How to build a consistency model: Learning flow maps via self-distillation

Building on the framework proposed in Boffi et al. (2024), we present a systematic approach for learning flow maps associated with flow and diffusion models. Flow map-based models, commonly known as consistency models, encompass recent efforts to improve the efficiency of generative models based on solutions to differential equations. By exploiting a relationship between the velocity field underlying a continuous-time flow and the instantaneous rate of change of the flow map, we show how to convert existing distillation schemes into direct training algorithms via self-distillation, eliminating the need for pre-trained models. We empirically evaluate several instantiations of our framework, finding that high-dimensional tasks like image synthesis benefit from objective functions that avoid temporal and spatial derivatives of the flow map, while lower-dimensional tasks can benefit from objectives incorporating higher-order derivatives to capture sharp features.