Welcome to the first note in a build-from-scratch series on generative AI — diffusion models, score matching, and flow matching. The goal is to derive each idea from first principles, implement it in code, and make it interactive so the intuition sticks.
The one idea
Every diffusion model rests on a single, almost suspiciously simple idea:
Gradually destroy the structure in your data by adding noise until nothing is left but pure randomness — then learn to reverse that process, one small step at a time.
If you can learn to undo noise, you can start from pure noise and generate data. That’s it. Everything else — DDPM, score-based SDEs, probability-flow ODEs, flow matching — is a different lens on this same forward/reverse pair.
The forward process
We corrupt a data point \(x_0\) over a continuous time \(t \in [0, 1]\) with a variance-preserving schedule:
\[x_t = \sqrt{\bar\alpha_t}\, x_0 + \sqrt{1 - \bar\alpha_t}\, \varepsilon, \qquad \varepsilon \sim \mathcal{N}(0, I),\]where \(\bar\alpha_t\) decays smoothly from \(1\) (all signal) to \(\approx 0\) (all noise). At \(t=0\) we have the data; as \(t \to 1\) every point relaxes into the same standard Gaussian ball, regardless of where it started.
Drag the slider below to see this happen — watch structured data (a spiral, two moons, or blobs) dissolve into Gaussian noise, and notice how the schedule \(\bar\alpha_t\) controls the pace:
A few things worth noticing as you play:
- It’s reversible in principle. Because we added a specific \(\varepsilon\), if we knew it we could subtract it back out. The whole learning problem is exactly this: predict the noise \(\varepsilon\) (equivalently, the score \(\nabla_x \log p_t(x)\)) from the corrupted \(x_t\).
- All paths end in the same place. Every distribution flows to \(\mathcal{N}(0, I)\). That shared endpoint is what lets us start generation from pure noise.
- The schedule matters. Most of the structure is destroyed in a surprisingly narrow band of \(t\) — which is why noise schedules and time-weighting are such a big deal in practice.
Where this series is going
From this single forward/reverse picture we’ll build up, with code and an interactive widget for each step:
- DDPM — the discrete-time denoising objective and why predicting \(\varepsilon\) works.
- Score-based models & the probability-flow ODE — the continuous-time (SDE) view and deterministic sampling.
- Flow matching & continuous normalizing flows — straightening the paths and learning velocity fields directly.
This first note is the conceptual on-ramp; the implementation-heavy sections are on their way (see below).