How Does a Diffusion Model Work, Part 2: Flow Matching and Diffusion

Send us Fan Mail Diffusion models have become the foundation of modern generative AI, powering state-of-the-art systems for image generation, video synthesis, protein design, and more. In the first episode of this series, we introduced generative modeling as a sampling problem and developed the probabilistic framework needed to think about how models learn complex data distributions. In this second episode, we move into the core mathematics behind **flow matching and diffusion modeling**, continuing through the MIT lecture notes *An Introduction to Flow Matching and Diffusion Models* by Peter Holderrieth and Ezra Erives. We explore how a simple, easy-to-sample distribution such as Gaussian noise can be continuously transformed into a complicated data distribution, and how this idea leads naturally to both deterministic and stochastic approaches to generative modeling. Topics covered include: - The basic idea behind flow matching - Transforming a noise distribution into a data distribution - Probability paths and time-dependent distributions - Vector fields and continuous-time generative processes - Ordinary differential equations (ODEs) in generative modeling - Stochastic differential equations (SDEs) and the role of injected noise - The relationship between diffusion models and continuous flows - Forward and reverse diffusion processes - How trained diffusion and flow models are used during inference - Numerical inference algorithms for solving the resulting ODEs and SDEs - Why the number of inference steps affects speed and generation quality - How these mathematical formulations connect to practical modern diffusion systems The goal of this episode is to build intuition for what a diffusion model is actually doing mathematically: rather than simply “removing noise,” we can view generation as learning a dynamical process that transports probability from a simple starting distribution toward the complex distribution of real data. This framework provides the foundation for later discussions of score matching, denoising objectives, probability flow ODEs, modern sampling algorithms, and the architectures used in state-of-the-art generative models. Lecture Notes: https://diffusion.csail.mit.edu/docs/lecture-notes.pdf Website: https://arkitekt-ai.com Contact: [support@arkitekt-ai.com](mailto:support@arkitekt-ai.com)