Title: Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces

Abstract: This study introduces a machine-learning framework designed for addressing Bayesian inverse problems within the function-space context. Leveraging one-step generative transport, the approach develops an amortized neural operator that approximates the posterior distribution—conditioned on fresh observations—by pushing forward a Gaussian source. We demonstrate that white-noise sources are ill-suited for the function-space limit, necessitating the use of a prior-aligned Gaussian Random Field (GRF) as the source distribution. This selection is validated by the Lipschitz regularity of the resulting one-step conditional posterior transport and supported by numerical experiments involving linear inverse and PDE-based inverse problems. Distinct from methods distilled from MCMC, our model is trained exclusively using prior samples alongside simulated, noisy partial observations. Upon completion of training, the algorithm produces a $64\times64$ posterior sample in approximately $10^{-3}$ seconds. This efficiency bypasses the need for repeated forward-model evaluations inherent in MCMC, as well as the iterative network evaluations required by multistep generative samplers, all while accurately capturing essential posterior summaries.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC