Flow Annealing Posterior Sampling for Function-Space Regression and Inverse Problems
Provides a principled, efficient posterior sampling method for function-space regression and inverse problems, benefiting scientific computing and machine learning practitioners.
FAPS is the first function-space posterior sampling framework unifying stochastic-process regression and PDE inverse problems, using pretrained flow-matching priors and a low-rank covariance preconditioner. It outperforms functional regression baselines and matches diffusion-based samplers in PDE inverse problems while reducing sampling cost.
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.