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

Ziyun Li, Ben Dai, Huancheng Hu, Henrik Boström, Soon Hoe Lim

Keywords: Flow matching

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.
Code: https://github.com/ziyunli-2023/EnfoPath-FM

Submission Number: 18