From Trajectories to Operators — A Unified Flow Map Perspective on Generative Modeling

In this post, we reframe continuous-time generative modeling from integrating trajectories to learning two-time operators (flow maps). This operator view unifies diffusion, flow matching, and consistency models, and suggests a practical diagnostic — semigroup-consistent jumps yield both step-robust generation and low compositional drift. We derive Eulerian/Lagrangian distillation objectives and use inpainting experiments to show why semigroup-consistent jumps can be both step-robust and composition-stable.