Skip to content

Reverse-Time SDE and Probability Flow ODE

This document connects the complete story of diffusion model sampling, showing how DDPM and DDIM emerge as discretizations of continuous-time processes.

Overview

Now we connect the whole story:

  • VP-SDE forward: Continuous-time noising process
  • Reverse-time SDE: Stochastic sampler (DDPM-like)
  • Probability flow ODE: Deterministic sampler (DDIM-like)
  • DDIM \(\eta\) parameter: Interpolates between ODE and SDE

Key insight: The same learned score function can be used for both stochastic and deterministic sampling.

Notation

Time Convention

  • Continuous time: \(t \in [0, T]\)
  • Convention: \(t = 0\) is clean data, \(t = T\) is pure noise

VP-SDE Forward Process

\[ dx = -\frac{1}{2}\beta(t) x\,dt + \sqrt{\beta(t)}\,dw \]

Signal Coefficient

\[ \bar{\alpha}(t) := \exp\left(-\int_0^t \beta(s)\,ds\right) \]

Forward marginal:

\[ q(x_t \mid x_0) = \mathcal{N}\left(\sqrt{\bar{\alpha}(t)} x_0, (1 - \bar{\alpha}(t)) I\right) \]

Score Function

\[ s(x, t) := \nabla_x \log p_t(x) \]

Learned network: \(s_\theta(x, t) \approx s(x, t)\)

Note: Equivalently, you can predict noise \(\epsilon_\theta\); we'll connect these later.

Step 1: Reverse-Time SDE for VP-SDE

General Theorem (Anderson, 1982)

For a forward Itô SDE:

\[ dx = f(x, t)\,dt + g(t)\,dw \]

the reverse-time SDE (running from \(T \to 0\)) is:

\[ dx = \left[f(x, t) - g(t)^2 \nabla_x \log p_t(x)\right]dt + g(t)\,d\bar{w} \]

where \(\bar{w}\) is reverse-time Brownian motion.

Apply to VP-SDE

For the VP-SDE:

  • \(f(x, t) = -\frac{1}{2}\beta(t) x\)
  • \(g(t) = \sqrt{\beta(t)}\)
  • \(g(t)^2 = \beta(t)\)

Therefore, the reverse VP-SDE is:

\[ \boxed{dx = \left[-\frac{1}{2}\beta(t) x - \beta(t) s(x, t)\right]dt + \sqrt{\beta(t)}\,d\bar{w}} \]

Interpretation

Drift terms:

  1. \(-\frac{1}{2}\beta(t) x\): Same "shrink toward zero" as forward process
  2. \(-\beta(t) s(x, t)\): Score correction that pushes toward high-density regions

Diffusion term: \(\sqrt{\beta(t)}\,d\bar{w}\) (same magnitude as forward, but reverse-time)

This is the continuous-time object that corresponds to DDPM sampling (stochastic).

Step 2: Probability Flow ODE

The Surprising Result (Song et al., 2021)

There exists a deterministic ODE whose solution has exactly the same marginal distributions \(p_t(x)\) as the reverse SDE.

That ODE is:

\[ \boxed{\frac{dx}{dt} = f(x, t) - \frac{1}{2}g(t)^2 s(x, t)} \]

Apply to VP-SDE

For the VP-SDE:

\[ \boxed{\frac{dx}{dt} = -\frac{1}{2}\beta(t) x - \frac{1}{2}\beta(t) s(x, t)} \]

Key Distinction

Property Reverse SDE Probability Flow ODE
Trajectories Stochastic Deterministic
Marginals \(p_t(x)\) \(p_t(x)\) (same!)
Noise Yes (\(\sqrt{\beta(t)}\,d\bar{w}\)) No
Sampling DDPM-like DDIM-like

This ODE is the continuous-time conceptual ancestor of DDIM.

Why This Matters

  • Same score function: Both use \(s(x, t)\)
  • Different dynamics: ODE has half the score correction, no noise
  • Same marginals: Generate from the same distribution
  • Different paths: Individual trajectories differ, but statistics match

Step 3: DDPM Sampling as SDE Discretization

Discretize the Reverse SDE

To generate samples, discretize time: \(t_N = T > t_{N-1} > \cdots > t_0 = 0\)

Let \(\Delta t_k = t_{k-1} - t_k\) (negative, since we go backward).

Euler–Maruyama Step

A simple Euler–Maruyama discretization (backward in time):

\[ x_{k-1} = x_k + \left[-\frac{1}{2}\beta(t_k) x_k - \beta(t_k) s_\theta(x_k, t_k)\right]\Delta t_k + \sqrt{\beta(t_k) |\Delta t_k|}\, z_k \]

where \(z_k \sim \mathcal{N}(0, I)\).

The DDPM Connection

The last term is the distinctive "DDPM-ness": fresh Gaussian noise at every step.

