Unfolding Generative Flows with Koopman Operators: Fast and Interpretable Sampling (2506.22304v1)

Abstract: Conditional Flow Matching (CFM) offers a simulation-free framework for training continuous-time generative models, bridging diffusion and flow-based approaches. However, sampling from CFM still relies on numerically solving non-linear ODEs which can be computationally expensive and difficult to interpret. Recent alternatives address sampling speed via trajectory straightening, mini-batch coupling or distillation. However, these methods typically do not shed light on the underlying \textit{structure} of the generative process. In this work, we propose to accelerate CFM and introduce an interpretable representation of its dynamics by integrating Koopman operator theory, which models non-linear flows as linear evolution in a learned space of observables. We introduce a decoder-free Koopman-CFM architecture that learns an embedding where the generative dynamics become linear, enabling closed-form, one-step sampling via matrix exponentiation. This results in significant speedups over traditional CFM as demonstrated on controlled 2D datasets and real-world benchmarks, MNIST, Fashion-MNIST (F-MNIST), and the Toronto Face Dataset (TFD). Unlike previous methods, our approach leads to a well-structured Koopman generator, whose spectral properties, eigenvalues, and eigenfunctions offer principled tools for analyzing generative behavior such as temporal scaling, mode stability, and decomposition in Koopman latent space. By combining sampling efficiency with analytical structure, Koopman-enhanced flow matching offers a potential step toward fast and interpretable generative modeling.