This paper explores amortized Bayesian inference for nonlinear inverse problems using only samples from the joint distribution of parameters and observations. Classical methods such as Markov Chain Monte Carlo (MCMC) require solving a new inference problem for each observation, making it computationally prohibitive, especially in infinite dimensions. In contrast, amortized Bayesian inversion learns a reusable map that rapidly generates posterior samples for new observations.
We learn an observation-dependent transport map that pushes a reference measure to an approximate posterior. The training minimizes the average energy distance between the posterior and the learned pushforward. Averaging over observations allows generalization across observation instances and efficient amortized inference. Moreover, the formulation is likelihood-free, requiring only samples from the joint distribution and avoiding likelihood evaluation.
Using an energy-distance objective removes the need for the invertibility of the transport map and for computation of Jacobian determinants, enabling flexible parameterizations in high- and infinite-dimensional settings. When the posterior has a density with respect to a Gaussian prior measure, we construct transport maps as the identity plus a learnable map valued in the prior's Cameron–Martin space. This guarantees that the learned posterior remains absolutely continuous with respect to the prior. In infinite dimensions, the transport map is parameterized using neural operators, allowing use at different grid resolutions.
We demonstrate this approach on finite-dimensional problems and PDE-based porous-medium flow and seismic inverse problems. The learned transport captures multimodality and dominant posterior modes while enabling fast sampling.
Blogger's Review: This study presents an innovative method for Bayesian inference that enhances computational efficiency by learning transport maps, particularly when dealing with high-dimensional and infinite-dimensional problems. It opens new avenues for solving complex inverse problems, showcasing the immense potential of deep learning techniques in scientific computing.