# An Introduction to Flow Matching and Diffusion Models

arXiv: https://arxiv.org/abs/2506.02070 · HTML: https://mlsanity.com/paper/2506.02070/

## Key facts

- Cited by 35 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 26 of its 51 references that are in the map.
- First posted on arXiv 2 Jun 2025.

## 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, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [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)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [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)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [eDiff-I: Text-to-Image Diffusion Models with an Ensemble of Expert Denoisers](https://mlsanity.com/paper/2211.01324/) (2022)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Movie Gen: A Cast of Media Foundation Models](https://mlsanity.com/paper/2410.13720/) (2024)
- [ByT5: Towards a token-free future with pre-trained byte-to-byte models](https://mlsanity.com/paper/2105.13626/) (2021)
- [Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design](https://mlsanity.com/paper/2402.04997/) (2024)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [UL2: Unifying Language Learning Paradigms](https://mlsanity.com/paper/2205.05131/) (2022)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Adversarial score matching and improved sampling for image generation](https://mlsanity.com/paper/2009.05475/) (2020)
- [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024)
- [Modeling Caption Diversity in Contrastive Vision-Language Pretraining](https://mlsanity.com/paper/2405.00740/) (2024)

## Built on by (most cited first, 30 of 35)

- [Diffusion Models in Simulation-Based Inference: A Tutorial Review](https://mlsanity.com/paper/2512.20685/) (2025)
- [DiffusionGemma Technical Report](https://mlsanity.com/paper/2608.00146/) (2026)
- [Guided Flow Policy: Learning from High-Value Actions in Offline Reinforcement Learning](https://mlsanity.com/paper/2512.03973/) (2025)
- [HyperDiT: Hyper-Connected Transformers for High-Fidelity Pixel-Space Diffusion](https://mlsanity.com/paper/2605.15741/) (2026)
- [Diffusion-Based Impedance Learning for Contact-Rich Manipulation Tasks](https://mlsanity.com/paper/2509.19696/) (2025)
- [Optimization-Guided Diffusion for Interactive Scene Generation](https://mlsanity.com/paper/2512.07661/) (2025)
- [Solving Room Impulse Response Inverse Problems Using Flow Matching with Analytic Wiener Denoiser](https://mlsanity.com/paper/2602.00652/) (2026)
- [SGMatch: Semantic-Guided Non-Rigid Shape Matching with Flow Regularization](https://mlsanity.com/paper/2603.12937/) (2026)
- [Entropy-Regularized Adjoint Matching for Offline Reinforcement Learning](https://mlsanity.com/paper/2605.06156/) (2026)
- [Back to the Feature: Explaining Video Classifiers with Video Counterfactual Explanations](https://mlsanity.com/paper/2511.20295/) (2025)
- [Max-Entropy Reinforcement Learning with Flow Matching and A Case Study on LQR](https://mlsanity.com/paper/2512.23870/) (2025)
- [Reducing Memorisation in Generative Models via Riemannian Bayesian Inference](https://mlsanity.com/paper/2602.00199/) (2026)
- [Guiding Vector Field Generation via Score-based Diffusion Model](https://mlsanity.com/paper/2604.24487/) (2026)
- [A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models](https://mlsanity.com/paper/2605.06829/) (2026)
- [MarkNull: Model-Agnostic Watermark Removal in AI-Generated Images via On-Manifold Latent Manipulation](https://mlsanity.com/paper/2608.10166/) (2026)
- [FlowCPO: A Unified Divergence View of Preference Alignment for Flow Models](https://mlsanity.com/paper/2609.09905/) (2026)
- [Secure and reversible face anonymization with diffusion models](https://mlsanity.com/paper/2510.01031/) (2025)
- [Contact Wasserstein Geodesics for Non-Conservative Schr"odinger Bridges](https://mlsanity.com/paper/2511.06856/) (2025)
- [What We Don't C: Manifold Disentanglement for Structured Discovery](https://mlsanity.com/paper/2511.09433/) (2025)
- [Can Continuous-Time Diffusion Models Generate and Solve Globally Constrained Discrete Problems? A Study on Sudoku](https://mlsanity.com/paper/2601.20363/) (2026)
- [Denoising Particle Filters: Learning State Estimation with Single-Step Objectives](https://mlsanity.com/paper/2602.19651/) (2026)
- [Towards Faster Language Model Inference Using Mixture-of-Experts Flow Matching](https://mlsanity.com/paper/2604.15009/) (2026)
- [Fashion130K: An E-commerce Fashion Dataset for Outfit Generation with Unified Multi-modal Condition](https://mlsanity.com/paper/2605.10127/) (2026)
- [SF-Flow: Sound field magnitude estimation via flow matching guided by sparse measurements](https://mlsanity.com/paper/2605.10398/) (2026)
- [A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models](https://mlsanity.com/paper/2605.22586/) (2026)
- [Flow Control: Steering Vision-Language-Action Models with Simple Real-Time Inputs](https://mlsanity.com/paper/2606.10180/) (2026)
- [Understanding Latent Flow Models for Tabular Data Synthesis: Targets, Paths, and Sampling](https://mlsanity.com/paper/2606.20878/) (2026)
- [Generative Models: Principles, Architectures, and Applications](https://mlsanity.com/paper/2608.08101/) (2026)
- [Uncovering Hidden Leptonic Correlations with Flow Matching and Autoencoders](https://mlsanity.com/paper/2608.15042/) (2026)
- [Quantum MeanFlow: single-shot generative sampling on NISQ hardware](https://mlsanity.com/paper/2609.02186/) (2026)

## Abstract (arXiv)

Diffusion and flow-based models have become the state of the art for generative AI across a wide range of data modalities, including images, videos, shapes, molecules, music, and more. This tutorial provides a self-contained introduction to diffusion and flow-based generative models from first principles. We systematically develop the necessary mathematical background in ordinary and stochastic differential equations and derive the core algorithms of flow matching and denoising diffusion models. We then provide a step-by-step guide to building image and video generators, including training methods, guidance, and architectural design. This course is ideal for machine learning researchers who want to develop a principled understanding of the theory and practice of generative AI.