EnfoPath: Energy-Informed Analysis of Generative Trajectories in Flow Matching

TL;DR

This paper introduces kinetic path energy (KPE), a new diagnostic for flow-based generative models that links the kinetic effort of sampling trajectories to semantic quality and data density, providing insights into generation difficulty.

Contribution

It proposes KPE as a novel, physics-inspired metric to analyze and interpret the trajectories of flow-based generative models, revealing their relation to sample quality and data distribution.

Findings

Higher KPE predicts better semantic quality.

Higher KPE correlates with lower data density.

Trajectory analysis uncovers the data distribution frontier.

Abstract

Flow-based generative models synthesize data by integrating a learned velocity field from a reference distribution to the target data distribution. Prior work has focused on endpoint metrics (e.g., fidelity, likelihood, perceptual quality) while overlooking a deeper question: what do the sampling trajectories reveal? Motivated by classical mechanics, we introduce kinetic path energy (KPE), a simple yet powerful diagnostic that quantifies the total kinetic effort along each generation path of ODE-based samplers. Through comprehensive experiments on CIFAR-10 and ImageNet-256, we uncover two key phenomena: ({i}) higher KPE predicts stronger semantic quality, indicating that semantically richer samples require greater kinetic effort, and ({ii}) higher KPE inversely correlates with data density, with informative samples residing in sparse, low-density regions. Together, these findings reveal that semantically informative samples naturally reside on the sparse frontier of the data distribution, demanding greater generative effort. Our results suggest that trajectory-level analysis offers a physics-inspired and interpretable framework for understanding generation difficulty and sample characteristics.