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Understanding the Reverse SDE and Probability Flow ODE

The Reverse-Time SDE

The Formula

\[ dx = \left[f(x,t) - g(t)^2 \nabla_x \log p_t(x)\right]dt + g(t)\,dw \]

Term-by-Term Interpretation

The infinitesimal change in \(x\) (the data) is the sum of:

  1. \(f(x,t)\,dt\) — The original drift from the forward process
  2. This term "continues" the deterministic flow from the forward SDE
  3. In VP-SDE: \(f(x,t) = -\frac{1}{2}\beta(t)x\) (shrinks toward origin)

  4. In VE-SDE: \(f(x,t) = 0\) (no drift)

  5. \(-g(t)^2 \nabla_x \log p_t(x)\,dt\) — The score correction (the key term!)

  6. Points toward regions of higher probability density
  7. \(\nabla_x \log p_t(x)\) is the score function: "which direction makes \(x\) more likely?"
  8. Scaled by \(g(t)^2\): stronger correction when noise is high

  9. This term reverses the diffusion: it "undoes" the noise

  10. \(g(t)\,dw\) — Stochastic noise (Brownian motion)

  11. Same diffusion coefficient as the forward process
  12. Maintains diversity in generated samples
  13. \(dw\) is a reverse-time Brownian increment

Intuition

Think of it as:

\[ dx = \underbrace{f(x,t)\,dt}_{\text{continue forward drift}} + \underbrace{(-g(t)^2 \nabla_x \log p_t(x))\,dt}_{\text{denoise: move toward data}} + \underbrace{g(t)\,dw}_{\text{add randomness}} \]

The score term is what makes generation possible. Without it, we'd just be running the forward process in reverse time, which wouldn't recover data from noise.

Where Does the Probability Flow ODE Come From?

The Formula

\[ dx = \left[f(x,t) - \tfrac{1}{2}g(t)^2 \nabla_x \log p_t(x)\right]dt \]

Note: No \(dw\) term — this is deterministic!

The Origin: Anderson's Result (1982)

The probability flow ODE comes from a deep result in stochastic calculus:

Key theorem (Anderson 1982): For any SDE, there exists a corresponding ODE that produces the same marginal distributions \(p_t(x)\) at each time \(t\).

This is not obvious! The SDE has randomness; the ODE has none. Yet they produce the same distribution over states.

The Derivation (High-Level)

  1. Start with the reverse SDE:

$$

dx = \left[f(x,t) - g(t)^2 \nabla_x \log p_t(x)\right]dt + g(t)\,dw $$

  1. The Fokker-Planck equation describes how probability density evolves:

$$

\frac{\partial p_t}{\partial t} = -\nabla \cdot (f \cdot p_t) + \frac{1}{2}g(t)^2 \nabla^2 p_t $$

  1. Key insight: The diffusion term \(\frac{1}{2}g(t)^2 \nabla^2 p_t\) can be rewritten as a drift term involving the score:

$$

\frac{1}{2}g(t)^2 \nabla^2 p_t = \nabla \cdot \left(\frac{1}{2}g(t)^2 \nabla p_t\right) = \nabla \cdot \left(\frac{1}{2}g(t)^2 p_t \nabla \log p_t\right) $$

  1. Absorb diffusion into drift: Combine the original drift with this "effective drift" from diffusion:

$$

\text{Effective drift} = f(x,t) - \frac{1}{2}g(t)^2 \nabla_x \log p_t(x) $$

  1. Result: An ODE with this effective drift produces the same Fokker-Planck equation, hence the same \(p_t(x)\).

Why Factor of ½?

Compare the two:

Equation Score coefficient Has noise?
Reverse SDE \(g(t)^2\) Yes (\(+g(t)dw\))
Probability Flow ODE \(\frac{1}{2}g(t)^2\) No

The factor of \(\frac{1}{2}\) compensates for the missing stochastic term. Roughly: - In the SDE, noise contributes to spreading probability mass - In the ODE, the score term must do all the work of shaping the distribution - But without noise, you need less "pull" from the score (hence \(\frac{1}{2}\))

Connection Between SDE and ODE

Same Marginals, Different Paths

Both equations produce samples from the same distribution \(p_0(x)\) (data distribution), but:

Property Reverse SDE Probability Flow ODE
Paths Random (different each run) Deterministic (same every time)
Diversity High (noise adds variety) Lower (same input → same output)
Speed Slower (needs small \(dt\)) Faster (can use larger steps)
Interpolation Not meaningful Latent space is well-defined

Why Have Both?

SDE sampling:

  • More diverse outputs
  • Better for creative generation
  • Theoretically "correct" reverse process

ODE sampling:

  • Faster (DDIM is based on this)
  • Deterministic: good for reproducibility
  • Enables latent space manipulation
  • Can use advanced ODE solvers (Runge-Kutta, etc.)

Practical Implications

For DDPM/DDIM

  • DDPM: Discretized reverse SDE (with noise)
  • DDIM: Discretized probability flow ODE (deterministic)
  • Both use the same trained model \(s_\theta(x,t)\)

For Fast Sampling

ODE formulation enables: - Larger time steps (less sensitive to discretization) - Adaptive step-size solvers - Fewer neural network evaluations (faster generation)

The Score Is Everything

In both formulations, the score function \(\nabla_x \log p_t(x)\) is the only learned quantity. Once you have it, you can choose either: - Stochastic sampling (SDE) - Deterministic sampling (ODE)

Summary

Question Answer
What does the reverse SDE compute? \(dx =\) (continue drift) + (denoise via score) + (add noise)
Where does the ODE come from? Fokker-Planck: absorb diffusion into an effective drift
Why factor of ½? Compensates for missing stochastic term
Which to use? SDE for diversity, ODE for speed/determinism

References