Flow Matching for Fast Posterior Sampling in Bayesian Inverse Problems
Abstract
Sampling from the posterior is the central task of computational Bayesian inverse problems. The standard workhorse in Bayesian inference---Markov chain Monte Carlo (MCMC)---is sequential, yields correlated samples, and must be rerun for each new observation. Conditional flow matching offers an amortized alternative: a transport map, trained once on joint samples of parameter and data, that yields independent approximate posterior samples for any observation at negligible online cost, without new likelihood evaluations. We give a careful, MCMC-literate assessment of flow matching for PDE-based inverse problems with function-valued parameters. Exploiting the flow's tractable density, we derive computable accuracy estimates for the learned posterior in total variation distance and Kullback--Leibler divergence and, moreover, propose a hybrid sampler that is made asymptotically exact by Metropolization. We validate the accuracy estimates and demonstrate amortization in several numerical examples, including electrical impedance tomography and a likelihood-free Lotka--Volterra model. Flow matching reproduces posterior means and credible bands closely at a fraction of the forward solves required by MCMC, but its accuracy degrades for concentrated posteriors and for observations that are atypical under the prior.
Related
- Accelerating posterior sampling in Bayesian inverse problems with diffusion models: an online-offline perspective
- Posterior Convergence without Force Convergence: Resolution-Stable Sampling for Rough Bayesian Inverse Problems
- Flow Annealing Posterior Sampling for Function-Space Regression and Inverse Problems
- Kinetic Langevin Meets Split Gibbs: Accelerated Posterior Sampling for Imaging Inverse Problems with Diffusion Priors
- On Prior-to-Posterior Stability in the Wasserstein Metric for Bayesian Inverse Problems
BibTeX
@misc{blechschmidt2026flow,
title = {{Flow Matching for Fast Posterior Sampling in Bayesian Inverse Problems}},
author = {Jan Blechschmidt and Oliver G. Ernst and Moritz Poguntke and Björn Sprungk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02377},
eprint = {2610.02377},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02377},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Flow Matching for Fast Posterior Sampling in Bayesian Inverse Problems
AU - Jan Blechschmidt
AU - Oliver G. Ernst
AU - Moritz Poguntke
AU - Björn Sprungk
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.02377
UR - https://arxiv.org/abs/2610.02377
AB - Sampling from the posterior is the central task of computational Bayesian inverse problems. The standard workhorse in Bayesian inference---Markov chain Monte Carlo (MCMC)---is sequential, yields correlated samples, and must be rerun for each new observation. Conditional flow matching offers an amortized alternative: a transport map, trained once on joint samples of parameter and data, that yields independent approximate posterior samples for any observation at negligible online cost, without new likelihood evaluations. We give a careful, MCMC-literate assessment of flow matching for PDE-based inverse problems with function-valued parameters. Exploiting the flow's tractable density, we derive computable accuracy estimates for the learned posterior in total variation distance and Kullback--Leibler divergence and, moreover, propose a hybrid sampler that is made asymptotically exact by Metropolization. We validate the accuracy estimates and demonstrate amortization in several numerical examples, including electrical impedance tomography and a likelihood-free Lotka--Volterra model. Flow matching reproduces posterior means and credible bands closely at a fraction of the forward solves required by MCMC, but its accuracy degrades for concentrated posteriors and for observations that are atypical under the prior.
ER -