Density-Level Backward Error for Low-NFE Generative Flows
Abstract
Low-NFE generative ODE sampling is constrained not only by solver order, but also by how a fixed budget of network evaluations is distributed along the flow. We develop a density-level backward-error framework that identifies the component of a solver's modified-equation defect that changes the evolving marginal, while excluding solenoidal trajectory distortion that is invisible in Wasserstein tangent geometry. The resulting residual induces a solver-order-aware compute-allocation rule and, through gauge freedom, an objective for selecting numerically favorable velocity representatives of the same density path. This separates our approach from schedule search driven by trajectory or endpoint surrogates and from higher-order solver design. Exact Gaussian and mixture models establish the mechanism, including cases where curvature-based allocation fails and continuum-optimal transport is not discretization-optimal. On a pretrained CIFAR-10 EDM probability-flow ODE, density-aware allocation consistently improves fixed-update Euler sampling in the low-NFE regime. Matched-seed convergence to a dense Euler reference confirms that the method accelerates approximation of the same high-NFE flow rather than changing the asymptotic sampler.
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BibTeX
@misc{obzherin2026densitylevel,
title = {{Density-Level Backward Error for Low-NFE Generative Flows}},
author = {Ewsey Obzherin and Valentin Khrulkov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29475},
eprint = {2610.29475},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29475},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Density-Level Backward Error for Low-NFE Generative Flows
AU - Ewsey Obzherin
AU - Valentin Khrulkov
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29475
UR - https://arxiv.org/abs/2610.29475
AB - Low-NFE generative ODE sampling is constrained not only by solver order, but also by how a fixed budget of network evaluations is distributed along the flow. We develop a density-level backward-error framework that identifies the component of a solver's modified-equation defect that changes the evolving marginal, while excluding solenoidal trajectory distortion that is invisible in Wasserstein tangent geometry. The resulting residual induces a solver-order-aware compute-allocation rule and, through gauge freedom, an objective for selecting numerically favorable velocity representatives of the same density path. This separates our approach from schedule search driven by trajectory or endpoint surrogates and from higher-order solver design. Exact Gaussian and mixture models establish the mechanism, including cases where curvature-based allocation fails and continuum-optimal transport is not discretization-optimal. On a pretrained CIFAR-10 EDM probability-flow ODE, density-aware allocation consistently improves fixed-update Euler sampling in the low-NFE regime. Matched-seed convergence to a dense Euler reference confirms that the method accelerates approximation of the same high-NFE flow rather than changing the asymptotic sampler.
ER -