首个统一函数空间回归与反问题的采样框架，高效准确地处理噪声数据和不确定性。
Flow Annealing Posterior Sampling for Function-Space Regression and Inverse Problems
- 基于预训练流匹配先验，通过似然引导实现后验采样
- 在多种随机过程与偏微分方程反问题中，后验样本一致性高、不确定性量化准
- 无需显式计算先验密度，降低推理成本，适合科学建模与复杂反问题
针对随机过程的严谨回归是长期存在的挑战，且与科学反问题密切相关。我们提出流退火后验采样（FAPS），据我们所知，这是首个统一函数空间回归与偏微分方程（PDE）反问题的后验采样框架。基于预训练的函数空间流匹配先验，FAPS 能从稀疏且含噪观测中实现似然引导的后验推断，支持可变查询离散化，并避免显式先验密度计算。其朗之万修正使用低秩协方差预条件器，利用跨离散化的主要函数空间相关性。在高斯与非高斯随机过程回归基准及多样化的 PDE 反问题上，FAPS 生成了具一致性的后验样本，不确定性量化准确，显著优于现有函数回归基线，且在测试时采样成本更低的情况下，其在含噪 PDE 反问题上的性能与基于扩散模型的采样器相当或更优。
原文摘要 · Abstract (English)
Principled regression for stochastic processes is a long-standing challenge with deep connections to scientific inverse problems. We introduce Flow Annealing Posterior Sampling (FAPS), to our knowledge the first function-space posterior sampling framework that unifies stochastic-process regression and PDE inverse problems. Built on pretrained function-space flow-matching priors, FAPS enables likelihood-guided posterior inference from sparse and noisy observations, supports variable query discretizations, and avoids explicit prior-density evaluation. Its Langevin correction uses a low-rank covariance preconditioner to exploit dominant function-space correlations across discretizations. Across Gaussian and non-Gaussian stochastic-process regression benchmarks and diverse PDE inverse problems, FAPS produces coherent posterior samples with accurate uncertainty quantification, significantly outperforming existing functional regression baselines and achieving competitive or better PDE noisy inverse performance than diffusion-based posterior samplers while reducing test-time sampling cost.
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