# Conditional score-based diffusion models for Bayesian inference in infinite dimensions

arXiv: https://arxiv.org/abs/2305.19147 · HTML: https://mlsanity.com/paper/2305.19147/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 29 of its 62 references that are in the map.
- First posted on arXiv 28 May 2023.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 29 of 29)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Conditional Image Generation with Score-Based Diffusion Models](https://mlsanity.com/paper/2111.13606/) (2021)
- [SNIPS: Solving Noisy Inverse Problems Stochastically](https://mlsanity.com/paper/2105.14951/) (2021)
- [From data to functa: Your data point is a function and you can treat it like one](https://mlsanity.com/paper/2201.12204/) (2022)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimation](https://mlsanity.com/paper/2106.05527/) (2021)
- [Semi-Amortized Variational Autoencoders](https://mlsanity.com/paper/1802.02550/) (2018)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [Neural Diffusion Processes](https://mlsanity.com/paper/2206.03992/) (2022)
- [Diffusion Generative Models in Infinite Dimensions](https://mlsanity.com/paper/2212.00886/) (2022)
- [Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference](https://mlsanity.com/paper/2006.06755/) (2020)
- [Continuous-Time Functional Diffusion Processes](https://mlsanity.com/paper/2303.00800/) (2023)
- [HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference](https://mlsanity.com/paper/1905.10687/) (2019)
- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [Spectral Diffusion Processes](https://mlsanity.com/paper/2209.14125/) (2022)
- [Learning by example: fast reliability-aware seismic imaging with normalizing flows](https://mlsanity.com/paper/2104.06255/) (2021)
- [Score-Based Generative Models for Molecule Generation](https://mlsanity.com/paper/2203.04698/) (2022)

## Built on by (most cited first, 30 of 38)

- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
- [Stochastic Optimal Control for Diffusion Bridges in Function Spaces](https://mlsanity.com/paper/2405.20630/) (2024)
- [Controllable seismic velocity synthesis using generative diffusion models](https://mlsanity.com/paper/2402.06277/) (2024)
- [Universal Functional Regression with Neural Operator Flows](https://mlsanity.com/paper/2404.02986/) (2024)
- [State-observation augmented diffusion model for nonlinear assimilation with unknown dynamics](https://mlsanity.com/paper/2407.21314/) (2024)
- [Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems](https://mlsanity.com/paper/2405.15676/) (2024)
- [Infinite-dimensional Diffusion Bridge Simulation via Operator Learning](https://mlsanity.com/paper/2405.18353/) (2024)
- [Probability-Flow ODE in Infinite-Dimensional Function Spaces](https://mlsanity.com/paper/2503.10219/) (2025)
- [Preconditioned Langevin Dynamics with Score-Based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problems](https://mlsanity.com/paper/2505.18276/) (2025)
- [ReGuidance: A Simple Diffusion Wrapper for Boosting Sample Quality on Hard Inverse Problems](https://mlsanity.com/paper/2506.10955/) (2025)
- [Scale-Adaptive Generative Flows for Multiscale Scientific Data](https://mlsanity.com/paper/2509.02971/) (2025)
- [Back-Projection Diffusion: Solving the Wideband Inverse Scattering Problem with Diffusion Models](https://mlsanity.com/paper/2408.02866/) (2024)
- [EquiReg: Equivariance Regularized Diffusion for Inverse Problems](https://mlsanity.com/paper/2505.22973/) (2025)
- [Supervised Guidance Training for Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2601.20756/) (2026)
- [Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamics](https://mlsanity.com/paper/2602.01449/) (2026)
- [Guided Unconditional and Conditional Generative Models for Super-Resolution and Inference of Quasi-Geostrophic Turbulence](https://mlsanity.com/paper/2507.00719/) (2025)
- [Ultra Fast PDE Solving via Physics Guided Few-step Diffusion](https://mlsanity.com/paper/2602.03627/) (2026)
- [Conditional neural control variates for variance reduction in Bayesian inverse problems](https://mlsanity.com/paper/2602.21357/) (2026)
- [Dual-space posterior sampling for Bayesian inference in constrained inverse problems](https://mlsanity.com/paper/2603.00393/) (2026)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [An Unconditional Representation of the Conditional Score in Infinite-Dimensional Linear Inverse Problems](https://mlsanity.com/paper/2405.15643/) (2024)
- [Non-asymptotic convergence bound of conditional diffusion models](https://mlsanity.com/paper/2508.10944/) (2025)
- [A Review of Diffusion-based Simulation-Based Inference: Foundations and Applications in Non-Ideal Data Scenarios](https://mlsanity.com/paper/2512.23748/) (2025)
- [On the role of memorization in learned priors for geophysical inverse problems](https://mlsanity.com/paper/2603.19629/) (2026)
- [GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains](https://mlsanity.com/paper/2605.03497/) (2026)
- [Deep Optimal Sensor Placement for Black Box Stochastic Simulations](https://mlsanity.com/paper/2410.12036/) (2024)
- [DiffuMural: Restoring Dunhuang Murals with Multi-scale Diffusion](https://mlsanity.com/paper/2504.09513/) (2025)
- [Score-based diffusion models for severely ill-posed problems in diffuse optical tomography](https://mlsanity.com/paper/2602.03449/) (2026)

## Abstract (arXiv)

Since their initial introduction, score-based diffusion models (SDMs) have been successfully applied to solve a variety of linear inverse problems in finite-dimensional vector spaces due to their ability to efficiently approximate the posterior distribution. However, using SDMs for inverse problems in infinite-dimensional function spaces has only been addressed recently, primarily through methods that learn the unconditional score. While this approach is advantageous for some inverse problems, it is mostly heuristic and involves numerous computationally costly forward operator evaluations during posterior sampling. To address these limitations, we propose a theoretically grounded method for sampling from the posterior of infinite-dimensional Bayesian linear inverse problems based on amortized conditional SDMs. In particular, we prove that one of the most successful approaches for estimating the conditional score in finite dimensions - the conditional denoising estimator - can also be applied in infinite dimensions. A significant part of our analysis is dedicated to demonstrating that extending infinite-dimensional SDMs to the conditional setting requires careful consideration, as the conditional score typically blows up for small times, contrarily to the unconditional score. We conclude by presenting stylized and large-scale numerical examples that validate our approach, offer additional insights, and demonstrate that our method enables large-scale, discretization-invariant Bayesian inference.