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.

翻译：我们提出了一种基于单步生成传输的机器学习算法，用于函数空间框架下的贝叶斯反问题。该方法建立在均值流理论基础上，通过以神经算子为骨干网络，学习一个完全条件化的摊销采样器，将参考高斯噪声映射为近似后验样本。我们证明，虽然白噪声参考在固定离散化下可能适用，但其与函数空间极限不兼容，导致基于偏微分方程产生的贝叶斯问题在推断中出现不稳定现象。为解决此问题，我们采用先验对齐的各向异性高斯参考分布，并建立了所得传输映射的Lipschitz正则性。本方法并非从MCMC蒸馏得到：训练过程仅依赖于先验样本及模拟的部分含噪观测数据。训练完成后，该方法可在$\sim 10^{-3}$秒内生成$64\times64$的后验样本，在匹配关键后验统计量的同时，避免了MCMC方法中重复的偏微分方程求解过程。