mlsanityData to 7 Oct 2026

Decoupled Diffusion Sampling for Inverse Problems on Function Spaces

arXiv 2601.23280 ↗First posted 30 Jan 2026cs.LG cs.NA math.NAcited by 9

On the map: Diffusion models → Inverse problems and image restoration

Abstract (arXiv)

We propose a data-efficient, physics-aware generative framework in function space for inverse PDE problems. Existing plug-and-play diffusion posterior samplers represent physics implicitly through joint coefficient-solution modeling, requiring substantial paired supervision. In contrast, our Decoupled Diffusion Inverse Solver (DDIS) employs a decoupled design: an unconditional diffusion learns the coefficient prior, while a neural operator explicitly models the forward PDE for guidance. This decoupling enables superior data efficiency and effective physics-informed learning, while naturally supporting Decoupled Annealing Posterior Sampling (DAPS) to avoid over-smoothing in Diffusion Posterior Sampling (DPS). Theoretically, we prove that DDIS avoids the guidance attenuation failure of joint models when training data is scarce. Empirically, DDIS achieves state-of-the-art performance under sparse observation, improving l2 error by 11% and spectral error by 54% on average; when data is limited to 1%, DDIS maintains accuracy with 40% advantage in l2 error compared to joint models.

Key facts. Cited by 9 papers in the map (data through 7 Oct 2026). Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models. Builds on 40 of its 72 references that are in the map. First posted on arXiv 30 Jan 2026.

Builds on

40 of 72 references.

Closest ancestors · 12

Same direction, newest first

  1. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEs22 May
  2. 2024A Survey on Diffusion Models for Inverse Problems30 Sep
  3. 2024Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing1 Jul
  4. 2024Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors29 May
  5. 2024Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction25 Mar
  6. 2023Monte Carlo guided Diffusion for Bayesian linear inverse problems15 Aug
  7. 2023Denoising Diffusion Models for Plug-and-Play Image Restoration15 May
  8. 2023A Variational Perspective on Solving Inverse Problems with Diffusion Models7 May
  9. 2022Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model1 Dec
  10. 2022Diffusion Posterior Sampling for General Noisy Inverse Problems29 Sep
  11. 2022Denoising Diffusion Restoration Models27 Jan
  12. 2021Deblurring via Stochastic Refinement5 Dec

Other references · 28

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2020Denoising Diffusion Implicit Modelscited by 9,655
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  4. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  5. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  6. 2021Palette: Image-to-Image Diffusion Modelscited by 1,296
  7. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  8. 2021Solving Inverse Problems in Medical Imaging with Score-Based Generative Modelscited by 514
  9. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  10. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
  11. 2021SNIPS: Solving Noisy Inverse Problems Stochasticallycited by 191
  12. 2022FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operatorscited by 188
16 more
  1. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstructioncited by 150
  2. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 100
  3. 2023Deep Learning for Day Forecasts from Sparse Observationscited by 70
  4. 2023CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problemscited by 54
  5. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraintscited by 46
  6. 2025Memorization and Regularization in Generative Diffusion Modelscited by 39
  7. 2024Gradient-Free Generation for Hard-Constrained Systemscited by 35
  8. 2024Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learningcited by 32
  9. 2024Generative Data Assimilation of Sparse Weather Station Observations at Kilometer Scalescited by 27
  10. 2017Deep Generative Adversarial Networks for Compressed Sensing Automates MRIcited by 15
  11. 2025Stochastic Process Learning via Operator Flow Matchingcited by 14
  12. 2025Fourier Neural Operators Explained: A Practical Perspectivecited by 14
  13. 2020Do CNNs solve the CT inverse problem?cited by 13
  14. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 11
  15. 2025Guiding diffusion models to reconstruct flow fields from sparse datacited by 11
  16. 2025PRISMA: Improving the Accuracy-Latency Frontier of Diffusion-based PDE Solvers Using Physics-Informed Spectral Attentioncited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 9.

9 papers.

  1. 2026Function-Space Decoupled Diffusion for Forward and Inverse Modeling in Carbon Capture and Storagecited by 2
  2. 2026GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domainscited by 1
  3. 2026Pseudo-differential-enhanced physics-informed neural networkscited by 0
  4. 2026Autoregressive One-Step Generative Modeling for Dynamical System Forecastingcited by 0
  5. 2026Physical Fidelity Reconstruction via Improved Consistency-Distilled Flow Matching for Dynamical Systemscited by 0
  6. 2026Christoffel-DPS: Optimal sensor placement in diffusion posterior sampling for arbitrary distributionscited by 0
  7. 2026Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observationscited by 0
  8. 2026Flow Annealing Posterior Sampling for Function-Space Regression and Inverse Problemscited by 0
  9. 2026PosteriorBench: From Point Estimates to Posterior Matching in Evaluating Generative Inverse Solverscited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026Backward-Consistent Diffusion Sampling for Sparsely Observed PDE Inverse Problems
  2. 2025Unifying and extending Diffusion Models through PDEs for solving Inverse Problems
  3. 2024ODE-DPS: ODE-based Diffusion Posterior Sampling for Inverse Problems in Partial Differential Equation
  4. 2026Latent Diffusion Posterior Sampling with Surrogate Likelihood Guidance for PDE Inverse Problems
  5. 2025DAPS++: Rethinking Diffusion Inverse Problems with Decoupled Posterior Annealing
  6. 2023Solving Inverse Physics Problems with Score Matching
  7. 2025Improving Diffusion-based Inverse Algorithms under Few-Step Constraint via Learnable Linear Extrapolation
  8. 2024Enhancing and Accelerating Diffusion-Based Inverse Problem Solving through Measurements Optimization