probability flow ode

**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**: $dz = [f(z,t) - frac{1}{2}g^2(t) abla_z log p_t(z)]dt$ (deterministic, same marginals). - **Score Function**: Requires the score $ abla_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.