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

• URL: http://arxiv.org/abs/2603.14798v1
• Date: Mon, 16 Mar 2026 03:52:28 GMT
• Title: Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces
• Abstract summary: We propose a machine-learning algorithm for inverse problems in the function-space regime based on one-step generative transport.<n>We show that white-noise references may be admissible at fixed discretization, but they become incompatible with the function-space limit.
• Score: 1.3709465727733763
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime based on one-step generative transport. Building on the Mean Flows, we learn a fully conditional amortized sampler with a neural-operator backbone that maps a reference Gaussian noise to approximate posterior samples. We show that while white-noise references may be admissible at fixed discretization, they become incompatible with the function-space limit, leading to instability in inference for Bayesian problems arising from PDEs. To address this issue, we adopt a prior-aligned anisotropic Gaussian reference distribution and establish the Lipschitz regularity of the resulting transport. Our method is not distilled from MCMC: training relies only on prior samples and simulated partial and noisy observations. Once trained, it generates a $64\times64$ posterior sample in $\sim 10^{-3}$s, avoiding the repeated PDE solves of MCMC while matching key posterior summaries.

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