# Continuous-Time Functional Diffusion Processes

arXiv: https://arxiv.org/abs/2303.00800 · HTML: https://mlsanity.com/paper/2303.00800/

## Key facts

- Cited by 39 papers in the map (data through 14 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.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 44)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [LION: Latent Point Diffusion Models for 3D Shape Generation](https://mlsanity.com/paper/2210.06978/) (2022)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem](https://mlsanity.com/paper/2206.04119/) (2022)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)

## Built on by (most cited first, 30 of 39)

- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Conditional score-based diffusion models for Bayesian inference in infinite dimensions](https://mlsanity.com/paper/2305.19147/) (2023)
- [Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation](https://mlsanity.com/paper/2303.04772/) (2023)
- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
- [∞-Diff: Infinite Resolution Diffusion with Subsampled Mollified States](https://mlsanity.com/paper/2303.18242/) (2023)
- [Warped Diffusion: Solving Video Inverse Problems with Image Diffusion Models](https://mlsanity.com/paper/2410.16152/) (2024)
- [FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling](https://mlsanity.com/paper/2506.07902/) (2025)
- [Stochastic Optimal Control for Diffusion Bridges in Function Spaces](https://mlsanity.com/paper/2405.20630/) (2024)
- [Conditioning non-linear and infinite-dimensional diffusion processes](https://mlsanity.com/paper/2402.01434/) (2024)
- [Functional Diffusion](https://mlsanity.com/paper/2311.15435/) (2023)
- [Geometric Neural Diffusion Processes](https://mlsanity.com/paper/2307.05431/) (2023)
- [Wasserstein Bounds for generative diffusion models with Gaussian tail targets](https://mlsanity.com/paper/2412.11251/) (2024)
- [Universal Functional Regression with Neural Operator Flows](https://mlsanity.com/paper/2404.02986/) (2024)
- [Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems](https://mlsanity.com/paper/2405.15676/) (2024)
- [Infinite-dimensional Diffusion Bridge Simulation via Operator Learning](https://mlsanity.com/paper/2405.18353/) (2024)
- [Flow Straight and Fast in Hilbert Space: Functional Rectified Flow](https://mlsanity.com/paper/2509.10384/) (2025)
- [Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guarantees](https://mlsanity.com/paper/2410.18153/) (2024)
- [Probability-Flow ODE in Infinite-Dimensional Function Spaces](https://mlsanity.com/paper/2503.10219/) (2025)
- [Preconditioned Langevin Dynamics with Score-Based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problems](https://mlsanity.com/paper/2505.18276/) (2025)
- [Functional Mean Flow in Hilbert Space](https://mlsanity.com/paper/2511.12898/) (2025)
- [Physics-informed diffusion models in spectral space](https://mlsanity.com/paper/2602.09708/) (2026)
- [Spectral Convolutional Conditional Neural Processes](https://mlsanity.com/paper/2404.13182/) (2024)
- [Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulation](https://mlsanity.com/paper/2410.13794/) (2024)
- [Functional Adjoint Sampler: Scalable Sampling on Infinite Dimensional Spaces](https://mlsanity.com/paper/2511.06239/) (2025)
- [Optimize Any Topology: A Foundation Model for Shape- and Resolution-Free Structural Topology Optimization](https://mlsanity.com/paper/2510.23667/) (2025)
- [Supervised Guidance Training for Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2601.20756/) (2026)
- [Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamics](https://mlsanity.com/paper/2602.01449/) (2026)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [DFU: scale-robust diffusion model for zero-shot super-resolution image generation](https://mlsanity.com/paper/2401.06144/) (2023)
- [Drift Estimation for Stochastic Differential Equations with Denoising Diffusion Models](https://mlsanity.com/paper/2602.17830/) (2026)

## 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.