# Parameterizing uncertainty by deep invertible networks, an application to reservoir characterization

arXiv: https://arxiv.org/abs/2004.07871 · HTML: https://mlsanity.com/paper/2004.07871/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Apr 2020.

## Built on by (most cited first, 11 of 11)

- [Solution of Physics-based Bayesian Inverse Problems with Deep Generative Priors](https://mlsanity.com/paper/2107.02926/) (2021)
- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [Preconditioned training of normalizing flows for variational inference in inverse problems](https://mlsanity.com/paper/2101.03709/) (2021)
- [The efficacy and generalizability of conditional GANs for posterior inference in physics-based inverse problems](https://mlsanity.com/paper/2202.07773/) (2022)
- [Invertible Neural Networks versus MCMC for Posterior Reconstruction in Grazing Incidence X-Ray Fluorescence](https://mlsanity.com/paper/2102.03189/) (2021)
- [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)
- [WISE: full-Waveform variational Inference via Subsurface Extensions](https://mlsanity.com/paper/2401.06230/) (2023)
- [Learned multiphysics inversion with differentiable programming and machine learning](https://mlsanity.com/paper/2304.05592/) (2023)
- [InvertibleNetworks.jl: A Julia package for scalable normalizing flows](https://mlsanity.com/paper/2312.13480/) (2023)
- [Survey of Deep Learning and Physics-Based Approaches in Computational Wave Imaging](https://mlsanity.com/paper/2410.08329/) (2024)

## Abstract (arXiv)

Uncertainty quantification for full-waveform inversion provides a probabilistic characterization of the ill-conditioning of the problem, comprising the sensitivity of the solution with respect to the starting model and data noise. This analysis allows to assess the confidence in the candidate solution and how it is reflected in the tasks that are typically performed after imaging (e.g., stratigraphic segmentation following reservoir characterization). Classically, uncertainty comes in the form of a probability distribution formulated from Bayesian principles, from which we seek to obtain samples. A popular solution involves Monte Carlo sampling. Here, we propose instead an approach characterized by training a deep network that "pushes forward" Gaussian random inputs into the model space (representing, for example, density or velocity) as if they were sampled from the actual posterior distribution. Such network is designed to solve a variational optimization problem based on the Kullback-Leibler divergence between the posterior and the network output distributions. This work is fundamentally rooted in recent developments for invertible networks. Special invertible architectures, besides being computational advantageous with respect to traditional networks, do also enable analytic computation of the output density function. Therefore, after training, these networks can be readily used as a new prior for a related inversion problem. This stands in stark contrast with Monte-Carlo methods, which only produce samples. We validate these ideas with an application to angle-versus-ray parameter analysis for reservoir characterization.