probability flow ODE

Every diffusion SDE has a deterministic twin: an ordinary differential equation whose solution trajectories carry exactly the same distribution at every instant, only without the random jiggle. The probability flow ODE is that twin. Picture the cloud of all possible samples flowing smoothly from noise to data along fixed streamlines, rather than each particle taking a noisy random walk. Because the path is deterministic, the same starting noise always produces the same output.

It is obtained by removing the noise term of the reverse SDE and halving the diffusion contribution to the drift, leaving a velocity field expressed entirely through the score. Since the dynamics are an ODE, the model is exactly a continuous normalizing flow, so its log-likelihood is computable by integrating the instantaneous change of variables along the trajectory. In practice one solves the ODE with a standard numerical integrator, and higher-order solvers reach good samples in far fewer function evaluations than the stochastic chain.

The probability flow ODE is the bridge connecting diffusion models to deterministic samplers like DDIM, to fast solvers such as DPM-Solver, and to exact likelihood evaluation, all from a single score network trained the usual way.