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 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 of the underlying approximate posterior in total-variation distance and Kullback-Leibler divergence and, moreover, propose a hybrid sampler that is asymptotically exact by Metropolization.
We validate the accuracy estimates and demonstrate the amortization in several numerical examples, including electrical impedance tomography and a likelihood-free Lotka-Volterra model.