# Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems

arXiv: https://arxiv.org/abs/2209.02772 · HTML: https://mlsanity.com/paper/2209.02772/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 39 references that are in the map.
- First posted on arXiv 6 Sep 2022.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [A Stein variational Newton method](https://mlsanity.com/paper/1806.03085/) (2018)
- [Momentum Residual Neural Networks](https://mlsanity.com/paper/2102.07870/) (2021)
- [Deep autoregressive neural networks for high-dimensional inverse problems in groundwater contaminant source identification](https://mlsanity.com/paper/1812.09444/) (2018)
- [Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems](https://mlsanity.com/paper/1912.12976/) (2019)
- [Learning normal form autoencoders for data-driven discovery of universal,parameter-dependent governing equations](https://mlsanity.com/paper/2106.05102/) (2021)
- [Physics-aware, probabilistic model order reduction with guaranteed stability](https://mlsanity.com/paper/2101.05834/) (2021)
- [Bayesian model and dimension reduction for uncertainty propagation: applications in random media](https://mlsanity.com/paper/1711.02475/) (2017)
- [Multimodal, high-dimensional, model-based, Bayesian inverse problems with applications in biomechanics](https://mlsanity.com/paper/1512.04481/) (2015)

## Built on by (most cited first, 25 of 25)

- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Monte Carlo guided Diffusion for Bayesian linear inverse problems](https://mlsanity.com/paper/2308.07983/) (2023)
- [Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems](https://mlsanity.com/paper/2402.11722/) (2024)
- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [Random Grid Neural Processes for Parametric Partial Differential Equations](https://mlsanity.com/paper/2301.11040/) (2023)
- [BiLO: Bilevel Local Operator Learning for PDE Inverse Problems](https://mlsanity.com/paper/2404.17789/) (2024)
- [Learning Hidden Physics and System Parameters with Deep Operator Networks](https://mlsanity.com/paper/2412.05133/) (2024)
- [Efficient Bayesian inference using physics-informed invertible neural networks for inverse problems](https://mlsanity.com/paper/2304.12541/) (2023)
- [Generative Learning for Forecasting the Dynamics of Complex Systems](https://mlsanity.com/paper/2402.17157/) (2024)
- [Learning to solve Bayesian inverse problems: An amortized variational inference approach using Gaussian and Flow guides](https://mlsanity.com/paper/2305.20004/) (2023)
- [Weak neural variational inference for solving Bayesian inverse problems without forward models: applications in elastography](https://mlsanity.com/paper/2407.20697/) (2024)
- [DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modeling](https://mlsanity.com/paper/2502.06250/) (2025)
- [Resolution invariant deep operator network for PDEs with complex geometries](https://mlsanity.com/paper/2402.00825/) (2024)
- [Bayesian ECG reconstruction using denoising diffusion generative models](https://mlsanity.com/paper/2401.05388/) (2023)
- [Closure Discovery for Coarse-Grained Partial Differential Equations Using Grid-based Reinforcement Learning](https://mlsanity.com/paper/2402.00972/) (2024)
- [Physics-aware deep learning framework for the limited aperture inverse obstacle scattering problem](https://mlsanity.com/paper/2403.19470/) (2024)
- [Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later](https://mlsanity.com/paper/2509.19929/) (2025)
- [Inverse Neural Operator for ODE Parameter Optimization](https://mlsanity.com/paper/2603.11854/) (2026)
- [GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems](https://mlsanity.com/paper/2607.01128/) (2026)
- [Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Data](https://mlsanity.com/paper/2505.11308/) (2025)
- [How Regularization Terms Make Invertible Neural Networks Bayesian Point Estimators](https://mlsanity.com/paper/2510.26704/) (2025)
- [Learning Generalizable Neural Operators for Inverse Problems](https://mlsanity.com/paper/2512.18120/) (2025)
- [DeepONet-accelerated Bayesian inversion for moving boundary problems](https://mlsanity.com/paper/2512.20268/) (2025)
- [GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media](https://mlsanity.com/paper/2602.14642/) (2026)
- [Posterior-First Neural PDE Simulation: Inferring Hidden Problem State from a Single Field](https://mlsanity.com/paper/2605.03247/) (2026)

## Abstract (arXiv)

Neural Operators offer a powerful, data-driven tool for solving parametric PDEs as they can represent maps between infinite-dimensional function spaces. In this work, we employ physics-informed Neural Operators in the context of high-dimensional, Bayesian inverse problems. Traditional solution strategies necessitate an enormous, and frequently infeasible, number of forward model solves, as well as the computation of parametric derivatives. In order to enable efficient solutions, we extend Deep Operator Networks (DeepONets) by employing a RealNVP architecture which yields an invertible and differentiable map between the parametric input and the branch-net output. This allows us to construct accurate approximations of the full posterior, irrespective of the number of observations and the magnitude of the observation noise, without any need for additional forward solves nor for cumbersome, iterative sampling procedures. We demonstrate the efficacy and accuracy of the proposed methodology in the context of inverse problems for three benchmarks: an anti-derivative equation, reaction-diffusion dynamics and flow through porous media.