Preconditioned training of normalizing flows for variational inference in inverse problems

TL;DR

This paper introduces a preconditioning approach using conditional normalizing flows to efficiently sample from complex posterior distributions in inverse problems, significantly reducing training costs.

Contribution

It proposes a novel preconditioning scheme that leverages low-fidelity normalizing flows to accelerate high-fidelity training in inverse problems.

Findings

Significant speed-ups in training times demonstrated in numerical experiments.

Effective sampling from complex, high-dimensional posteriors achieved.

Pretraining on low-fidelity models improves high-fidelity inference efficiency.

Abstract

Obtaining samples from the posterior distribution of inverse problems with expensive forward operators is challenging especially when the unknowns involve the strongly heterogeneous Earth. To meet these challenges, we propose a preconditioning scheme involving a conditional normalizing flow (NF) capable of sampling from a low-fidelity posterior distribution directly. This conditional NF is used to speed up the training of the high-fidelity objective involving minimization of the Kullback-Leibler divergence between the predicted and the desired high-fidelity posterior density for indirect measurements at hand. To minimize costs associated with the forward operator, we initialize the high-fidelity NF with the weights of the pretrained low-fidelity NF, which is trained beforehand on available model and data pairs. Our numerical experiments, including a 2D toy and a seismic compressed sensing example, demonstrate that thanks to the preconditioning considerable speed-ups are achievable compared to training NFs from scratch.