# Faster Uncertainty Quantification for Inverse Problems with Conditional Normalizing Flows

In inverse problems, we often have access to data consisting of paired samples (x,y)∼ p_X,Y(x,y) where y are partial observations of a physical system, and x represents the unknowns of the problem. Under these circumstances, we can employ supervised training to learn a solution x and its uncertainty from the observations y. We refer to this problem as the "supervised" case. However, the data y∼ p_Y(y) collected at one point could be distributed differently than observations y'∼ p_Y'(y'), relevant for a current set of problems. In the context of Bayesian inference, we propose a two-step scheme, which makes use of normalizing flows and joint data to train a conditional generator q_θ(x|y) to approximate the target posterior density p_X|Y(x|y). Additionally, this preliminary phase provides a density function q_θ(x|y), which can be recast as a prior for the "unsupervised" problem, e.g. when only the observations y'∼ p_Y'(y'), a likelihood model y'|x, and a prior on x' are known. We then train another invertible generator with output density q'_ϕ(x|y') specifically for y', allowing us to sample from the posterior p_X|Y'(x|y'). We present some synthetic results that demonstrate considerable training speedup when reusing the pretrained network q_θ(x|y') as a warm start or preconditioning for approximating p_X|Y'(x|y'), instead of learning from scratch. This training modality can be interpreted as an instance of transfer learning. This result is particularly relevant for large-scale inverse problems that employ expensive numerical simulations.

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