Unfolding Generative Flows with Koopman Operators: Fast and Interpretable Sampling

TL;DR

This paper introduces a novel approach to generative flow modeling by linearizing flow dynamics using Koopman operators, enabling fast, interpretable, and parallelizable sampling with competitive quality.

Contribution

It proposes lifting Conditional Flow Matching into Koopman space, allowing one-step sampling via matrix exponential and providing spectral interpretability of the generative process.

Findings

Achieves significant speedup in sampling compared to traditional CNFs.

Provides spectral analysis tools for understanding flow dynamics.

Maintains high sample quality comparable to existing methods.

Abstract

Continuous Normalizing Flows (CNFs) enable elegant generative modeling but remain bottlenecked by slow sampling: producing a single sample requires solving a nonlinear ODE with hundreds of function evaluations. Recent approaches such as Rectified Flow and OT-CFM accelerate sampling by straightening trajectories, yet the learned dynamics remain nonlinear black boxes, limiting both efficiency and interpretability. We propose a fundamentally different perspective: globally linearizing flow dynamics via Koopman theory. By lifting Conditional Flow Matching (CFM) into a higher-dimensional Koopman space, we represent its evolution with a single linear operator. This yields two key benefits. First, sampling becomes one-step and parallelizable, computed in closed form via the matrix exponential. Second, the Koopman operator provides a spectral blueprint of generation, enabling novel interpretability through its eigenvalues and modes. We derive a practical, simulation-free training objective that enforces infinitesimal consistency with the teacher's dynamics and show that this alignment preserves fidelity along the full generative path, distinguishing our method from boundary-only distillation. Empirically, our approach achieves competitive sample quality with dramatic speedups, while uniquely enabling spectral analysis of generative flows.