Self-Consistent Flow: Unifying Velocity and Endpoint Prediction for Rectified Flow Models

Rectified flow models can be parameterized to predict either the instantaneous velocity (V-Flow) or the data endpoint (X-Flow). While these targets are mathematically equivalent in an ideal setting, they exhibit different empirical behaviors. This paper investigates why these differences occur and proposes a unified framework to combine their respective advantages.

The authors analyze the learning errors associated with each parameterization. They find that endpoint prediction (X-Flow) provides a clearer training signal, making it more stable during the initial stages of optimization. Conversely, velocity prediction (V-Flow) is more stable during the final stages of sampling as the trajectory approaches the data manifold. To leverage both, the authors introduce SC-Flow, which uses a single neural network backbone to predict both targets. A binary flag toggles the output, and a lightweight consistency loss enforces that the velocity and endpoint predictions remain analytically coherent throughout training.

The study demonstrates that X-Flow is easier to train because it effectively operates within a lower-dimensional latent space, reducing the impact of high-dimensional noise. However, deriving velocity from an endpoint predictor leads to amplified errors as the sampling time approaches the data manifold. By jointly training the model with a consistency objective, SC-Flow achieves superior optimization stability and produces straighter, higher-quality generation paths. Experiments on ImageNet show that SC-Flow significantly improves FID scores compared to standard rectified flow baselines without requiring major architectural changes or increasing inference costs.