continuous normalizing flows (CNF)

A continuous normalizing flow replaces the stack of discrete invertible layers with a single ordinary differential equation. Instead of asking how each layer warps a point, you specify a velocity field, learned by a neural network, and let a point flow along it for a fixed time. The composition of infinitely many infinitesimal transformations is the solution of the ODE, so the flow is as deep as the integrator chooses to be, with one set of weights shared across all time.

The decisive simplification is the instantaneous change of variables: as the point flows, its log-density changes at a rate equal to minus the divergence of the velocity field, so the total log-likelihood is recovered by integrating that divergence along the trajectory. This replaces the expensive log-determinant of a Jacobian with a trace, which can be estimated cheaply by the Hutchinson stochastic trace estimator. Training originally backpropagated through the ODE solver via the adjoint method, but modern practice trains the same velocity field by simulation-free flow matching, which is far more stable and scalable.

CNFs are the unifying object behind probability flow ODEs and flow matching: a diffusion's deterministic sampler is precisely a CNF, and most large continuous generators today are CNFs trained by regression rather than by maximum likelihood.