FIRM: FLOW-BASED IMAGING VIA REGULARIZED MINIMIZATION

Flow matching methods for inverse problems typically incorporate measurements through network conditioning or guidance during sampling. Neither approach explicitly applies the forward operator within the learned conditional velocity field. We develop a principled measurement-conditional velocity parameterization that does. For a linear interpolation path, we express the optimal velocity through the posterior mean $\mathrm{E}[x_1|x_t,y]$ and show that this mean is the unique minimizer of a variational objective with an explicit data-consistency term. The velocity defined by this minimizer provably transports the source distribution to the measurement-conditioned posterior. This result leads to a forward operator-aware velocity field that alternates data-consistency steps with learned updates, is trained end-to-end, and requires no separate guidance during sampling. Across five imaging tasks, our method achieves leading reconstruction quality with up to $50\times$ fewer network evaluations than competitive flow-based methods. Varying the number of sampling steps also controls the distortion-perception trade-off without retraining.