# Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces

arXiv: https://arxiv.org/abs/2603.14798 · HTML: https://mlsanity.com/paper/2603.14798/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 38 references that are in the map.
- First posted on arXiv 16 Mar 2026.

## Builds on (most cited first, 24 of 24)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Improved Mean Flows: On the Challenges of Fastforward Generative Models](https://mlsanity.com/paper/2512.02012/) (2025)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [DMPlug: A Plug-in Method for Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2405.16749/) (2024)
- [On conditional diffusion models for PDE simulations](https://mlsanity.com/paper/2410.16415/) (2024)
- [Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching](https://mlsanity.com/paper/2405.18816/) (2024)
- [Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2302.10130/) (2023)
- [Mathematical analysis of singularities in the diffusion model under the submanifold assumption](https://mlsanity.com/paper/2301.07882/) (2023)
- [Wasserstein Bounds for generative diffusion models with Gaussian tail targets](https://mlsanity.com/paper/2412.11251/) (2024)
- [Pathway to O(√d) Complexity bound under Wasserstein metric of flow-based models](https://mlsanity.com/paper/2512.06702/) (2025)
- [PODNO: Proper Orthogonal Decomposition Neural Operators](https://mlsanity.com/paper/2504.18513/) (2025)

## Built on by (most cited first, 1 of 1)

- [Posterior sampling by source-space MCMC via prior-based few-step transport maps](https://mlsanity.com/paper/2610.01034/) (2026)

## Abstract (arXiv)

We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime. Based on one-step generative transport, the method learns an amortized neural operator whose pushforward of a Gaussian source approximates the posterior distribution conditioned on each new observation. We show that white-noise sources are incompatible with the function-space limit, and therefore adopt a prior-aligned GRF as the source. We justify this choice through the Lipschitz regularity of the resulting one-step conditional posterior transport and numerical experiments on linear inverse and PDE-based inverse problems. The method is not distilled from MCMC: it is trained only with prior samples and simulated partial noisy observations. Once trained, it generates a 64×64 posterior sample in ~ 10-3s, avoiding repeated forward-model evaluations in MCMC and repeated network evaluations in multistep generative samplers while matching key posterior summaries.