mlsanityData to 14 Oct 2026

Unifying and extending Diffusion Models through PDEs for solving Inverse Problems

arXiv 2504.07437 ↗First posted 10 Apr 2025cs.LG stat.CO stat.MLcited by 7

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

Abstract (arXiv)

Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these models have been derived using principles of variational inference, denoising, statistical signal processing, and stochastic differential equations. In contrast to the conventional presentation, in this study we derive diffusion models using ideas from linear partial differential equations and demonstrate that this approach has several benefits that include a constructive derivation of the forward and reverse processes, a unified derivation of multiple formulations and sampling strategies, and the discovery of a new class of variance preserving models. We also apply the conditional version of these models to solve canonical conditional density estimation problems and challenging inverse problems. These problems help establish benchmarks for systematically quantifying the performance of different formulations and sampling strategies in this study and for future studies. Finally, we identify and implement a mechanism through which a single diffusion model can be applied to measurements obtained from multiple measurement operators. Taken together, the contents of this manuscript provide a new understanding of and several new directions in the application of diffusion models to solving physics-based inverse problems.

Key facts. Cited by 7 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 30 of its 42 references that are in the map. First posted on arXiv 10 Apr 2025.

Builds on

30 of 42 references.

Closest ancestors · 8

Same direction, newest first

  1. 2024A Survey on Diffusion Models for Inverse Problems30 Sep
  2. 2024Spatially-Aware Diffusion Models with Cross-Attention for Global Field Reconstruction with Sparse Observations30 Aug
  3. 2024Back-Projection Diffusion: Solving the Wideband Inverse Scattering Problem with Diffusion Models5 Aug
  4. 2024Conditional score-based diffusion models for solving inverse problems in mechanics19 Jun
  5. 2023A Variational Perspective on Solving Inverse Problems with Diffusion Models7 May
  6. 2023Score-Based Diffusion Models as Principled Priors for Inverse Imaging23 Apr
  7. 2022Diffusion Posterior Sampling for General Noisy Inverse Problems29 Sep
  8. 2022Improving Diffusion Models for Inverse Problems using Manifold Constraints2 Jun

Other references · 22

  1. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  5. 2022Elucidating the Design Space of Diffusion-Based Generative Modelscited by 3,045
  6. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,316
  7. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  8. 2020Improved Techniques for Training Score-Based Generative Modelscited by 1,043
  9. 2021CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputationcited by 582
  10. 2021Solving Inverse Problems in Medical Imaging with Score-Based Generative Modelscited by 521
  11. 2022Diffusion models as plug-and-play priorscited by 231
  12. 2021Conditional Image Generation with Score-Based Diffusion Modelscited by 197
10 more
  1. 2024Physics-Informed Diffusion Modelscited by 90
  2. 2023CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problemscited by 56
  3. 2023Provable Probabilistic Imaging using Score-Based Generative Priorscited by 51
  4. 2025Memorization and Regularization in Generative Diffusion Modelscited by 40
  5. 2020Solving inverse problems using conditional invertible neural networkscited by 37
  6. 2021Solution of Physics-based Bayesian Inverse Problems with Deep Generative Priorscited by 29
  7. 2022The efficacy and generalizability of conditional GANs for posterior inference in physics-based inverse problemscited by 15
  8. 2024Zero-Shot Uncertainty Quantification using Diffusion Probabilistic Modelscited by 14
  9. 2023A dimension-reduced variational approach for solving physics-based inverse problems using generative adversarial network priors and normalizing flowscited by 12
  10. 2023Solution of physics-based inverse problems using conditional generative adversarial networks with full gradient penaltycited by 7

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 3; 2026: 4.

7 papers.

  1. 2025Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approachcited by 18
  2. 2026D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matchingcited by 6
  3. 2026Closed-form conditional diffusion models for data assimilationcited by 3
  4. 2025Generative Algorithms for Wildfire Progression Reconstruction from Multi-Modal Satellite Active Fire Measurements and Terrain Heightcited by 2
  5. 2026Conditional flow matching for physics-constrained inverse problems with finite training datacited by 2
  6. 2025Time-dependent density estimation using binary classifierscited by 0
  7. 2026Generative AI for material design: A mechanics perspective from burgers to mattercited by 0

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  3. 2025Exact Evaluation of the Accuracy of Diffusion Models for Inverse Problems with Gaussian Data Distributions
  4. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEs
  5. 2025Solving Diffusion Inverse Problems with Restart Posterior Sampling
  6. 2024Divide-and-Conquer Posterior Sampling for Denoising Diffusion Priors
  7. 2025A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm
  8. 2024Enhancing Diffusion Models for Inverse Problems with Covariance-Aware Posterior Sampling