FM-DeepRV: Deep Learning for Bayesian Inference with Flow Matching

Bayesian inference with Gaussian-process (GP) priors is intractable at scale. Drawing a GP sample inside a Hamiltonian Monte Carlo (HMC) sampler requires a Cholesky factorisation of the kernel matrix, an $\mathcal{O}(N^3)$ operation that is repeated at every proposal whenever kernel hyperparameters are inferred jointly with the latent field. DeepRV sidesteps this by training a neural decoder offline to approximate the factorisation directly, replacing the Cholesky solve with a single forward pass. However, a decoder trained with mean squared error gives no way to trade inference cost against fidelity. We propose \textbf{FM-DeepRV}, which replaces this decoder with a \emph{flow-matching} network: a velocity field trained to transport Gaussian noise to GP samples and decoded by integrating the field for $K$ steps, giving a tunable quality/compute knob that DeepRV lacks. We ablate the resulting design space, including coupling (whitened pairing versus independent), path, objective (conditional flow matching versus MeanFlow), and backbone (gMLP versus a coordinate-token Diffusion Tranformer(DiT)), on synthetic Mat\'ern GPs, and validate FM-DeepRV (whitened coupling, MeanFlow) on three real tasks: climate reanalysis (ERA5), satellite scanline gap-filling (Landsat-7), and areal-Poisson change-of-support epidemiological downscaling (London LSOA). FM-DeepRV matches or beats the exact-GP HMC at one to two network evaluations per step ($K{=}1$–$2$).