# Reflected Flow Matching

arXiv: https://arxiv.org/abs/2405.16577 · HTML: https://mlsanity.com/paper/2405.16577/

## Key facts

- Cited by 15 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 31 of its 43 references that are in the map.
- First posted on arXiv 26 May 2024.

## 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 31)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Junction Tree Variational Autoencoder for Molecular Graph Generation](https://mlsanity.com/paper/1802.04364/) (2018)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Flow Matching for Scalable Simulation-Based Inference](https://mlsanity.com/paper/2305.17161/) (2023)
- [Guided Flows for Generative Modeling and Decision Making](https://mlsanity.com/paper/2311.13443/) (2023)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Subspace Diffusion Generative Models](https://mlsanity.com/paper/2205.01490/) (2022)
- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [Mirror Diffusion Models for Constrained and Watermarked Generation](https://mlsanity.com/paper/2310.01236/) (2023)
- [Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning](https://mlsanity.com/paper/2012.12137/) (2020)

## Built on by (most cited first, 15 of 15)

- [Unveil Inversion and Invariance in Flow Transformer for Versatile Image Editing](https://mlsanity.com/paper/2411.15843/) (2024)
- [Stochastic Sampling from Deterministic Flow Models](https://mlsanity.com/paper/2410.02217/) (2024)
- [Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domains](https://mlsanity.com/paper/2510.08929/) (2025)
- [Field Matching: an Electrostatic Paradigm to Generate and Transfer Data](https://mlsanity.com/paper/2502.02367/) (2025)
- [Zero-Flow Encoders](https://mlsanity.com/paper/2602.00797/) (2026)
- [ReFORM: Reflected Flows for On-support Offline RL via Noise Manipulation](https://mlsanity.com/paper/2602.05051/) (2026)
- [Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis](https://mlsanity.com/paper/2509.24531/) (2025)
- [Latent Stochastic Interpolants](https://mlsanity.com/paper/2506.02276/) (2025)
- [Constraint-Aware Flow Matching via Randomized Exploration](https://mlsanity.com/paper/2508.13316/) (2025)
- [FlowDet: Unifying Object Detection and Generative Transport Flows](https://mlsanity.com/paper/2512.16771/) (2025)
- [Set-Supervised Diffusion Policy: Learning Action-Chunking Diffusion through Corrections](https://mlsanity.com/paper/2606.01865/) (2026)
- [PolyFlow: Safe and Efficient Polytope-Constrained Flow Matching with Constraint Embedding and Projection-free Update](https://mlsanity.com/paper/2606.13400/) (2026)
- [Constrained Flow Matching via Lagrangian Dual Flows](https://mlsanity.com/paper/2607.04513/) (2026)
- [Primal-Dual Flow Matching for Sample-Wise Constrained Generation](https://mlsanity.com/paper/2610.12699/) (2026)
- [Deterministic Prediction as Global Velocity: Flow Matching for Probabilistic Time-Series Forecast](https://mlsanity.com/paper/2610.18621/) (2026)

## Abstract (arXiv)

Continuous normalizing flows (CNFs) learn an ordinary differential equation to transform prior samples into data. Flow matching (FM) has recently emerged as a simulation-free approach for training CNFs by regressing a velocity model towards the conditional velocity field. However, on constrained domains, the learned velocity model may lead to undesirable flows that result in highly unnatural samples, e.g., oversaturated images, due to both flow matching error and simulation error. To address this, we add a boundary constraint term to CNFs, which leads to reflected CNFs that keep trajectories within the constrained domains. We propose reflected flow matching (RFM) to train the velocity model in reflected CNFs by matching the conditional velocity fields in a simulation-free manner, similar to the vanilla FM. Moreover, the analytical form of conditional velocity fields in RFM avoids potentially biased approximations, making it superior to existing score-based generative models on constrained domains. We demonstrate that RFM achieves comparable or better results on standard image benchmarks and produces high-quality class-conditioned samples under high guidance weight.