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

TL;DR

This paper introduces a novel neural-operator-based one-step generative method for Bayesian inverse problems in function spaces, addressing instability issues with white-noise references and enabling fast, accurate posterior sampling without MCMC.

Contribution

It proposes a prior-aligned anisotropic Gaussian reference and demonstrates a stable, efficient neural-operator approach for Bayesian inverse problems in PDEs, improving over traditional methods.

Findings

Achieves rapid posterior sampling in 10^{-3} seconds.

Addresses instability caused by white-noise references in function-space limits.

Matches key posterior summaries without repeated PDE solves.

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.