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

Researchers introduce Flow Annealing Posterior Sampling (FAPS), a new function-space framework that unifies stochastic-process regression with PDE inverse problems using pretrained flow-matching priors. The method enables probabilistic inference from sparse observations while maintaining computational efficiency and accurate uncertainty quantification, outperforming existing baselines.

FAPS represents a meaningful advance in computational statistics and scientific machine learning by bridging two traditionally separate domains: stochastic-process regression and inverse problem solving. The framework leverages modern flow-matching techniques—a recent advancement in generative modeling—to construct flexible prior distributions over function spaces, then applies likelihood-guided posterior inference to condition these priors on observational data.

The work emerges from growing recognition that inverse problems in scientific computing require principled uncertainty quantification, not just point estimates. Traditional approaches either treat regression and PDE inverse problems separately or lack rigorous probabilistic foundations. FAPS unifies these by operating directly in function space rather than finite-dimensional approximations, enabling seamless handling of variable discretizations and sparse observations without explicit prior density calculations.

For the scientific computing and machine learning communities, FAPS offers practical advantages: it reduces sampling costs compared to diffusion-based alternatives while maintaining competitive accuracy, supports noisy and incomplete data, and provides coherent posterior samples with reliable uncertainty estimates. The low-rank covariance preconditioner exploits dominant correlations across discretizations, making the approach scalable for realistic problem sizes.

The broader impact extends to any field requiring principled uncertainty quantification in inverse problems—geophysics, medical imaging, materials science, and climate modeling. As organizations increasingly integrate machine learning into scientific workflows, tools that combine flexibility with statistical rigor become critical. The framework's ability to work with pretrained priors also suggests potential for transfer learning and few-shot learning in scientific domains.

• →FAPS unifies stochastic-process regression and PDE inverse problems in a single probabilistic framework for the first time
• →The method reduces test-time sampling costs while maintaining or exceeding accuracy of existing diffusion-based approaches
• →Low-rank covariance preconditioning enables efficient posterior inference across variable function-space discretizations
• →Framework supports sparse, noisy observations without requiring explicit prior-density evaluation
• →Coherent posterior samples with accurate uncertainty quantification outperform existing functional regression baselines