Lecture 16 — Generative Models: Flow Matching & ODEs

1. Target Data Distribution (Smiley Face)

STATIC

The target distribution is a 2D smiley face built from sampled points along a circle (face outline), two dots (eyes), and an arc (mouth), each perturbed with small Gaussian noise.

Points: 1200

2. Flow Matching — Trajectories

ANIMATED

Particles follow straight-line (linear interpolation) paths from noise (t=0) to data (t=1): x_t = (1−t) x₀ + t x₁

Speed: 1.0 t = 0.00

Noise (source) Data (target) Current particles

3. Velocity Vector Field

INTERACTIVE

The learned velocity field v(x_t, t) = x₁ − x₀ shown as arrows on a grid. Arrows point from the current interpolated position toward the target. Drag the slider to see how the field changes with time.

t: 0.00 Speed 1.0×

4. Sample Generation (ODE Integration)

ANIMATED

Starting from noise, we integrate dx/dt = v(x, t) via Euler's method to generate new samples that match the target distribution.

Speed: 1.0 Euler steps: 50 t = 0.00

5. Simple ODE: dx/dt = −x

INTERACTIVE

Analytical solution: x(t) = x(0) · e^−t. Drag the slider to change the initial condition.

x(0): 2.0

6. Euler's Method Visualization

INTERACTIVE

Compare the true solution x(t) = x(0) e^−t with Euler steps x(t+Δt) = x(t) + Δt · f(x, t). Larger step sizes yield less accurate (and eventually unstable) approximations.

Δt: 0.50 x(0): 2.0

True solution Euler approximation

7. Multiple Euler Step Sizes

VISUALIZATION

Euler approximations with step sizes Δt ∈ {0.1, 0.5, 1.0, 1.5, 2.0, 2.5}. For Δt > 2 the method becomes unstable and oscillates.

x(0): 2.0