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

The article introduces Flow Annealing Posterior Sampling (FAPS), a novel framework for function-space posterior sampling that integrates stochastic-process regression with PDE inverse problems. FAPS utilizes pretrained function-space flow-matching priors and incorporates a Langevin correction with a low-rank covariance preconditioner, demonstrating improved posterior sample coherence and uncertainty quantification across various benchmarks. This approach significantly outperforms existing functional regression methods and offers competitive performance against diffusion-based samplers while reducing sampling costs, making it a valuable tool for practitioners addressing inverse problems in AI and scientific computing.