# Solution of Physics-based Bayesian Inverse Problems with Deep Generative Priors

arXiv: https://arxiv.org/abs/2107.02926 · HTML: https://mlsanity.com/paper/2107.02926/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 78 references that are in the map.
- First posted on arXiv 6 Jul 2021.

## Builds on (most cited first, 23 of 23)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [TensorFlow Distributions](https://mlsanity.com/paper/1711.10604/) (2017)
- [NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport](https://mlsanity.com/paper/1903.03704/) (2019)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [Hamiltonian Monte Carlo for Hierarchical Models](https://mlsanity.com/paper/1312.0906/) (2013)
- [Generalizing Hamiltonian Monte Carlo with Neural Networks](https://mlsanity.com/paper/1711.09268/) (2017)
- [Solving Bayesian Inverse Problems via Variational Autoencoders](https://mlsanity.com/paper/1912.04212/) (2019)
- [Nonparametric Density Estimation & Convergence Rates for GANs under Besov IPM Losses](https://mlsanity.com/paper/1902.03511/) (2019)
- [Constraint-Aware Neural Networks for Riemann Problems](https://mlsanity.com/paper/1904.12794/) (2019)
- [Improved generator objectives for GANs](https://mlsanity.com/paper/1612.02780/) (2016)
- [Parameterizing uncertainty by deep invertible networks, an application to reservoir characterization](https://mlsanity.com/paper/2004.07871/) (2020)
- [Enforcing Deterministic Constraints on Generative Adversarial Networks for Emulating Physical Systems](https://mlsanity.com/paper/1911.06671/) (2019)
- [GAN-based Priors for Quantifying Uncertainty](https://mlsanity.com/paper/2003.12597/) (2020)

## Built on by (most cited first, 30 of 30)

- [The efficacy and generalizability of conditional GANs for posterior inference in physics-based inverse problems](https://mlsanity.com/paper/2202.07773/) (2022)
- [Solving the Inverse Problem of Electrocardiography for Cardiac Digital Twins: A Survey](https://mlsanity.com/paper/2406.11445/) (2024)
- [Bayesian Posterior Perturbation Analysis with Integral Probability Metrics](https://mlsanity.com/paper/2303.01512/) (2023)
- [Solution of physics-based inverse problems using conditional generative adversarial networks with full gradient penalty](https://mlsanity.com/paper/2306.04895/) (2023)
- [Unifying and extending Diffusion Models through PDEs for solving Inverse Problems](https://mlsanity.com/paper/2504.07437/) (2025)
- [Conditional score-based diffusion models for solving inverse problems in mechanics](https://mlsanity.com/paper/2406.13154/) (2024)
- [Deep Learning and Computational Physics (Lecture Notes)](https://mlsanity.com/paper/2301.00942/) (2023)
- [VI-DGP: A variational inference method with deep generative prior for solving high-dimensional inverse problems](https://mlsanity.com/paper/2302.11173/) (2023)
- [Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities](https://mlsanity.com/paper/2406.13036/) (2024)
- [A Primer on Variational Inference for Physics-Informed Deep Generative Modelling](https://mlsanity.com/paper/2409.06560/) (2024)
- [Error Analysis of Bayesian Inverse Problems with Generative Priors](https://mlsanity.com/paper/2601.17374/) (2026)
- [Empowering Bayesian Neural Networks with Functional Priors through Anchored Ensembling for Mechanics Surrogate Modeling Applications](https://mlsanity.com/paper/2409.05234/) (2024)
- [Guided Unconditional and Conditional Generative Models for Super-Resolution and Inference of Quasi-Geostrophic Turbulence](https://mlsanity.com/paper/2507.00719/) (2025)
- [Conditional flow matching for physics-constrained inverse problems with finite training data](https://mlsanity.com/paper/2603.14135/) (2026)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [Dimension-reduced KRnet maps for high-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2303.00573/) (2023)
- [Learning WENO for entropy stable schemes to solve conservation laws](https://mlsanity.com/paper/2403.14848/) (2024)
- [Integrating Generative and Physics-Based Models for Ptychographic Imaging with Uncertainty Quantification](https://mlsanity.com/paper/2412.10882/) (2024)
- [Sequential Bayesian Design for Efficient Surrogate Construction in the Inversion of Darcy Flows](https://mlsanity.com/paper/2507.17713/) (2025)
- [Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later](https://mlsanity.com/paper/2509.19929/) (2025)
- [Learning end-to-end inversion of circular Radon transforms in the partial radial setup](https://mlsanity.com/paper/2308.14144/) (2023)
- [Bayesian imaging inverse problem with SA-Roundtrip prior via HMC-pCN sampler](https://mlsanity.com/paper/2310.17817/) (2023)
- [Multi-fidelity Hamiltonian Monte Carlo](https://mlsanity.com/paper/2405.05033/) (2024)
- [Fully data-driven inverse hyperelasticity with hyper-network neural ODE fields](https://mlsanity.com/paper/2506.08146/) (2025)
- [Learning from Imperfect Data: Robust Inference of Dynamic Systems using Simulation-based Generative Model](https://mlsanity.com/paper/2507.10884/) (2025)
- [Multi-Dataset Inverse Problem Solving with Distributed Generative AI](https://mlsanity.com/paper/2608.26283/) (2026)
- [Generative Priors Conditioned on Natural Language for Bayesian Inversion in PDEs](https://mlsanity.com/paper/2609.32941/) (2026)
- [High-Dimensional Simulation-Based Inference in Latent Spaces](https://mlsanity.com/paper/2609.37381/) (2026)
- [Posterior sampling by source-space MCMC via prior-based few-step transport maps](https://mlsanity.com/paper/2610.01034/) (2026)
- [Latent Annealed Posterior Sampling with a Variance-Preserving Bridge for Sparsely Observed Random Fields](https://mlsanity.com/paper/2610.20659/) (2026)

## Abstract (arXiv)

Inverse problems are ubiquitous in nature, arising in almost all areas of science and engineering ranging from geophysics and climate science to astrophysics and biomechanics. One of the central challenges in solving inverse problems is tackling their ill-posed nature. Bayesian inference provides a principled approach for overcoming this by formulating the inverse problem into a statistical framework. However, it is challenging to apply when inferring fields that have discrete representations of large dimensions (the so-called "curse of dimensionality") and/or when prior information is available only in the form of previously acquired solutions. In this work, we present a novel method for efficient and accurate Bayesian inversion using deep generative models. Specifically, we demonstrate how using the approximate distribution learned by a Generative Adversarial Network (GAN) as a prior in a Bayesian update and reformulating the resulting inference problem in the low-dimensional latent space of the GAN, enables the efficient solution of large-scale Bayesian inverse problems. Our statistical framework preserves the underlying physics and is demonstrated to yield accurate results with reliable uncertainty estimates, even in the absence of information about underlying noise model, which is a significant challenge with many existing methods. We demonstrate the effectiveness of proposed method on a variety of inverse problems which include both synthetic as well as experimentally observed data.