mlsanityData to 13 Oct 2026

Functional Flow Matching

arXiv 2305.17209 ↗First posted 26 May 2023cs.LG stat.MLcited by 65

On the map: Diffusion models → Flow matching, time series and science

Abstract (arXiv)

We propose Functional Flow Matching (FFM), a function-space generative model that generalizes the recently-introduced Flow Matching model to operate in infinite-dimensional spaces. Our approach works by first defining a path of probability measures that interpolates between a fixed Gaussian measure and the data distribution, followed by learning a vector field on the underlying space of functions that generates this path of measures. Our method does not rely on likelihoods or simulations, making it well-suited to the function space setting. We provide both a theoretical framework for building such models and an empirical evaluation of our techniques. We demonstrate through experiments on several real-world benchmarks that our proposed FFM method outperforms several recently proposed function-space generative models.

Key facts. Cited by 65 papers in the map (data through 13 Oct 2026). Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. Builds on 34 of its 71 references that are in the map. First posted on arXiv 26 May 2023.

Builds on

34 of 71 references.

Closest ancestors · 5

Same direction, newest first

  1. 2023Diffusion Models for Time Series Applications: A Survey1 May
  2. 2022Flow Matching for Generative Modeling6 Oct
  3. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow7 Sep
  4. 2021CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation7 Jul
  5. 2021Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting28 Jan

Other references · 29

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,952
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,028
  3. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,213
  4. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,339
  5. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,077
  6. 2021Improved Denoising Diffusion Probabilistic Modelscited by 3,751
  7. 2018Neural Ordinary Differential Equationscited by 3,750
  8. 2016Gaussian Error Linear Units (GELUs)cited by 3,472
  9. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,857
  10. 2022Imagen Video: High Definition Video Generation with Diffusion Modelscited by 1,768
  11. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,348
  12. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,317
17 more
  1. 2022Building Normalizing Flows with Stochastic Interpolantscited by 979
  2. 2021ILVR: Conditioning Method for Denoising Diffusion Probabilistic Modelscited by 635
  3. 2023Scaling up GANs for Text-to-Image Synthesiscited by 492
  4. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  5. 2022It's Raw! Audio Generation with State-Space Modelscited by 171
  6. 2021Neural SDEs as Infinite-Dimensional GANscited by 161
  7. 2020Multivariate Probabilistic Time Series Forecasting via Conditioned Normalizing Flowscited by 143
  8. 2023Score-based Diffusion Models in Function Spacecited by 81
  9. 2022Matching Normalizing Flows and Probability Paths on Manifoldscited by 57
  10. 2022Generative Adversarial Neural Operatorscited by 51
  11. 2022Diffusion Generative Models in Infinite Dimensionscited by 49
  12. 2021ScoreGrad: Multivariate Probabilistic Time Series Forecasting with Continuous Energy-based Generative Modelscited by 43
  13. 2023Continuous-Time Functional Diffusion Processescited by 39
  14. 2022Deep Latent State Space Models for Time-Series Generationcited by 37
  15. 2023Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generationcited by 37
  16. 2022Generalized Variational Inference in Function Spaces: Gaussian Measures meet Bayesian Deep Learningcited by 21
  17. 2022Energy-Based Models for Functional Data using Path Measure Tiltingcited by 5

Built on by

Cited by year (2026 to 13 Oct)

Citing papers per year: 2023: 1; 2024: 10; 2025: 21; 2026: 33.

65 papers.

