The probability flow ODE is a deterministic ordinary differential equation that generates the same marginal probability densities p_t as a corresponding stochastic differential equation, but without any noise. For a general Itô SDE dX_t = v(X_t, t)dt + σ(X_t, t)dW_t with diffusion tensor D = σσ^T, the probability flow ODE is:
dX_t = [v(X_t, t) - ½(∇·D)(X_t, t) - ½D(X_t, t)∇log p_t(X_t)] dt
This ODE preserves the same time-evolving density as the SDE, enabling deterministic sampling and exact likelihood computation. The construction works because replacing the stochastic noise with a score-dependent correction term yields the same Fokker-Planck equation.
In the context of Brownian random bridges, the probability flow ODE takes a particularly revealing form: dξ_t = ½[(ξ_t - x)/t + (E[Y|ξ_t] - ξ_t)/(T-t)]dt. This is the arithmetic mean of two distinct forces: an inertial source velocity (ξ_t - x)/t pushing outward from the initial point, and a conditional target velocity (E[Y|ξ_t] - ξ_t)/(T-t) pulling toward the target distribution. The target-pull component has the same form as the rectified flow velocity, identifying the bridge probability flow ODE as a “half-speed” variant of rectified flow tempered by source-dependent drift. In the zero-volatility limit (σ → 0), the source-push and target-pull velocities coincide, and the ODE reduces to pure rectified flow.
Key Details
- In score-based generative modelling (Song et al. 2021), the probability flow ODE enables exact likelihood computation via the instantaneous change of variables formula
- The ODE is also called the “neural ODE” when the score function is parameterized by a neural network
- For VP-SDE: dX_t = [-½β(t)X_t - ½β(t)∇log p_t(X_t)]dt
- For the Gaussian random bridge: the half-speed structure means the source-push exactly cancels the score contribution in the reverse-time drift
- The concept generalizes to the Kolmogorov forward (Fokker-Planck) equation framework: the probability flow ODE converts the diffusion equation into a Liouville (transport) equation