The Velocity Deficit: Initial Energy Injection for Flow Matching

2026-05-14

AI Summary

This work addresses a critical issue in high-dimensional flow matching: the systematic underestimation of velocity magnitude at trajectory initialization induces integration lag, preventing generated samples from accurately reaching the data manifold. The study uncovers an asymmetric mechanism wherein velocity contraction is detrimental at the start but beneficial near the end of trajectories. To mitigate this without retraining, the authors propose a joint strategy combining a Scale Scheduling Corrector (SSC) with Magnitude-Aware Flow Matching (MAFM), implementable with just a single line of code. The approach substantially improves performance—on ImageNet-1k, it reduces FID from 13.68 to 7.58 (a 44.6% improvement), achieves a 5× speedup, and yields 50-step generation quality surpassing the original 250-step baseline; on MS-COCO text-to-image generation, FID also improves by approximately 22%.

Abstract

While Flow Matching theoretically guarantees constant-velocity trajectories, we identify a critical breakdown in high-dimensional practice: the Velocity Deficit. We show that the MSE objective systematically underestimates velocity magnitude, causing generated samples to fail to reach the data manifold-a phenomenon we term Integration Lag. To rectify this, we propose Initial Energy Injection, instantiated via two complementary methods: the training-based Magnitude-Aware Flow Matching (MAFM) and the training-free Scale Schedule Corrector (SSC). Both are grounded in our discovery of a crucial asymmetry: velocity contraction causes harmful kinetic stagnation at the trajectory's start, yet acts as a beneficial denoising mechanism at its end. Empirically, SSC yields significant efficiency gains with zero retraining and just one line of code. On ImageNet-1k (256x256), it improves FID by 44.6% (from 13.68 to 7.58) and achieves a 5x speedup, enabling a 50-step generator (FID 7.58) to beat a 250-step baseline (FID 8.65). Furthermore, our methods generalize to Text-to-Image tasks and high-resolution generation, improving FID on MS-COCO by ~22%.

Problem

• Velocity Deficit
• Flow Matching
• Integration Lag
• velocity magnitude underestimation
• data manifold

Innovation

• Velocity Deficit
• Initial Energy Injection
• Flow Matching
• Scale Schedule Corrector
• Integration Lag