Flow Matching for Fast Posterior Sampling in Bayesian Inverse Problems
Jan Blechschmidt, Oliver G. Ernst, Moritz Poguntke, Björn Sprungk
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.
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