Probability Flow ODE is the deterministic ODE whose trajectories have the same marginal distributions as a given stochastic differential equation — replacing the stochastic dynamics with a deterministic flow that transports probability mass in the same way, enabling exact likelihood computation and efficient sampling.

How the Probability Flow ODE Works

• Forward SDE: $dz = f(z,t)dt + g(t)dW_t$ (stochastic process from data to noise).
• Probability Flow ODE: $dz = [f(z,t) - frac{1}{2}g^2(t)\nabla_z log p_t(z)]dt$ (deterministic, same marginals).
• Score Function: Requires the score $\nabla_z log p_t(z)$, estimated by a trained score network.
• Reversibility: Integrating the ODE backward generates samples from the data distribution.

Why It Matters

• Exact Likelihood: The probability flow ODE enables exact log-likelihood computation via the instantaneous change of variables formula.
• DDIM: The DDIM sampler for diffusion models is the discretized probability flow ODE.
• Faster Sampling: Deterministic ODE allows adaptive step sizes and fewer function evaluations than SDE sampling.

Probability Flow ODE is the deterministic twin of diffusion — a noise-free ODE that produces the same distribution as the stochastic diffusion process.