# A Theory on Flow Matching with Neural Networks

arXiv: https://arxiv.org/abs/2606.10089 · HTML: https://mlsanity.com/paper/2606.10089/

## Key facts

- Cited by 1 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 17 of its 52 references that are in the map.
- First posted on arXiv 8 Jun 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, 17 of 17)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (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)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [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)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Pyramidal Flow Matching for Efficient Video Generative Modeling](https://mlsanity.com/paper/2410.05954/) (2024)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization](https://mlsanity.com/paper/2401.15604/) (2024)
- [Flow matching achieves almost minimax optimal convergence](https://mlsanity.com/paper/2405.20879/) (2024)
- [LAFMA: A Latent Flow Matching Model for Text-to-Audio Generation](https://mlsanity.com/paper/2406.08203/) (2024)
- [A Theoretical Analysis of Discrete Flow Matching Generative Models](https://mlsanity.com/paper/2509.22623/) (2025)

## Built on by (most cited first, 1 of 1)

- [Learning Discriminative Geometry for Drifting Models](https://mlsanity.com/paper/2610.04703/) (2026)

## Abstract (arXiv)

In this work, we develop theoretical foundation for flow matching with neural-network-parameterized conditional velocity fields. We establish convergence guarantees for gradient descent in the over-parameterized 2-layered ReLU neural network regime. We derive generalization bounds for the conditional velocity-field matching objective. Building on these results, we provide Wasserstein-distance guarantees for the samples generated by the induced flow. Our analysis is based on generalization bound for multi-task representation learning with unbounded losses, which may be of independent interest beyond flow-based generative modeling. These theoretical results are validated through extensive experiments on both synthetic and real-world image benchmarks.