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.

Machine Learning (stat.ML); Artificial Intelligence (cs.AI); Machine Learning (cs.LG)

arXiv:2606.22346 [stat.ML]

(or arXiv:2606.22346v2 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2606.22346 Focus to learn more arXiv-issued DOI via DataCite

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