# Guided Diffusion Sampling on Function Spaces with Applications to PDEs

arXiv: https://arxiv.org/abs/2505.17004 · HTML: https://mlsanity.com/paper/2505.17004/

## Key facts

- Cited by 36 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 50 of its 65 references that are in the map.
- First posted on arXiv 22 May 2025.

## Placed in

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

## Builds on (most cited first, 30 of 50)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (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)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (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)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contraction](https://mlsanity.com/paper/2112.05146/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [A Variational Perspective on Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2305.04391/) (2023)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Solving Inverse Problems with Latent Diffusion Models via Hard Data Consistency](https://mlsanity.com/paper/2307.08123/) (2023)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [GenCast: Diffusion-based ensemble forecasting for medium-range weather](https://mlsanity.com/paper/2312.15796/) (2023)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing](https://mlsanity.com/paper/2407.01521/) (2024)
- [Solving Linear Inverse Problems Provably via Posterior Sampling with Latent Diffusion Models](https://mlsanity.com/paper/2307.00619/) (2023)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023)

## Built on by (most cited first, 30 of 36)

- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Mesh-Informed Neural Operator : A Transformer Generative Approach](https://mlsanity.com/paper/2506.16656/) (2025)
- [Decoupled Diffusion Sampling for Inverse Problems on Function Spaces](https://mlsanity.com/paper/2601.23280/) (2026)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Physics-informed diffusion models in spectral space](https://mlsanity.com/paper/2602.09708/) (2026)
- [EquiReg: Equivariance Regularized Diffusion for Inverse Problems](https://mlsanity.com/paper/2505.22973/) (2025)
- [PRISMA: Improving the Accuracy-Latency Frontier of Diffusion-based PDE Solvers Using Physics-Informed Spectral Attention](https://mlsanity.com/paper/2512.01370/) (2025)
- [Supervised Guidance Training for Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2601.20756/) (2026)
- [Latent Generative Solvers for Generalizable Long-Term Physics Simulation](https://mlsanity.com/paper/2602.11229/) (2026)
- [Residual-augmented flow matching operators for probabilistic partial differential equations](https://mlsanity.com/paper/2512.12749/) (2025)
- [Ultra Fast PDE Solving via Physics Guided Few-step Diffusion](https://mlsanity.com/paper/2602.03627/) (2026)
- [Function-Space Decoupled Diffusion for Forward and Inverse Modeling in Carbon Capture and Storage](https://mlsanity.com/paper/2602.12274/) (2026)
- [Ambient Physics: Training Neural PDE Solvers with Partial Observations](https://mlsanity.com/paper/2602.13873/) (2026)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [Operator Flow Matching for Timeseries Forecasting](https://mlsanity.com/paper/2510.15101/) (2025)
- [Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equations](https://mlsanity.com/paper/2512.14980/) (2025)
- [Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver](https://mlsanity.com/paper/2602.18248/) (2026)
- [GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains](https://mlsanity.com/paper/2605.03497/) (2026)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026)
- [Recursive Flow Matching](https://mlsanity.com/paper/2605.26535/) (2026)
- [PosteriorBench: From Point Estimates to Posterior Matching in Evaluating Generative Inverse Solvers](https://mlsanity.com/paper/2609.20794/) (2026)
- [Reconstructing Multi-Scale Physical Fields from Extremely Sparse Measurements with an Autoencoder-Diffusion Cascade](https://mlsanity.com/paper/2512.01572/) (2025)
- [Stochastic Interpolants in Hilbert Spaces](https://mlsanity.com/paper/2602.01988/) (2026)
- [GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media](https://mlsanity.com/paper/2602.14642/) (2026)
- [pADAM: A Plug-and-Play All-in-One Diffusion Architecture for Multi-Physics Learning](https://mlsanity.com/paper/2603.16757/) (2026)
- [Large-Scale 3D Ground-Motion Synthesis with Physics-Inspired Latent Operator Flow Matching](https://mlsanity.com/paper/2603.17403/) (2026)
- [Modelling Gas-Phase Reaction Kinetics with Guided Particle Diffusion Sampling](https://mlsanity.com/paper/2604.16461/) (2026)
- [Learning Interpretable PDE Representations for Generative Reconstructions with Structured Sparsity](https://mlsanity.com/paper/2604.23867/) (2026)
- [(HB-ARFM) History-Bootstrapped Flow Matching for Inverse Boiling Reconstruction](https://mlsanity.com/paper/2606.00349/) (2026)
- [Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations](https://mlsanity.com/paper/2606.03936/) (2026)

## Abstract (arXiv)

We propose a general framework for conditional sampling in PDE-based inverse problems, targeting the recovery of whole solutions from extremely sparse or noisy measurements. This is accomplished by a function-space diffusion model and plug-and-play guidance for conditioning. Our method first trains an unconditional, discretization-agnostic denoising model using neural operator architectures. At inference, we refine the samples to satisfy sparse observation data via a gradient-based guidance mechanism. Through rigorous mathematical analysis, we extend Tweedie's formula to infinite-dimensional Banach spaces, providing the theoretical foundation for our posterior sampling approach. Our method (FunDPS) accurately captures posterior distributions in function spaces under minimal supervision and severe data scarcity. Across five PDE tasks with only 3% observation, our method achieves an average 32% accuracy improvement over state-of-the-art fixed-resolution diffusion baselines while reducing sampling steps by 4x. Furthermore, multi-resolution fine-tuning ensures strong cross-resolution generalizability. To the best of our knowledge, this is the first diffusion-based framework to operate independently of discretization, offering a practical and flexible solution for forward and inverse problems in the context of PDEs. Code is available at https://github.com/neuraloperator/FunDPS