Flow Annealing Posterior Sampling for Function-Space Regression and Inverse Problems
Abstract
Principled regression for stochastic processes is a long-standing challenge with deep connections to scientific inverse problems. We introduce Flow Annealing Posterior Sampling (FLAPS), 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, FLAPS 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, FLAPS produces coherent posterior samples with well-calibrated uncertainty quantification, significantly outperforming existing functional regression baselines and achieving competitive or better noisy PDE inverse performance than diffusion-based posterior samplers while reducing test-time sampling cost.
Related
- Accelerating posterior sampling in Bayesian inverse problems with diffusion models: an online-offline perspective
- Flow Matching for Fast Posterior Sampling in Bayesian Inverse Problems
- Kinetic Langevin Meets Split Gibbs: Accelerated Posterior Sampling for Imaging Inverse Problems with Diffusion Priors
- Posterior Convergence without Force Convergence: Resolution-Stable Sampling for Rough Bayesian Inverse Problems
- Transport along Local Annealing Paths: Wasserstein Geometry and Multimodal Sampling
BibTeX
@misc{shi2026flow,
title = {{Flow Annealing Posterior Sampling for Function-Space Regression and Inverse Problems}},
author = {Yaozhong Shi and Zachary E. Ross and Yisong Yue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.22346},
eprint = {2606.22346},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.22346},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Flow Annealing Posterior Sampling for Function-Space Regression and Inverse Problems
AU - Yaozhong Shi
AU - Zachary E. Ross
AU - Yisong Yue
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2606.22346
UR - https://arxiv.org/abs/2606.22346
AB - Principled regression for stochastic processes is a long-standing challenge with deep connections to scientific inverse problems. We introduce Flow Annealing Posterior Sampling (FLAPS), 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, FLAPS 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, FLAPS produces coherent posterior samples with well-calibrated uncertainty quantification, significantly outperforming existing functional regression baselines and achieving competitive or better noisy PDE inverse performance than diffusion-based posterior samplers while reducing test-time sampling cost.
ER -