# Faster Uncertainty Quantification for Inverse Problems with Conditional Normalizing Flows

arXiv: https://arxiv.org/abs/2007.07985 · HTML: https://mlsanity.com/paper/2007.07985/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jul 2020.

## Built on by (most cited first, 9 of 9)

- [Preconditioned training of normalizing flows for variational inference in inverse problems](https://mlsanity.com/paper/2101.03709/) (2021)
- [Globally injective and bijective neural operators](https://mlsanity.com/paper/2306.03982/) (2023)
- [Deep Bayesian inference for seismic imaging with tasks](https://mlsanity.com/paper/2110.04825/) (2021)
- [Learning by example: fast reliability-aware seismic imaging with normalizing flows](https://mlsanity.com/paper/2104.06255/) (2021)
- [Deep Variational Inverse Scattering](https://mlsanity.com/paper/2212.04309/) (2022)
- [Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview](https://mlsanity.com/paper/2402.12072/) (2024)
- [Deep Invertible Approximation of Topologically Rich Maps between Manifolds](https://mlsanity.com/paper/2210.00577/) (2022)
- [Variational Sparse Paired Autoencoders (vsPAIR) for Inverse Problems and Uncertainty Quantification](https://mlsanity.com/paper/2602.02948/) (2026)
- [Fine-tuned Normalizing Flows for ALICE Zero Degree Calorimeter Fast Simulation](https://mlsanity.com/paper/2608.12795/) (2026)

## Abstract (arXiv)

In inverse problems, we often have access to data consisting of paired samples (x,y)~ pX,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~ pY(y) collected at one point could be distributed differently than observations y'~ pY'(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 qθ(x|y) to approximate the target posterior density pX|Y(x|y). Additionally, this preliminary phase provides a density function qθ(x|y), which can be recast as a prior for the "unsupervised" problem, e.g.~when only the observations y'~ pY'(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 pX|Y'(x|y'). We present some synthetic results that demonstrate considerable training speedup when reusing the pretrained network qθ(x|y') as a warm start or preconditioning for approximating pX|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.