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The Machinery of Generation TL;DR This 3-part introduction to flow matching is largely inspired by reference [1]. If you do not have time to complete the full course and companion code, this series is a fast, practical alternative: it distills the core ideas with high-quality visualizations and cleaner code to make the material easier to absorb. 1. Generation as Sampling and Transformation The starting point of generative modeling is the sampling problem. We are given training samples ...

The goal In the previous section, generation was framed as a transport problem: start from a simple source distribution and move samples through an ODE until they match the data distribution. In flow matching, that transport is governed by a time-dependent vector field. The central question of this section is therefore: what vector field should we use as the training target? For the running toy example, the source distribution is ...

Constructing the Training Loss Section 1 framed generative modeling as sampling through transformation: start from a simple distribution and transport samples toward the data distribution by integrating an ODE. In that picture, the learned vector field is the local motion rule, and the induced flow is the global transformation that turns noise into data. Section 2 then showed how to construct the relevant target vector field. By introducing conditional and marginal probability paths, we obtained a closed-form expression for the tractable conditional vector field $u_t^{\text{target}}(x \mid z)$ and saw, via the marginalization trick, how these conditional fields assemble into the marginal vector field $u_t^{\text{target}}(x)$ that actually governs the distribution-level transport. ...

VAEs are an especially important model to study if you want to understand modern generative modeling for two reasons. First, they introduced a clean probabilistic latent-variable framework for generation—showing how to learn a distribution over hidden representations and sample from it in a principled way. Second, VAEs remain central to state-of-the-art generative systems today: in many diffusion-based models (notably latent diffusion and Stable Diffusion [1]), a VAE is the component that compresses images into a latent space and decodes generated latents back into images, making high-quality generation practical and efficient. ...