# Bayesian multiscale deep generative model for the solution of high-dimensional inverse problems

arXiv: https://arxiv.org/abs/2102.03169 · HTML: https://mlsanity.com/paper/2102.03169/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 75 references that are in the map.
- First posted on arXiv 4 Feb 2021.

## Builds on (most cited first, 30 of 31)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Residual Dense Network for Image Super-Resolution](https://mlsanity.com/paper/1802.08797/) (2018)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [Robustly Disentangled Causal Mechanisms: Validating Deep Representations for Interventional Robustness](https://mlsanity.com/paper/1811.00007/) (2018)
- [A deep-learning-based surrogate model for data assimilation in dynamic subsurface flow problems](https://mlsanity.com/paper/1908.05823/) (2019)
- [Deep convolutional encoder-decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media](https://mlsanity.com/paper/1807.00882/) (2018)
- [Stochastic seismic waveform inversion using generative adversarial networks as a geological prior](https://mlsanity.com/paper/1806.03720/) (2018)
- [Solving inverse problems using conditional invertible neural networks](https://mlsanity.com/paper/2007.15849/) (2020)
- [Structured Uncertainty Prediction Networks](https://mlsanity.com/paper/1802.07079/) (2018)
- [Training-image based geostatistical inversion using a spatial generative adversarial neural network](https://mlsanity.com/paper/1708.04975/) (2017)
- [Deep autoregressive neural networks for high-dimensional inverse problems in groundwater contaminant source identification](https://mlsanity.com/paper/1812.09444/) (2018)
- [Inversion using a new low-dimensional representation of complex binary geological media based on a deep neural network](https://mlsanity.com/paper/1710.09196/) (2017)
- [Towards a Robust Parameterization for Conditioning Facies Models Using Deep Variational Autoencoders and Ensemble Smoother](https://mlsanity.com/paper/1812.06900/) (2018)
- [Multi-Resolution A*](https://mlsanity.com/paper/2004.06684/) (2020)
- [Structured Bayesian Gaussian process latent variable model: applications to data-driven dimensionality reduction and high-dimensional inversion](https://mlsanity.com/paper/1807.04302/) (2018)
- [A Deep-Learning-Based Geological Parameterization for History Matching Complex Models](https://mlsanity.com/paper/1807.02716/) (2018)
- [Integration of adversarial autoencoders with residual dense convolutional networks for estimation of non-Gaussian hydraulic conductivities](https://mlsanity.com/paper/1906.11828/) (2019)
- [Predictive Collective Variable Discovery with Deep Bayesian Models](https://mlsanity.com/paper/1809.06913/) (2018)
- [Beyond black-boxes in Bayesian inverse problems and model validation: applications in solid mechanics of elastography](https://mlsanity.com/paper/1803.00930/) (2018)
- [Multilevel Dimension-Independent Likelihood-Informed MCMC for Large-Scale Inverse Problems](https://mlsanity.com/paper/1910.12431/) (2019)

## Built on by (most cited first, 12 of 12)

- [A domain-decomposed VAE method for Bayesian inverse problems](https://mlsanity.com/paper/2301.05708/) (2023)
- [Deep adaptive sampling for surrogate modeling without labeled data](https://mlsanity.com/paper/2402.11283/) (2024)
- [Bayesian neural networks for predicting uncertainty in full-field material response](https://mlsanity.com/paper/2406.14838/) (2024)
- [VI-DGP: A variational inference method with deep generative prior for solving high-dimensional inverse problems](https://mlsanity.com/paper/2302.11173/) (2023)
- [Deep Learning for Simultaneous Inference of Hydraulic and Transport Properties](https://mlsanity.com/paper/2110.12367/) (2021)
- [The Deep Latent Space Particle Filter for Real-Time Data Assimilation with Uncertainty Quantification](https://mlsanity.com/paper/2406.02204/) (2024)
- [Noise-Adaptive Diffusion Sampling for Inverse Problems Without Task-Specific Tuning](https://mlsanity.com/paper/2604.16919/) (2026)
- [Dimension-reduced KRnet maps for high-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2303.00573/) (2023)
- [Bayesian Inverse Problems Meet Flow Matching: Efficient and Flexible Inference via Transformers](https://mlsanity.com/paper/2503.01375/) (2025)
- [Variational Autoencoder for Generating Broader-Spectrum prior Proposals in Markov chain Monte Carlo Methods](https://mlsanity.com/paper/2507.00020/) (2025)
- [GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media](https://mlsanity.com/paper/2602.14642/) (2026)
- [Deep Adaptive Dimension Reduction for Bayesian Inference in Inverse Problems](https://mlsanity.com/paper/2605.29373/) (2026)

## Abstract (arXiv)

Estimation of spatially-varying parameters for computationally expensive forward models governed by partial differential equations is addressed. A novel multiscale Bayesian inference approach is introduced based on deep probabilistic generative models. Such generative models provide a flexible representation by inferring on each scale a low-dimensional latent encoding while allowing hierarchical parameter generation from coarse- to fine-scales. Combining the multiscale generative model with Markov Chain Monte Carlo (MCMC), inference across scales is achieved enabling us to efficiently obtain posterior parameter samples at various scales. The estimation of coarse-scale parameters using a low-dimensional latent embedding captures global and notable parameter features using an inexpensive but inaccurate solver. MCMC sampling of the fine-scale parameters is enabled by utilizing the posterior information in the immediate coarser-scale. In this way, the global features are identified in the coarse-scale with inference of low-dimensional variables and inexpensive forward computation, and the local features are refined and corrected in the fine-scale. The developed method is demonstrated with two types of permeability estimation for flow in heterogeneous media. One is a Gaussian random field (GRF) with uncertain length scales, and the other is channelized permeability with the two regions defined by different GRFs. The obtained results indicate that the method allows high-dimensional parameter estimation while exhibiting stability, efficiency and accuracy.