Co-Evolving Paths and Flows via Path-Flow Alignment

2026-10-06

AI Summary

This study addresses the limitations of flow matching, which is constrained by fixed interpolation paths and prone to path overfitting during joint training. To overcome these issues, this work proposes a path-flow alignment loss that jointly optimizes an endpoint-preserving path network and a flow network to enable their co-evolution. Furthermore, a stochastic path regularizer is designed to introduce an explicit entropy lower bound, thereby suppressing low-entropy bottlenecks and ensuring training stability. The proposed approach is compatible with model-guided training and requires no modifications to the inference architecture. Experiments on ImageNet demonstrate significant improvements in FID scores, establishing a more flexible and efficient paradigm for path optimization within flow matching frameworks.

Abstract

We study path-flow alignment as a unified training objective for flow matching. Instead of fixing the interpolation path and learning only the velocity field, we jointly train an endpoint-preserving path network and a flow network using the same alignment loss: the flow learns to match the path velocity, and the path learns to align its velocity to the current flow. Although every fixed learned path defines a valid flow-matching objective, the alignment loss alone is not a reliable criterion for path learning. We identify path overfitting, a failure mode in which the alignment loss decreases while sample quality worsens. We find that this failure is associated with low-entropy bottlenecks in the induced probability path, where the learned path routes samples through overly concentrated intermediate marginals. Motivated by this diagnosis, we introduce a stochastic path regularizer that hides part of the source information from the path network while preserving exact endpoints. The resulting regularization gives an explicit entropy floor for the stochastic training-path marginals and empirically suppresses the bottleneck in the learned sampler, making joint path-flow training effective. On ImageNet-256x256 with SiT backbones, our method consistently improves FID across model scales, extends to model-guidance training, and leaves the inference-time architecture and sampler unchanged. Code is available at https://github.com/lizeyu090312/traj_opt_paper

Problem

• flow matching
• path-flow alignment
• path overfitting
• low-entropy bottleneck
• generative modeling

Innovation

• Path-Flow Alignment
• Flow Matching
• Path Overfitting
• Stochastic Path Regularizer
• Entropy Floor