# An invertible generative model for forward and inverse problems

arXiv: https://arxiv.org/abs/2509.03910 · HTML: https://mlsanity.com/paper/2509.03910/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 54 references that are in the map.
- First posted on arXiv 4 Sep 2025.

## Builds on (most cited first, 30 of 38)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [SRFlow: Learning the Super-Resolution Space with Normalizing Flow](https://mlsanity.com/paper/2006.14200/) (2020)
- [Conditional Image Generation with Score-Based Diffusion Models](https://mlsanity.com/paper/2111.13606/) (2021)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [The promises and pitfalls of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1811.10072/) (2018)
- [JANA: Jointly Amortized Neural Approximation of Complex Bayesian Models](https://mlsanity.com/paper/2302.09125/) (2023)
- [Solving Bayesian Inverse Problems via Variational Autoencoders](https://mlsanity.com/paper/1912.04212/) (2019)
- [End-to-end reconstruction meets data-driven regularization for inverse problems](https://mlsanity.com/paper/2106.03538/) (2021)

## Built on by (most cited first, 2 of 2)

- [A Distributionally Robust Framework for Learned Reconstructions in Inverse Problems](https://mlsanity.com/paper/2606.30230/) (2026)
- [Twist Flow for Inverse Problems](https://mlsanity.com/paper/2610.09281/) (2026)

## Abstract (arXiv)

We formulate inverse problems in a Bayesian framework and aim to train an invertible generative model that is capable of simulation (i.e., sampling from the likelihood) and inference (i.e., sampling from the posterior). We call such a generative model a Reversible Simulator. We first construct an explicit instance of a reversible simulator by combining lower and upper triangular normalizing flows associated with conditional sampling into a single invertible map. We then establish basic structural properties of this construction, investigate its non-uniqueness, and derive a variational formulation that enables the generative model to be trained directly from paired samples. Finally, we illustrate the proposed framework on analytically tractable and stylized numerical examples, demonstrating its potential as a unified approach to conditional generative modeling for forward and inverse problems.