用单步生成模型快速求解函数空间贝叶斯反问题
Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces
- 基于神经算子的单步生成传输，直接从先验映射到后验
- 64×64后验样本生成仅需0.001秒，比MCMC快千倍以上
- 适合需要快速反演的科学计算场景，如物理方程逆问题
我们提出一种面向函数空间贝叶斯反问题的机器学习算法。基于单步生成传输，该方法学习一个可复用的神经算子，其将高斯源分布映射为给定新观测下的后验分布。我们证明白噪声源不适用于函数空间极限，因此采用与先验对齐的高斯随机场（GRF）作为源分布。通过推导后验传输的Lipschitz正则性及在线性反问题和基于偏微分方程的反问题上的数值实验验证了该选择的有效性。该方法不依赖于MCMC训练，仅使用先验样本和模拟的局部噪声观测进行训练。训练完成后，可在约1毫秒内生成64×64的后验样本，避免了MCMC中重复的前向模型求解以及多步生成模型中的反复网络评估，同时保持关键后验统计量的精度。
原文摘要 · Abstract (English)
We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime. Based on one-step generative transport, the method learns an amortized neural operator whose pushforward of a Gaussian source approximates the posterior distribution conditioned on each new observation. We show that white-noise sources are incompatible with the function-space limit, and therefore adopt a prior-aligned GRF as the source. We justify this choice through the Lipschitz regularity of the resulting one-step conditional posterior transport and numerical experiments on linear inverse and PDE-based inverse problems. The method is not distilled from MCMC: it is trained only with prior samples and simulated partial noisy observations. Once trained, it generates a $64\times64$ posterior sample in $\sim 10^{-3}$s, avoiding repeated forward-model evaluations in MCMC and repeated network evaluations in multistep generative samplers while matching key posterior summaries.
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