We develop an amortized framework for Bayesian inference in nonlinear inverse problems, including function-space problems where the unknown is a field. Classical methods such as MCMC typically solve a new sampling problem for every new observation, which can be expensive when inference must be repeated many times. Instead, we learn a single observation-dependent transport map that, after an offline training stage, can rapidly generate approximate posterior samples for new data. The method is likelihood-free: it requires only samples from the joint distribution of parameters and observations, rather than explicit evaluation of the likelihood or posterior density. The map is trained using an averaged energy-distance objective, which avoids the need for invertibility or Jacobian determinants and therefore allows flexible neural-network and neural-operator parameterizations. For infinite-dimensional Bayesian inverse problems with Gaussian priors, we design the transport as an identity perturbation in the Cameron–Martin space, preserving the absolute-continuity structure of the posterior with respect to the prior. Numerical experiments on finite-dimensional, Darcy-flow, and seismic wave-equation inverse problems show that the learned maps capture important posterior features, including multimodality and dominant modes, while enabling fast posterior sampling for new observations.
See the preprint here.