# Stochastic Process Learning via Operator Flow Matching

arXiv: https://arxiv.org/abs/2501.04126 · HTML: https://mlsanity.com/paper/2501.04126/

## Key facts

- Cited by 14 papers in the map (data through 14 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.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 38)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Pyramidal Flow Matching for Efficient Video Generative Modeling](https://mlsanity.com/paper/2410.05954/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [MetaSDF: Meta-learning Signed Distance Functions](https://mlsanity.com/paper/2006.09662/) (2020)
- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Doubly Stochastic Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/1705.08933/) (2017)
- [Convolutional Conditional Neural Processes](https://mlsanity.com/paper/1910.13556/) (2019)
- [Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1512.07666/) (2015)
- [Generative Models as Distributions of Functions](https://mlsanity.com/paper/2102.04776/) (2021)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [Generative Adversarial Neural Operators](https://mlsanity.com/paper/2205.03017/) (2022)
- [Seismic wave propagation and inversion with Neural Operators](https://mlsanity.com/paper/2108.05421/) (2021)
- [Neural Diffusion Processes](https://mlsanity.com/paper/2206.03992/) (2022)
- [Diffusion Generative Models in Infinite Dimensions](https://mlsanity.com/paper/2212.00886/) (2022)

## Built on by (most cited first, 14 of 14)

- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Mesh-Informed Neural Operator : A Transformer Generative Approach](https://mlsanity.com/paper/2506.16656/) (2025)
- [Decoupled Diffusion Sampling for Inverse Problems on Function Spaces](https://mlsanity.com/paper/2601.23280/) (2026)
- [Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computing](https://mlsanity.com/paper/2604.07366/) (2026)
- [Is Flow Matching Just Trajectory Replay for Sequential Data?](https://mlsanity.com/paper/2602.08318/) (2026)
- [Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experiments](https://mlsanity.com/paper/2507.17887/) (2025)
- [Residual-augmented flow matching operators for probabilistic partial differential equations](https://mlsanity.com/paper/2512.12749/) (2025)
- [Discretization and Statistical Consistency of Functional Flow Matching](https://mlsanity.com/paper/2608.04531/) (2026)
- [FuncGenFoil: Airfoil Generation and Editing Model in Function Space](https://mlsanity.com/paper/2502.10712/) (2025)
- [Generative Neural Operators through Diffusion Last Layer](https://mlsanity.com/paper/2602.04139/) (2026)
- [Large-Scale 3D Ground-Motion Synthesis with Physics-Inspired Latent Operator Flow Matching](https://mlsanity.com/paper/2603.17403/) (2026)
- [Flow Annealing Posterior Sampling for Function-Space Regression and Inverse Problems](https://mlsanity.com/paper/2606.22346/) (2026)
- [Orca: Neural Operators for Causal Reasoning in Continuous Time](https://mlsanity.com/paper/2607.27867/) (2026)
- [Deep operator learning for efficient sampling from invariant measures of stochastic differential equations](https://mlsanity.com/paper/2609.11376/) (2026)

## 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.