# Variational Autoencoding of PDE Inverse Problems

arXiv: https://arxiv.org/abs/2006.15641 · HTML: https://mlsanity.com/paper/2006.15641/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jun 2020.

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

- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Fully probabilistic deep models for forward and inverse problems in parametric PDEs](https://mlsanity.com/paper/2208.04856/) (2022)
- [Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution](https://mlsanity.com/paper/2103.14629/) (2021)
- [Random Grid Neural Processes for Parametric Partial Differential Equations](https://mlsanity.com/paper/2301.11040/) (2023)
- [InVAErt networks: a data-driven framework for model synthesis and identifiability analysis](https://mlsanity.com/paper/2307.12586/) (2023)
- [Disentangling Generative Factors of Physical Fields Using Variational Autoencoders](https://mlsanity.com/paper/2109.07399/) (2021)
- [A Primer on Variational Inference for Physics-Informed Deep Generative Modelling](https://mlsanity.com/paper/2409.06560/) (2024)
- [Theory Guided and Interpretable Neural Operator Design for Partial Differential Equation Learning](https://mlsanity.com/paper/2609.33715/) (2026)

## Abstract (arXiv)

Specifying a governing physical model in the presence of missing physics and recovering its parameters are two intertwined and fundamental problems in science. Modern machine learning allows one to circumvent these, via emulators and surrogates, but in doing so disregards prior knowledge and physical laws that are especially important for small data regimes, interpretability, and decision making. In this work we fold the mechanistic model into a flexible data-driven surrogate to arrive at a physically structured decoder network. This provides accelerated inference for the Bayesian inverse problem, and can act as a drop-in regulariser that encodes a-priori physical information. We employ the variational form of the PDE problem and introduce stochastic local approximations as a form of model based data augmentation. We demonstrate both the accuracy and increased computational efficiency of the framework on real world settings and structured spatial processes.