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  1. 2024Informed Correctors for Discrete Diffusion Modelscited by 57
  2. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraintscited by 48
  3. 2024Flow Matching with Gaussian Process Priors for Probabilistic Time Series Forecastingcited by 41
  4. 2024Gradient-Free Generation for Hard-Constrained Systemscited by 37
  5. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 33
  6. 2024Flow matching achieves almost minimax optimal convergencecited by 28
  7. 2024Dynamic Conditional Optimal Transport through Simulation-Free Flowscited by 24
  8. 2024Trajectory Flow Matching with Applications to Clinical Time Series Modelingcited by 22
  9. 2024Warped Diffusion: Solving Video Inverse Problems with Image Diffusion Modelscited by 21
  10. 2025Stochastic Process Learning via Operator Flow Matchingcited by 14
  11. 2023Functional Diffusioncited by 13
  12. 2024Extended Flow Matching: a Method of Conditional Generation with Generalized Continuity Equationcited by 11
38 more
  1. 2025Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generationcited by 11
  2. 2025On the minimax optimality of Flow Matching through the connection to kernel density estimationcited by 9
  3. 2025Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problemscited by 8
  4. 2025Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generationcited by 7
  5. 2025Flow Matching Neural Processescited by 6
  6. 2025Probability-Flow ODE in Infinite-Dimensional Function Spacescited by 5
  7. 2025Distribution estimation via Flow Matching with Lipschitz guaranteescited by 5
  8. 2025Scale-Adaptive Generative Flows for Multiscale Scientific Datacited by 5
  9. 2025Flow Straight and Fast in Hilbert Space: Functional Rectified Flowcited by 5
  10. 2025Functional Mean Flow in Hilbert Spacecited by 5
  11. 2026Project and Generate: Divergence-Free Neural Operators for Incompressible Flowscited by 5
  12. 2026Topological Flow Matchingcited by 5
  13. 2024Spectral Convolutional Conditional Neural Processescited by 4
  14. 2024Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulationcited by 4
  15. 2025Functional Adjoint Sampler: Scalable Sampling on Infinite Dimensional Spacescited by 4
  16. 2026Constraint-Aware Flow Matching: Decision Aligned End-to-End Training for Constrained Samplingcited by 3
  17. 2025Generative Unordered Flow for Set-Structured Data Generationcited by 2
  18. 2025A Jump-Diffusion Framework for Irregular Time Series Generationcited by 2
  19. 2025Residual-augmented flow matching operators for probabilistic partial differential equationscited by 2
  20. 2026GenCP: Towards Generative Modeling Paradigm of Coupled Physicscited by 2
  21. 2026ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidancecited by 2
  22. 2026Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Oncecited by 2
  23. 2025Pairwise Optimal Transports for Training All-to-All Flow-Based Condition Transfer Modelcited by 1
  24. 2025PIRF: Physics-Informed Reward Fine-Tuning for Diffusion Modelscited by 1
  25. 2025Operator Flow Matching for Timeseries Forecastingcited by 1
  26. 2026Particle-Guided Diffusion for Gas-Phase Reaction Kineticscited by 1
  27. 2026Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spacescited by 1
  28. 2026Optimal-Transport-Guided Functional Flow Matching for Turbulent Field Generation in Hilbert Spacecited by 1
  29. 2026Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inferencecited by 1
  30. 2026GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domainscited by 1
  31. 2026Generative Modeling of Quantum Distribution with Functional Flow Matchingcited by 1
  32. 2026LatentFlow: A General Framework for Conditioning Stochastic Processescited by 1
  33. 2026Discretization and Statistical Consistency of Functional Flow Matchingcited by 1
  34. 2025FuncGenFoil: Airfoil Generation and Editing Model in Function Spacecited by 0
  35. 2026Nearly Optimal Bayesian Inference for Structural Missingnesscited by 0
  36. 2026Generative Neural Operators through Diffusion Last Layercited by 0
  37. 2026Correcting Source Mismatch in Flow Matching with Radial-Angular Transportcited by 0
  38. 2026Prior-Fitted Functional Flow: In-Context Generative Models for Pharmacokineticscited by 0

50 most cited of 65 All 65 (JSON)

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  1. 2026A Flow Matching Algorithm for Many-Shot Adaptation to Unseen Distributions
  2. 2024Explicit Flow Matching: On The Theory of Flow Matching Algorithms with Applications
  3. 2026Flow Matching with Arbitrary Auxiliary Paths
  4. 2024Convergence Analysis of Flow Matching in Latent Space with Transformers
  5. 2024Local Flow Matching Generative Models
  6. 2025Smooth Flow Matching for Synthesizing Functional Data
  7. 2026One-Sided Quantile Coupling for Flow Matching
  8. 2024Categorical Flow Matching on Statistical Manifolds