Finite-Step Geometry of Marginal-Preserving Velocity Fields in Generative Flows

Ewsey Obzherin, Valentin Khrulkov

Generative flows can admit different velocity fields that induce the same continuous marginal evolution but behave differently after numerical discretization. We study this finite-step non-equivalence while keeping the marginal path and numerical method fixed. Projected modified equations characterize the component of local integration error that changes the marginal law and give sufficient conditions for higher marginal convergence order. For affine fixed-shape distributions, we characterize exact finite-step reachability and derive spectral obstructions and sharp nearest-reachable Wasserstein errors. For diffusion probability-flow ODEs, we construct skew-score perturbations that preserve the exact marginal path under exact scores and quantify their leakage under score error. Controlled Gaussian and mixture experiments recover the predicted order, reachability, and obstruction phenomena.