Flow-Based Generative Models

From Normalizing Flows to Flow Matching · UFRJ

Graduate-level course, offered jointly to two programmes: PGMAT (semester, includes an initial deep-learning bootcamp) and PESC (trimester, joining at the main course). Two 90-minute sessions per week.

Version 2026.2.

Course motivation

Ten years of work on flow-based generative modelling converge on a single simulation-free framework — Flow Matching. Diffusion models turn out to be a family of particular choices inside it. This course teaches the general framework first and then arrives at diffusion from it, rather than the other way round.

Diffusion is therefore never a separate theory here. It enters through three short detours off the Flow Matching path, each making one precise point:

  • D1 — the paths. Pick a particular family of Gaussian probability paths and Flow Matching reproduces exactly the noising process of a diffusion model: the VP and VE processes are specific noise schedules, and DDPM’s “forward process” is a choice of interpolation.
  • D2 — the vocabulary. Score matching, noise prediction and velocity prediction are three names for objects we will already have built. The Rosetta Stone handout translates them term by term, which is what makes the diffusion literature readable in either dialect.
  • D3 — the sampler. Once the vector field is trained, how you draw samples is an independent decision: deterministic ODE, stochastic SDE, or few-step schemes. Diffusion samplers become one option among several, not a commitment.

What the course covers

  1. Discrete-time flows (U1): exact likelihood via change of variables; architectural ingenuity against expressivity constraints.
  2. Continuous time (U2): Neural ODEs and CNFs lift those constraints, but training now requires simulating the model — the pain point.
  3. Flow Matching (U3): simulation-free training of the vector field; the resolution — with the diffusion detours D1–D3.
  4. Frontiers (U6): manifolds, discrete data, graphs — student seminars.

Materials

Every session comes in two forms: lecture notes to read, and slides to follow in class (with a printable PDF). Pick a session in the sidebar; its notes page opens with links to the deck and the PDF. The Rosetta Stone — the dictionary between the Flow Matching and diffusion vocabularies — is a living handout that grows through U3.