Inverse problems are ubiquitous across scientific applications ranging from seismic imaging to astrophysics to nondestructive testing, yet classical solvers often struggle with nonconvexity and repeated evaluations of expensive physical models. Diffusion models can amortize this computation offline, but largely data-driven conditional models underutilize known physics and suffer from ill-conditioned transport maps when the target distribution is concentrated. Plug-and-play (PnP) methods explicitly leverage the physical model to guide a learned prior, improving data consistency but reintroducing costly iterative computations. We instead propose AdjointFlow, a framework built on stochastic interpolants that injects the known physics into the generative transport itself. We design a data-dependent, adjoint-based initialization to reduce transport distance to the target, modify the transport velocity to remain regular as the endpoint distributions concentrate, and parameterize the network to build the initialization directly into the learned velocity. Together, these enable accurate reconstruction at substantially lower cost than generative models starting from Gaussian noise. Experiments on inverse scattering, seismic imaging, and black hole imaging show that AdjointFlow consistently produces more accurate reconstructions than conditional diffusion models while being up to five orders of magnitude faster at inference than PnP methods. Optional inference-time refinement that queries the physical model can further improve data consistency, matching or exceeding the accuracy of PnP methods without retraining.

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