Discrete MeanFlow: One-Step Generation via Conditional Transition Kernels

Abstract

MeanFlow enables one-step generation in continuous spaces by learning an average velocity over a time interval rather than the instantaneous velocity field of flow matching. However, discrete state spaces lack smooth trajectories or spatial derivatives, so the continuous formulation does not directly apply. We introduce Discrete MeanFlow, a theoretical finite-state formulation that replaces point motion with probability-mass transport. Our key object is the conditional transition kernel of a continuous-time Markov chain (CTMC), from which we define a mean discrete rate that measures average transition-probability change over a finite interval. We prove a Discrete MeanFlow identity that relates this finite-interval rate to the instantaneous CTMC generator at the endpoint, using the Kolmogorov forward equation in place of the spatial chain rule in continuous MeanFlow. We parameterize the transition kernel directly using a boundary-by-construction design that guarantees valid probability outputs and exact boundary conditions without auxiliary losses. Because the learned kernel defines a probability distribution, generation requires only a single forward pass followed by direct sampling, eliminating the need for iterative denoising, ODE integration, or multi-step refinement. We validate the identity on exact finite-state Markov chains, demonstrating that the learned kernel recovers the analytical ground truth with high precision. We also investigate the training objective and the limits of factorization on synthetic sequence tasks across varying alphabet sizes and sequence lengths.