Flower: A Flow-Matching Solver for Inverse Problems

Abstract

We introduce Flower, a solver for linear inverse problems. It leverages a pre-trained flow model to produce reconstructions that are consistent with the observed measurements. Flower operates through an iterative procedure over three steps: (i) a flow-consistent destination estimation, where the velocity network predicts a denoised target; (ii) a refinement step that projects the estimated destination onto a feasible set defined by the forward operator; and (iii) a time-progression step that re-projects the refined destination along the flow trajectory. We provide a theoretical analysis that demonstrates how Flower approximates Bayesian posterior sampling, thereby unifying perspectives from plug-and-play methods and generative inverse solvers. On the practical side, Flower achieves state-of-the-art reconstruction quality while using nearly identical hyperparameters across various linear inverse problems. Our code is available at https://github.com/mehrsapo/Flower.