Co-Evolving Paths and Flows via Path-Flow Alignment

Zeyu Michael Li, William Xingxu Chen, Xiang Cheng

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…