# Solving inverse problems using conditional invertible neural networks

arXiv: https://arxiv.org/abs/2007.15849 · HTML: https://mlsanity.com/paper/2007.15849/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jul 2020.

## Built on by (most cited first, 30 of 37)

- [Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models](https://mlsanity.com/paper/2108.12956/) (2021)
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- [Diagnosing and Fixing Manifold Overfitting in Deep Generative Models](https://mlsanity.com/paper/2204.07172/) (2022)
- [Conditional Injective Flows for Bayesian Imaging](https://mlsanity.com/paper/2204.07664/) (2022)
- [Variational Bayesian Optimal Experimental Design with Normalizing Flows](https://mlsanity.com/paper/2404.13056/) (2024)
- [Bayesian multiscale deep generative model for the solution of high-dimensional inverse problems](https://mlsanity.com/paper/2102.03169/) (2021)
- [Bi-fidelity Variational Auto-encoder for Uncertainty Quantification](https://mlsanity.com/paper/2305.16530/) (2023)
- [Variational Sequential Optimal Experimental Design using Reinforcement Learning](https://mlsanity.com/paper/2306.10430/) (2023)
- [Inverse Aerodynamic Design of Gas Turbine Blades using Probabilistic Machine Learning](https://mlsanity.com/paper/2108.10163/) (2021)
- [Unifying and extending Diffusion Models through PDEs for solving Inverse Problems](https://mlsanity.com/paper/2504.07437/) (2025)
- [Efficient Bayesian inference using physics-informed invertible neural networks for inverse problems](https://mlsanity.com/paper/2304.12541/) (2023)
- [Conditional score-based diffusion models for solving inverse problems in mechanics](https://mlsanity.com/paper/2406.13154/) (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)
- [Geodesy of irregular small bodies via neural density fields: geodesyNets](https://mlsanity.com/paper/2105.13031/) (2021)
- [Uncertainty quantification for ptychography using normalizing flows](https://mlsanity.com/paper/2111.00745/) (2021)
- [VI-DGP: A variational inference method with deep generative prior for solving high-dimensional inverse problems](https://mlsanity.com/paper/2302.11173/) (2023)
- [Conditional Pseudo-Reversible Normalizing Flow for Surrogate Modeling in Quantifying Uncertainty Propagation](https://mlsanity.com/paper/2404.00502/) (2024)
- [Invertible Surrogate Models: Joint surrogate modelling and reconstruction of Laser-Wakefield Acceleration by invertible neural networks](https://mlsanity.com/paper/2106.00432/) (2021)
- [Flow-based Visual Quality Enhancer for Super-resolution Magnetic Resonance Spectroscopic Imaging](https://mlsanity.com/paper/2207.10181/) (2022)
- [Fast light-field 3D microscopy with out-of-distribution detection and adaptation through Conditional Normalizing Flows](https://mlsanity.com/paper/2306.06408/) (2023)
- [On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networks](https://mlsanity.com/paper/2308.09367/) (2023)
- [Two-Stage Surrogate Modeling for Data-Driven Design Optimization with Application to Composite Microstructure Generation](https://mlsanity.com/paper/2401.02008/) (2024)
- [ISR: Invertible Symbolic Regression](https://mlsanity.com/paper/2405.06848/) (2024)
- [DAWN-FM: Data-Aware and Noise-Informed Flow Matching for Solving Inverse Problems](https://mlsanity.com/paper/2412.04766/) (2024)
- [On Learning the Tail Quantiles of Driving Behavior Distributions via Quantile Regression and Flows](https://mlsanity.com/paper/2305.13106/) (2023)
- [Probabilistic Forecasting of Irregular Time Series via Conditional Flows](https://mlsanity.com/paper/2402.06293/) (2024)
- [Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview](https://mlsanity.com/paper/2402.12072/) (2024)
- [Uncertainty-Aware Surrogate-based Amortized Bayesian Inference for Computationally Expensive Models](https://mlsanity.com/paper/2505.08683/) (2025)
- [Learning Gradient Flow: Using Equation Discovery to Accelerate Engineering Optimization](https://mlsanity.com/paper/2602.13513/) (2026)

## Abstract (arXiv)

Inverse modeling for computing a high-dimensional spatially-varying property field from indirect sparse and noisy observations is a challenging problem. This is due to the complex physical system of interest often expressed in the form of multiscale PDEs, the high-dimensionality of the spatial property of interest, and the incomplete and noisy nature of observations. To address these challenges, we develop a model that maps the given observations to the unknown input field in the form of a surrogate model. This inverse surrogate model will then allow us to estimate the unknown input field for any given sparse and noisy output observations. Here, the inverse mapping is limited to a broad prior distribution of the input field with which the surrogate model is trained. In this work, we construct a two- and three-dimensional inverse surrogate models consisting of an invertible and a conditional neural network trained in an end-to-end fashion with limited training data. The invertible network is developed using a flow-based generative model. The developed inverse surrogate model is then applied for an inversion task of a multiphase flow problem where given the pressure and saturation observations the aim is to recover a high-dimensional non-Gaussian permeability field where the two facies consist of heterogeneous permeability and varying length-scales. For both the two- and three-dimensional surrogate models, the predicted sample realizations of the non-Gaussian permeability field are diverse with the predictive mean being close to the ground truth even when the model is trained with limited data.