mlsanityData to 7 Oct 2026

Continuous-Time Functional Diffusion Processes

arXiv 2303.00800 ↗First posted 1 Mar 2023cs.LG stat.MLcited by 39

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

We introduce Functional Diffusion Processes (FDPs), which generalize score-based diffusion models to infinite-dimensional function spaces. FDPs require a new mathematical framework to describe the forward and backward dynamics, and several extensions to derive practical training objectives. These include infinite-dimensional versions of Girsanov theorem, in order to be able to compute an ELBO, and of the sampling theorem, in order to guarantee that functional evaluations in a countable set of points are equivalent to infinite-dimensional functions. We use FDPs to build a new breed of generative models in function spaces, which do not require specialized network architectures, and that can work with any kind of continuous data. Our results on real data show that FDPs achieve high-quality image generation, using a simple MLP architecture with orders of magnitude fewer parameters than existing diffusion models.

Key facts. Cited by 39 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 44 of its 81 references that are in the map. First posted on arXiv 1 Mar 2023.

Builds on

44 of 81 references.

Closest ancestors · 18

Same direction, newest first

  1. 2023Score-based Diffusion Models in Function Space14 Feb
  2. 2022Diffusion Generative Models in Infinite Dimensions1 Dec
  3. 2022From Points to Functions: Infinite-dimensional Representations in Diffusion Models25 Oct
  4. 2022Spectral Diffusion Processes28 Sep
  5. 2022Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise19 Aug
  6. 2022How Much is Enough? A Study on Diffusion Times in Score-based Generative Models10 Jun
  7. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  8. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  9. 2021Tackling the Generative Learning Trilemma with Denoising Diffusion GANs15 Dec
  10. 2021Score-Based Generative Modeling with Critically-Damped Langevin Diffusion14 Dec
  11. 2021Variational Diffusion Models1 Jul
  12. 2021Score-based Generative Modeling in Latent Space10 Jun
6 more
  1. 2021Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimation10 Jun
  2. 2021Cascaded Diffusion Models for High Fidelity Image Generation30 May
  3. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  4. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  5. 2020Denoising Diffusion Probabilistic Models19 Jun
  6. 2020Improved Techniques for Training Score-Based Generative Models16 Jun

Other references · 26

  1. 2017Attention Is All You Needcited by 59,932
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  3. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,539
  4. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,322
  5. 2017Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networkscited by 4,861
  6. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  7. 2014Deep Learning Face Attributes in the Wildcited by 3,679
  8. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  9. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,214
  10. 2022Imagen Video: High Definition Video Generation with Diffusion Modelscited by 1,765
  11. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,339
  12. 2021Diffusion Probabilistic Models for 3D Point Cloud Generationcited by 734
14 more
  1. 2022Equivariant Diffusion for Molecule Generation in 3Dcited by 666
  2. 2022LION: Latent Point Diffusion Models for 3D Shape Generationcited by 552
  3. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  4. 2021Choose a Transformer: Fourier or Galerkincited by 333
  5. 2020AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradientscited by 282
  6. 2022Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problemcited by 229
  7. 2023WIRE: Wavelet Implicit Neural Representationscited by 220
  8. 2022The Role of ImageNet Classes in Fr'echet Inception Distancecited by 211
  9. 2022From data to functa: Your data point is a function and you can treat it like onecited by 183
  10. 2022Generative Modelling With Inverse Heat Dissipationcited by 164
  11. 2021DiffSinger: Singing Voice Synthesis via Shallow Diffusion Mechanismcited by 154
  12. 2022COIN++: Neural Compression Across Modalitiescited by 81
  13. 2022Neural Diffusion Processescited by 50
  14. 2022Zero-shot Blind Image Denoising via Implicit Neural Representationscited by 12

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 7; 2024: 12; 2025: 10; 2026: 10.

39 papers.

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By area:

  1. 2023Functional Flow Matchingcited by 63
  2. 2023Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generationcited by 37
  3. 2023Conditional score-based diffusion models for Bayesian inference in infinite dimensionscited by 37
  4. 2024Dynamic Conditional Optimal Transport through Simulation-Free Flowscited by 24
  5. 2023∞-Diff: Infinite Resolution Diffusion with Subsampled Mollified Statescited by 21
  6. 2024Warped Diffusion: Solving Video Inverse Problems with Image Diffusion Modelscited by 21
  7. 2025FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modelingcited by 19
  8. 2024Stochastic Optimal Control for Diffusion Bridges in Function Spacescited by 16
  9. 2024Conditioning non-linear and infinite-dimensional diffusion processescited by 15
  10. 2023Functional Diffusioncited by 13
  11. 2023Geometric Neural Diffusion Processescited by 10
  12. 2024Wasserstein Bounds for generative diffusion models with Gaussian tail targetscited by 10
27 more
  1. 2024Universal Functional Regression with Neural Operator Flowscited by 9
  2. 2024Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problemscited by 7
  3. 2024Infinite-dimensional Diffusion Bridge Simulation via Operator Learningcited by 7
  4. 2024Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guaranteescited by 5
  5. 2025Probability-Flow ODE in Infinite-Dimensional Function Spacescited by 5
  6. 2025Preconditioned Langevin Dynamics with Score-Based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problemscited by 5
  7. 2025Flow Straight and Fast in Hilbert Space: Functional Rectified Flowcited by 5
  8. 2025Functional Mean Flow in Hilbert Spacecited by 5
  9. 2024Spectral Convolutional Conditional Neural Processescited by 4
  10. 2024Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulationcited by 4
  11. 2025Functional Adjoint Sampler: Scalable Sampling on Infinite Dimensional Spacescited by 4
  12. 2026Supervised Guidance Training for Infinite-Dimensional Diffusion Modelscited by 3
  13. 2026Physics-informed diffusion models in spectral spacecited by 3
  14. 2025Optimize Any Topology: A Foundation Model for Shape- and Resolution-Free Structural Topology Optimizationcited by 2
  15. 2026Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamicscited by 2
  16. 2023DFU: scale-robust diffusion model for zero-shot super-resolution image generationcited by 1
  17. 2026Drift Estimation for Stochastic Differential Equations with Denoising Diffusion Modelscited by 1
  18. 2026Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inferencecited by 1
  19. 2026LatentFlow: A General Framework for Conditioning Stochastic Processescited by 1
  20. 2024Deep Optimal Sensor Placement for Black Box Stochastic Simulationscited by 0
  21. 2025FuncGenFoil: Airfoil Generation and Editing Model in Function Spacecited by 0
  22. 2025Neural Bridge Processescited by 0
  23. 2025FunPhase: A Periodic Functional Autoencoder for Motion Generation via Phase Manifoldscited by 0
  24. 2026Infinite-dimensional generative diffusions via Doob's h-transformcited by 0
  25. 2026SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximationcited by 0
  26. 2026Dimension-Uniform Discretization Analysis of Preconditioned Annealed Langevin Dynamics for Multimodal Gaussian Mixturescited by 0
  27. 2026Approximating Measures on Function Spaces: Transport and Truncationcited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2023Infinite-Dimensional Diffusion Models
  2. 2022FP-Diffusion: Improving Score-based Diffusion Models by Enforcing the Underlying Score Fokker-Planck Equation
  3. 2024Fixed Point Diffusion Models
  4. 2025Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction
  5. 2024Input-Adaptive Generative Dynamics in Diffusion Models
  6. 2024Non-Normal Diffusion Models
  7. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Models
  8. 2025Geometric Regularity in Deterministic Sampling Dynamics of Diffusion-based Generative Models