Trajectory Consistency for One-Step Generation on Euler Mean Flows

TL;DR

This paper introduces Euler Mean Flows (EMF), a flow-based generative framework that ensures long-range trajectory consistency with minimal sampling, reducing training time and memory while improving sample quality.

Contribution

The paper presents a novel EMF framework that replaces complex trajectory constraints with a linear surrogate, enabling efficient one-step generation with better stability and lower computational costs.

Findings

Improved sample quality in image synthesis and geometry generation.

Approximately 50% reduction in training time and memory usage.

Enhanced optimization stability for long-horizon flow-map compositions.

Abstract

We propose \emph{Euler Mean Flows (EMF)}, a flow-based generative framework for one-step and few-step generation that enforces long-range trajectory consistency with minimal sampling cost. The key idea of EMF is to replace the trajectory consistency constraint, which is difficult to supervise and optimize over long time scales, with a principled linear surrogate that enables direct data supervision for long-horizon flow-map compositions. We derive this approximation from the semigroup formulation of flow-based models and show that, under mild regularity assumptions, it faithfully approximates the original consistency objective while being substantially easier to optimize. This formulation leads to a unified, JVP-free training framework that supports both $u$-prediction and $x_1$-prediction variants, avoiding explicit Jacobian computations and significantly reducing memory and computational overhead. Experiments on image synthesis, particle-based geometry generation, and functional generation demonstrate improved optimization stability and sample quality under fixed sampling budgets, together with approximately $50\%$ reductions in training time and memory consumption compared to existing one-step methods for image generation.