mlsanityData to 14 Oct 2026

Diffusion Generative Models in Infinite Dimensions

arXiv 2212.00886 ↗First posted 1 Dec 2022cs.LG stat.MLcited by 49

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion generative models have recently been applied to domains where the available data can be seen as a discretization of an underlying function, such as audio signals or time series. However, these models operate directly on the discretized data, and there are no semantics in the modeling process that relate the observed data to the underlying functional forms. We generalize diffusion models to operate directly in function space by developing the foundational theory for such models in terms of Gaussian measures on Hilbert spaces. A significant benefit of our function space point of view is that it allows us to explicitly specify the space of functions we are working in, leading us to develop methods for diffusion generative modeling in Sobolev spaces. Our approach allows us to perform both unconditional and conditional generation of function-valued data. We demonstrate our methods on several synthetic and real-world benchmarks.

Key facts. Cited by 49 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 33 of its 57 references that are in the map. First posted on arXiv 1 Dec 2022.

Builds on

33 of 57 references.

Closest ancestors · 6

Same direction, newest first

  1. 2022Riemannian Diffusion Models16 Aug
  2. 2022Riemannian Score-Based Generative Modelling6 Feb
  3. 2021ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models6 Aug
  4. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  5. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  6. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 27

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  2. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,585
  3. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,355
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  5. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  6. 2022Video Diffusion Modelscited by 2,305
  7. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,354
  8. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  9. 2020WaveGrad: Estimating Gradients for Waveform Generationcited by 622
  10. 2021CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputationcited by 582
  11. 2022GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generationcited by 516
  12. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
15 more
  1. 2021Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecastingcited by 337
  2. 2018Neural Processescited by 311
  3. 2019Attentive Neural Processescited by 283
  4. 2022Diffusion Probabilistic Modeling for Video Generationcited by 238
  5. 2022Diffusion-based Time Series Imputation and Forecasting with Structured State Space Modelscited by 198
  6. 2022From data to functa: Your data point is a function and you can treat it like onecited by 187
  7. 2019Functional Variational Bayesian Neural Networkscited by 127
  8. 2021Generative Models as Distributions of Functionscited by 89
  9. 2020All You Need is a Good Functional Prior for Bayesian Deep Learningcited by 61
  10. 2015On Sparse variational methods and the Kullback-Leibler divergence between stochastic processescited by 57
  11. 2022Generative Adversarial Neural Operatorscited by 52
  12. 2022Neural Diffusion Processescited by 50
  13. 2020Understanding Variational Inference in Function-Spacecited by 40
  14. 2022Generalized Variational Inference in Function Spaces: Gaussian Measures meet Bayesian Deep Learningcited by 21
  15. 2021Variational Gaussian Processes: A Functional Analysis Viewcited by 3

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 10; 2024: 13; 2025: 17; 2026: 9.

49 papers.

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By area:

  1. 2023Score-based Diffusion Models in Function Spacecited by 82
  2. 2023Functional Flow Matchingcited by 67
  3. 2023Continuous-Time Functional Diffusion Processescited by 39
  4. 2023Conditional score-based diffusion models for Bayesian inference in infinite dimensionscited by 38
  5. 2023Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generationcited by 37
  6. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 36
  7. 2023Conditional Optimal Transport on Function Spacescited by 26
  8. 2024Dynamic Conditional Optimal Transport through Simulation-Free Flowscited by 25
  9. 2023∞-Diff: Infinite Resolution Diffusion with Subsampled Mollified Statescited by 21
  10. 2024Warped Diffusion: Solving Video Inverse Problems with Image Diffusion Modelscited by 21
  11. 2025FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modelingcited by 20
  12. 2025Stochastic Process Learning via Operator Flow Matchingcited by 14
37 more
  1. 2025Mesh-Informed Neural Operator : A Transformer Generative Approachcited by 11
  2. 2023Geometric Neural Diffusion Processescited by 10
  3. 2024Wasserstein Bounds for generative diffusion models with Gaussian tail targetscited by 10
  4. 2024BioDiffusion: A Versatile Diffusion Model for Biomedical Signal Synthesiscited by 9
  5. 2024Universal Functional Regression with Neural Operator Flowscited by 9
  6. 2025Generating Full-field Evolution of Physical Dynamics from Irregular Sparse Observationscited by 9
  7. 2024∞-Brush: Controllable Large Image Synthesis with Diffusion Models in Infinite Dimensionscited by 8
  8. 2025Learning Reconfigurable Representations for Multimodal Federated Learning with Missing Datacited by 8
  9. 2026PerturbDiff: Functional Diffusion for Single-Cell Perturbation Modelingcited by 8
  10. 2024Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problemscited by 7
  11. 2023PriorCVAE: scalable MCMC parameter inference with Bayesian deep generative modellingcited by 6
  12. 2025Flow Straight and Fast in Hilbert Space: Functional Rectified Flowcited by 6
  13. 2025Flow Matching Neural Processescited by 6
  14. 2025Probability-Flow ODE in Infinite-Dimensional Function Spacescited by 5
  15. 2025Preconditioned Langevin Dynamics with Score-Based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problemscited by 5
  16. 2025Scale-Adaptive Generative Flows for Multiscale Scientific Datacited by 5
  17. 2025Functional Mean Flow in Hilbert Spacecited by 5
  18. 2026Physics-informed diffusion models in spectral spacecited by 5
  19. 2024SigDiffusions: Score-Based Diffusion Models for Time Series via Log-Signature Embeddingscited by 4
  20. 2024Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulationcited by 4
  21. 2024Regularized KL-Divergence for Well-Defined Function-Space Variational Inference in Bayesian neural networkscited by 3
  22. 2025Optimize Any Topology: A Foundation Model for Shape- and Resolution-Free Structural Topology Optimizationcited by 3
  23. 2025PRISMA: Improving the Accuracy-Latency Frontier of Diffusion-based PDE Solvers Using Physics-Informed Spectral Attentioncited by 3
  24. 2026Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamicscited by 3
  25. 2024Can neural operators always be continuously discretized?cited by 2
  26. 2025Residual-augmented flow matching operators for probabilistic partial differential equationscited by 2
  27. 2023DFU: scale-robust diffusion model for zero-shot super-resolution image generationcited by 1
  28. 2025A Review of Diffusion-based Simulation-Based Inference: Foundations and Applications in Non-Ideal Data Scenarioscited by 1
  29. 2026Optimal-Transport-Guided Functional Flow Matching for Turbulent Field Generation in Hilbert Spacecited by 1
  30. 2026Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inferencecited by 1
  31. 2024Mean-field Chaos Diffusion Modelscited by 0
  32. 2024Deep Optimal Sensor Placement for Black Box Stochastic Simulationscited by 0
  33. 2025FuncGenFoil: Airfoil Generation and Editing Model in Function Spacecited by 0
  34. 2026Generative Neural Operators through Diffusion Last Layercited by 0
  35. 2026Infinite-dimensional generative diffusions via Doob's h-transformcited by 0
  36. 2026Dimension-Uniform Discretization Analysis of Preconditioned Annealed Langevin Dynamics for Multimodal Gaussian Mixturescited by 0
  37. 2026Improving Function Space Flow Matching with Kernel Optimal Transportcited by 0

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  2. 2024GUD: Generation with Unified Diffusion
  3. 2026Stochastic Interpolants in Hilbert Spaces
  4. 2025Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction
  5. 2023Functional Diffusion
  6. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models
  7. 2023Diffusion Models for Generative Artificial Intelligence: An Introduction for Applied Mathematicians
  8. 2022Convergence of denoising diffusion models under the manifold hypothesis