The interactive film is in the book's Companion

Play it, pause it, turn it and drag it with Mathematics for AI and Machine Learning.

Get the book

Probability Flow ODE: Deterministic Paths, Identical Marginals

Chapter 20 · Mathematics for AI and Machine Learning · Advanced · 1:34

An interactive live film: noisy stochastic diffusion paths vs smooth deterministic probability flow streamlines; at every time slice, the marginal distributions strictly coincide.

Narration

Diffusion models generate data through stochastic differential equations. Every sample path is noisy, random, and jagged.

Surprisingly, every diffusion process has a twin: the probability flow ordinary differential equation. It is completely deterministic and smooth.

The rest of the narration is in the book's Companion.

More from Chapter 20

Interactive1:45Flow Matching: Velocity Fields and the Continuity EquationChapter 20Preview1:28From a Residual Network to a Neural ODEChapter 20Preview1:27Gradient Flow: Gradient Descent With Infinitely Small StepsChapter 20Preview1:26Memory by Orthogonal Projection (HiPPO)Chapter 20Preview1:36Momentum Methods: Second-Order ODEChapter 20Preview1:40Backward Error Analysis: What Gradient Descent FollowsChapter 20Preview1:46Rectified Flow: Straightening by RetrainingChapter 20