Moment Z-Flow: Learning Path Moments for One-Step Generation

One-step generation requires a representation of finite-time transport that can be predicted in a single network evaluation. We introduce Moment Z-Flow, which jointly learns normalized temporal moments of a velocity path and combines them through a normalized generating-function readout. Training combines local moment supervision with cocycle consistency across intervals. A complex exponential representation organizes the moments into a weighted cocycle under interval composition. The complex coordinate controls temporal amplitude and phase; the deployed readout is exactly implementable as a fixed real-linear combination and preserves constant velocity. At 100k updates on CIFAR-10, our MeanFlow-based implementation achieves FID-50K 3.54, compared with 3.84 for the MeanFlow reference. Activating the moment readout improves the same-checkpoint center FID from 4.33 to 3.54 and from 4.43 to 3.68 across two training seeds. Without cocycle supervision, the two-seed mean FID is 4.04, compared with 3.61 for the full method. The sCM + Z-Flow extension achieves a reported CIFAR-10 FID of 3.58 at 100k updates. On ImageNet-256, Z-Flow achieves a one-step FID of 2.44 at 400k training steps. These results support joint path-moment prediction and calibrated readout as a practical approach to fast generative transport.