Trajectory Consistency for One-Step Generation on Euler Mean Flows

TL;DR

This work tackles the challenge of long-range trajectory consistency in one-step flow-based generative models. It introduces Euler Mean Flows (EMF), a local linearization of the semigroup constraint that yields a surrogate loss $L^E( heta)$ (and its $x_1$-prediction variant) which can be data-supervised without Jacobian-vector products. The authors prove that, under mild assumptions, EMF approximates the original trajectory-consistency objective up to $O( abla)$-level error and provide a unified, JVP-free training framework with theoretical guarantees. Empirically, EMF delivers improved optimization stability and sample quality across latent-space and pixel-space image generation, as well as SDF, point clouds, and functional generation, while reducing training time and memory by roughly 50% relative to existing one-step methods. The approach also enables efficient one-step generation in sparse domains, broadening applicability to diverse geometric and functional tasks.

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.

Figures (26)

• Figure 1: Illustration of trajectory consistency and the Euler Mean Flow (EMF) method. Left: Multiple flow maps can satisfy trajectory consistency, but only the solid path correctly transports noise to the data distribution, highlighting the necessity of data supervision. Middle: Two existing approaches for learning long-range trajectories, including continuous-equation-based methods and progressive extension. Right: Our EMF reformulates the trajectory consistency equation via a local linear approximation and introduces direct data supervision for long-range dynamics through the resulting linearized segment.
• Figure 2: We present 1-step generation results of our EMF method for functional image generation (top left), SDF generation conditioned on 64 surface points (bottom left), unconditional point cloud generation on ShapeNet chang2015shapenet (bottom right), and ImageNet deng2009imagenet class-conditional generation (right).
• Figure 3: Auxiliary Branch for $u_{t\to t}^\theta(x)$ Prediction
• Figure 4: Training loss comparison between Euler Mean Flow and Mean Flow.
• Figure 5: Latent image unconditional generation result trained on CelebA-HQ dataset liu2015faceattributes. First, second and third rows shows 1-step, 2-steps and 4-steps generation respectively.

Theorems & Definitions (11)

• Theorem 4.1: Non-existence of conditional flow maps
• Theorem 4.2: Local Linear Approximation
• Lemma 1
• Theorem 4.3: Surrogate Loss Validity
• Theorem 4.4: Surrogate Loss Validity for $x_1$-Prediction
• proof
• proof
• proof
• Lemma 2
• proof