2D Score Field Playground

1 · Define the training goal

Draw a distribution on the canvas (or pick a preset). Strokes are infinitely thin curves on purpose: real data lives on a thin set, so the density is zero almost everywhere and the gradient of log-density is undefined off the curve. That is why we need noise.

0 points

2 · Add noise (forward process)

Slide t from 0 (clean data) to 1 (pure noise). Each point is mixed with its own fixed Gaussian noise, so you can watch the thin curve smear into a fat blob and finally into a Gaussian.

Show clean data for reference

Once noised, every region of the plane has non-zero density, so a "which way is more probable" direction exists everywhere. That direction is what the network will learn.

3 · Train the network

Training uses a random t each step.

Network architecture

step 0 · loss –

Inspect what it learned at any noise level:

Show noisy points at this t

4 · Sample from noise (reverse process)

Steps

η=1 is DDPM (fresh noise each step, stochastic: the reverse SDE); η=0 is DDIM (deterministic: the probability-flow ODE).

Start points

Replay

The start points menu picks where the samples begin (a display aid: the model was trained to start from Gaussian noise).

Show training data Show learned field at current t

5 · Reflow (straighten the paths)

The model from step 3 follows curved paths from noise to data, because many straight training lines cross and get averaged. Reflow: (a) run the trained model's ODE from random noise z to get its endpoint x₁; (b) train a second model on the straight line between each (z, x₁) pair. Pairs don't cross much, so the new field is nearly straight. The new model starts from the old weights.

step 0 · loss –

Original ODE paths (curved) Straight lines between the pairs (reflow target)

Show points on the straight lines at this t Show reflow model field at this t

6 · Compare original vs reflowed

Both models start from the same points and use the same number of Euler steps. A straighter field survives big steps: with 1–4 steps the original lands off the data while the reflowed model stays close.

Steps

Replay

In path mode the thin line is each model's exact ODE trajectory (40 fine steps) and the dotted polyline is what the chosen number of Euler steps actually does.

Show training data