Flow Annealing Posterior Sampling for Function-Space Regression and Inverse Problems

2026-06-21

AI Summary

This work addresses the challenges of posterior sampling and inaccurate uncertainty quantification in stochastic process regression and partial differential equation (PDE) inverse problems under sparse, noisy observations. We propose Flow Annealing Posterior Sampling (FAPS), a novel framework that leverages a pre-trained function-space flow-matching prior and performs likelihood-guided posterior inference without requiring explicit prior density evaluation. FAPS supports variable query discretizations and introduces a low-rank covariance-preconditioned Langevin correction to capture dominant cross-discretization functional correlations. Our method unifies treatment of both Gaussian and non-Gaussian regression tasks alongside diverse PDE inverse problems, achieving superior accuracy in uncertainty quantification compared to existing baselines while maintaining lower test-time sampling costs than diffusion-based approaches.

Abstract

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.

Problem

• function-space regression
• stochastic processes
• inverse problems
• posterior sampling
• uncertainty quantification

Innovation

• function-space sampling
• posterior inference
• flow matching
• inverse problems
• uncertainty quantification