mlsanityData to 7 Oct 2026

Stochastic Process Learning via Operator Flow Matching

arXiv 2501.04126 ↗First posted 7 Jan 2025cs.LGcited by 14

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

Abstract (arXiv)

Expanding on neural operators, we propose a novel framework for stochastic process learning across arbitrary domains. In particular, we develop operator flow matching (OFM) for learning stochastic process priors on function spaces. OFM provides the probability density of the values of any collection of points and enables mathematically tractable functional regression at new points with mean and density estimation. Our method outperforms state-of-the-art models in stochastic process learning, functional regression, and prior learning.

Key facts. Cited by 14 papers in the map (data through 7 Oct 2026). Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. Builds on 38 of its 59 references that are in the map. First posted on arXiv 7 Jan 2025.

Builds on

38 of 59 references.

Closest ancestors · 4

Same direction, newest first

  1. 2023Functional Flow Matching26 May
  2. 2023Improving and generalizing flow-based generative models with minibatch optimal transport1 Feb
  3. 2022Flow Matching for Generative Modeling6 Oct
  4. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow7 Sep

Other references · 34

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  3. 2018Neural Ordinary Differential Equationscited by 3,733
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  5. 2016Density estimation using Real NVPcited by 2,029
  6. 2022DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Stepscited by 1,961
  7. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  8. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  9. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
  10. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 584
  11. 2024Pyramidal Flow Matching for Efficient Video Generative Modelingcited by 354
  12. 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 293
22 more
  1. 2019Attentive Neural Processescited by 280
  2. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  3. 2020MetaSDF: Meta-learning Signed Distance Functionscited by 214
  4. 2018When Gaussian Process Meets Big Data: A Review of Scalable GPscited by 205
  5. 2017Doubly Stochastic Variational Inference for Deep Gaussian Processescited by 143
  6. 2019Convolutional Conditional Neural Processescited by 139
  7. 2015Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networkscited by 121
  8. 2021Generative Models as Distributions of Functionscited by 89
  9. 2023Score-based Diffusion Models in Function Spacecited by 80
  10. 2020A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challengescited by 51
  11. 2021Seismic wave propagation and inversion with Neural Operatorscited by 51
  12. 2022Generative Adversarial Neural Operatorscited by 51
  13. 2022Neural Diffusion Processescited by 50
  14. 2022Diffusion Generative Models in Infinite Dimensionscited by 49
  15. 2022Real-time high-resolution CO2 geological storage prediction using nested Fourier neural operatorscited by 47
  16. 2023Infinite-Dimensional Diffusion Modelscited by 38
  17. 2020Transforming Gaussian Processes With Normalizing Flowscited by 28
  18. 2024Flowing from Words to Pixels: A Noise-Free Framework for Cross-Modality Evolutioncited by 26
  19. 2024Dynamic Conditional Optimal Transport through Simulation-Free Flowscited by 24
  20. 2020Deep Sigma Point Processescited by 16
  21. 2023Broadband Ground Motion Synthesis via Generative Adversarial Neural Operators: Development and Validationcited by 12
  22. 2024Universal Functional Regression with Neural Operator Flowscited by 9

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 5; 2026: 9.

14 papers.

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  1. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 33
  2. 2025Mesh-Informed Neural Operator : A Transformer Generative Approachcited by 11
  3. 2026Decoupled Diffusion Sampling for Inverse Problems on Function Spacescited by 9
  4. 2026Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computingcited by 4
  5. 2025Residual-augmented flow matching operators for probabilistic partial differential equationscited by 2
  6. 2026Is Flow Matching Just Trajectory Replay for Sequential Data?cited by 2
  7. 2025Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experimentscited by 1
  8. 2026Discretization and Statistical Consistency of Functional Flow Matchingcited by 1
  9. 2025FuncGenFoil: Airfoil Generation and Editing Model in Function Spacecited by 0
  10. 2026Generative Neural Operators through Diffusion Last Layercited by 0
  11. 2026Large-Scale 3D Ground-Motion Synthesis with Physics-Inspired Latent Operator Flow Matchingcited by 0
  12. 2026Flow Annealing Posterior Sampling for Function-Space Regression and Inverse Problemscited by 0
2 more
  1. 2026Orca: Neural Operators for Causal Reasoning in Continuous Timecited by 0
  2. 2026Deep operator learning for efficient sampling from invariant measures of stochastic differential equationscited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2025Flow Matching Neural Processes
  2. 2026Improving Function Space Flow Matching with Kernel Optimal Transport
  3. 2026Random Process Flow Matching: Generative Implicit Representations of Multivariate Random Fields
  4. 2025Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans
  5. 2026Latent Process Generator Matching
  6. 2025Multitask Learning with Stochastic Interpolants
  7. 2024Operator-Informed Score Matching for Markov Diffusion Models
  8. 2025Functional Mean Flow in Hilbert Space