# Deep Adaptive Dimension Reduction for Bayesian Inference in Inverse Problems

arXiv: https://arxiv.org/abs/2605.29373 · HTML: https://mlsanity.com/paper/2605.29373/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 44 references that are in the map.
- First posted on arXiv 28 May 2026.

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

- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [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)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [Protein Conformation Generation via Force-Guided SE(3) Diffusion Models](https://mlsanity.com/paper/2403.14088/) (2024)
- [The Expressive Power of a Class of Normalizing Flow Models](https://mlsanity.com/paper/2006.00392/) (2020)
- [Solving Bayesian Inverse Problems via Variational Autoencoders](https://mlsanity.com/paper/1912.04212/) (2019)
- [Adaptive deep density approximation for Fokker-Planck equations](https://mlsanity.com/paper/2103.11181/) (2021)
- [Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2210.03008/) (2022)
- [Learning in latent spaces improves the predictive accuracy of deep neural operators](https://mlsanity.com/paper/2304.07599/) (2023)
- [Bayesian multiscale deep generative model for the solution of high-dimensional inverse problems](https://mlsanity.com/paper/2102.03169/) (2021)
- [Adaptive operator learning for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2310.17844/) (2023)
- [Hierarchical Bayesian Level Set Inversion](https://mlsanity.com/paper/1601.03605/) (2016)
- [Deep adaptive sampling for surrogate modeling without labeled data](https://mlsanity.com/paper/2402.11283/) (2024)
- [VI-DGP: A variational inference method with deep generative prior for solving high-dimensional inverse problems](https://mlsanity.com/paper/2302.11173/) (2023)
- [Estimating Committor Functions via Deep Adaptive Sampling on Rare Transition Paths](https://mlsanity.com/paper/2501.15522/) (2025)
- [Stable Derivative Free Gaussian Mixture Variational Inference for Bayesian Inverse Problems](https://mlsanity.com/paper/2501.04259/) (2025)

## Abstract (arXiv)

Solving high-dimensional PDE-governed inverse problems is often challenging due to complex non-Gaussian posterior distributions, expensive forward model evaluations, and misspecified prior information. To address these issues, we propose a deep adaptive dimension-reduction Bayesian inference framework based on the Variational Flow (VF) model. Since standard normalizing flows are restricted by bijective mappings and cannot directly reduce dimensions, VF overcomes this limitation by integrating VAE-based nonlinear dimension reduction with dual normalizing flows for the latent prior and encoder. This design provides a strictly higher evidence lower bound than VAE and allows more flexible approximation of complex posterior distributions. We further introduce an iterative prior updating strategy that gradually moves the prior mean toward high-probability posterior regions, avoiding manual prior tuning. These components form a closed adaptive loop together with an adaptively fine-tuned Fourier Neural Operator (FNO) surrogate: VF generates posterior-concentrated samples to refine the surrogate, while the updated surrogate further improves posterior inference. Numerical experiments on a 100-dimensional Rosenbrock problem and three standard PDE-governed inverse problems show that our method delivers competitive or superior accuracy compared with MCMC, UKI, and SVGD baselines across all tested configurations, with the most pronounced advantages emerging in challenging scenarios such as high-noise observations and high-dimensional parameter spaces.