sampling](https://mlsanity.com/paper/2311.13581/) (2023)
- [STARFlow: Scaling Latent Normalizing Flows for High-resolution Image Synthesis](https://mlsanity.com/paper/2506.06276/) (2025)
- [Flow Matching with Semidiscrete Couplings](https://mlsanity.com/paper/2509.25519/) (2025)
- [On Fitting Flow Models with Large Sinkhorn Couplings](https://mlsanity.com/paper/2506.05526/) (2025)
- [Predictive Sampling with Forecasting Autoregressive Models](https://mlsanity.com/paper/2002.09928/) (2020)
- [Diffusion Models with Deterministic Normalizing Flow Priors](https://mlsanity.com/paper/2309.01274/) (2023)
- [Bidirectional Normalizing Flow: From Data to Noise and Back](https://mlsanity.com/paper/2512.10953/) (2025)
- [Distilling the Knowledge from Conditional Normalizing Flows](https://mlsanity.com/paper/2106.12699/) (2021)

## Built on by (most cited first, 3 of 3)

- [Normalizing Trajectory Models](https://mlsanity.com/paper/2605.08078/) (2026)
- [Beyond Random Couplings: Contrastive Noise Alignment in Generative Flows](https://mlsanity.com/paper/2609.18488/) (2026)
- [Global Transport Couplings for Classifier-Free Guided Flows](https://mlsanity.com/paper/2610.07555/) (2026)

## Abstract (arXiv)

Flow models have rapidly become the go-to method for training and deploying large-scale generators, owing their success to inference-time flexibility via adjustable integration steps. A crucial ingredient in flow training is the choice of coupling measure for sampling noise/data pairs that define the flow matching (FM) regression loss. While FM training defaults usually to independent coupling, recent works show that adaptive couplings informed by noise/data distributions (e.g., via optimal transport, OT) improve both model training and inference. We radicalize this insight by shifting the paradigm: rather than computing adaptive couplings directly, we use distilled couplings from a different, pretrained model capable of placing noise and data spaces in bijection -- a property intrinsic to normalizing flows (NF) through their maximum likelihood and invertibility requirements. Leveraging recent advances in NF image generation via auto-regressive (AR) blocks, we propose Normalized Flow Matching (NFM), a new method that distills the quasi-deterministic coupling of pretrained NF models to train student flow models. These students achieve the best of both worlds: significantly outperforming flow models trained with independent or even OT couplings, while also improving on the teacher AR-NF model.