# Enhanced uncertainty quantification variational autoencoders for the solution of Bayesian inverse problems

arXiv: https://arxiv.org/abs/2502.13105 · HTML: https://mlsanity.com/paper/2502.13105/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 44 references that are in the map.
- First posted on arXiv 18 Feb 2025.

## Builds on (most cited first, 16 of 16)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [Evolutional Deep Neural Network](https://mlsanity.com/paper/2103.09959/) (2021)
- [Multimodal Generative Learning Utilizing Jensen-Shannon-Divergence](https://mlsanity.com/paper/2006.08242/) (2020)
- [Bayesian parameter estimation using conditional variational autoencoders for gravitational-wave astronomy](https://mlsanity.com/paper/1909.06296/) (2019)
- [A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion](https://mlsanity.com/paper/1308.1313/) (2013)
- [Solving Bayesian Inverse Problems via Variational Autoencoders](https://mlsanity.com/paper/1912.04212/) (2019)
- [Learning Bayesian posteriors with neural networks for gravitational-wave inference](https://mlsanity.com/paper/1909.05966/) (2019)
- [Constraining Variational Inference with Geometric Jensen-Shannon Divergence](https://mlsanity.com/paper/2006.10599/) (2020)
- [Physics-informed Neural Network Estimation of Material Properties in Soft Tissue Nonlinear Biomechanical Models](https://mlsanity.com/paper/2312.09787/) (2023)
- [TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems](https://mlsanity.com/paper/2105.12033/) (2021)
- [Mitigating the Influence of the Boundary on PDE-based Covariance Operators](https://mlsanity.com/paper/1610.05280/) (2016)
- [TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problems](https://mlsanity.com/paper/2412.07010/) (2024)

## Abstract (arXiv)

Among other uses, neural networks are a powerful tool for solving deterministic and Bayesian inverse problems in real-time, where variational autoencoders, a specialized type of neural network, enable the Bayesian estimation of model parameters and their distribution from observational data allowing real-time inverse uncertainty quantification. In this work, we build upon existing research [Goh, H. et al., Proceedings of Machine Learning Research, 2022] by proposing a novel loss function to train variational autoencoders for Bayesian inverse problems. When the forward map is affine, we provide a theoretical proof of the convergence of the latent states of variational autoencoders to the posterior distribution of the model parameters. We validate this theoretical result through numerical tests and we compare the proposed variational autoencoder with the existing one in the literature both in terms of accuracy and generalization properties. Finally, we test the proposed variational autoencoder on a Laplace equation, with comparison to the original one and Markov Chains Monte Carlo.