DDPM typically presents this as a Gaussian transition:

\[ p_\theta(x_{k-1} \mid x_k) = \mathcal{N}(x_{k-1}; \mu_\theta(x_k, t_k), \sigma_k^2 I) \]

But mathematically, a one-step Euler–Maruyama update is exactly a Gaussian transition:

  • Mean: \(x_k + \text{drift} \cdot \Delta t\)
  • Variance: \(\text{diffusion}^2 \cdot |\Delta t|\)

Key Insight

\[ \boxed{\text{DDPM sampling} \approx \text{Euler–Maruyama discretization of reverse SDE}} \]

Step 4: DDIM Sampling as ODE Discretization

Discretize the Probability Flow ODE

Now discretize the probability flow ODE:

\[ \frac{dx}{dt} = -\frac{1}{2}\beta(t) x - \frac{1}{2}\beta(t) s_\theta(x, t) \]

Euler Step

A simple Euler discretization:

\[ x_{k-1} = x_k + \left[-\frac{1}{2}\beta(t_k) x_k - \frac{1}{2}\beta(t_k) s_\theta(x_k, t_k)\right]\Delta t_k \]

Notice: No randomness term!

The DDIM Connection

\[ \boxed{\text{DDIM sampling} \approx \text{Euler discretization of probability flow ODE}} \]

Key difference from DDPM: Deterministic trajectories, no added noise.

Step 5: DDIM Update in \(\bar{\alpha}\) Notation

This bridges back to the discrete DDPM/DDIM formulas you see in code.

Discrete-Time Formulation

In discrete-time DDPM notation (steps \(t \in \{1, \ldots, T\}\)):

\[ x_t = \sqrt{\bar{\alpha}_t} x_0 + \sqrt{1 - \bar{\alpha}_t} \epsilon \]

Predict \(x_0\)

A network predicts \(\epsilon_\theta(x_t, t)\). Form an estimate of \(x_0\):

\[ \boxed{\hat{x}_0(x_t, t) = \frac{x_t - \sqrt{1 - \bar{\alpha}_t}\, \epsilon_\theta(x_t, t)}{\sqrt{\bar{\alpha}_t}}} \]

DDIM Deterministic Update (\(\eta = 0\))

\[ \boxed{x_{t-1} = \sqrt{\bar{\alpha}_{t-1}}\, \hat{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1}}\, \epsilon_\theta(x_t, t)} \]

Why This Form is Natural

  1. Keeps the noise direction: Uses \(\epsilon_\theta(x_t, t)\) predicted at time \(t\)
  2. Changes the noise scale: From \(\sqrt{1 - \bar{\alpha}_t}\) to \(\sqrt{1 - \bar{\alpha}_{t-1}}\)
  3. No new randomness: Deterministic update

Interpretation: Follow one consistent flow line rather than resampling noise at each step.

This is exactly what a deterministic probability flow ODE discretization does.

Step 6: The \(\eta\) Parameter (Interpolating ODE and SDE)

DDIM is often written with a parameter \(\eta \in [0, 1]\) that controls extra noise:

\[ x_{t-1} = \sqrt{\bar{\alpha}_{t-1}}\, \hat{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1} - \sigma_t^2}\, \epsilon_\theta(x_t, t) + \sigma_t z \]

where \(z \sim \mathcal{N}(0, I)\) and \(\sigma_t\) is chosen based on \(\eta\):

The Spectrum

\(\eta\) \(\sigma_t\) Behavior Corresponds to
\(0\) \(0\) Deterministic Probability flow ODE
\(1\) DDPM variance Stochastic Reverse SDE
\((0, 1)\) Intermediate Hybrid Interpolation

Intuitive Mapping

  • Probability flow ODE \(\Leftrightarrow\) DDIM (\(\eta = 0\))
  • Reverse SDE \(\Leftrightarrow\) DDPM (stochastic, "full noise")

Key insight: The \(\eta\) parameter lets you trade off between:

  • Determinism (faster, reproducible, good for interpolation)
  • Stochasticity (more diverse samples, better mode coverage)

Summary: The Conceptual Triangle

Here's the clean mental model:

The Learned Object

Score field: \(s_\theta(x, t) \approx \nabla_x \log p_t(x)\)

Two Ways to Sample

You can generate samples by evolving \(x\) backward using either:

  1. Reverse SDE (stochastic)
  2. Adds noise every step
  3. DDPM-like sampling

  4. Equation: \(dx = [f - g^2 s]\,dt + g\,d\bar{w}\)

  5. Probability Flow ODE (deterministic)

  6. No added noise
  7. DDIM-like sampling
  8. Equation: \(\frac{dx}{dt} = f - \frac{1}{2}g^2 s\)

Key Insights

  • Same score: Both use \(s_\theta(x, t)\)
  • Different dynamics: SDE adds noise, ODE doesn't
  • Same marginals: Generate from the same distribution \(p_t(x)\)
  • Different trajectories: Individual paths differ

The Complete Picture

Forward SDE → Marginals p_t(x) → Score s(x,t) → Reverse SDE (DDPM)
                                              ↘
                                                Probability Flow ODE (DDIM)

References

  1. Anderson, B. D. O. (1982). Reverse-time diffusion equation models. Stochastic Processes and their Applications.
  2. Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., & Poole, B. (2021). Score-Based Generative Modeling through Stochastic Differential Equations. ICLR.
  3. Song, J., Meng, C., & Ermon, S. (2021). Denoising Diffusion Implicit Models. ICLR.