mlsanityData to 7 Oct 2026

Stochastic Optimal Control for Diffusion Bridges in Function Spaces

arXiv 2405.20630 ↗First posted 31 May 2024cs.LGcited by 16

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Recent advancements in diffusion models and diffusion bridges primarily focus on finite-dimensional spaces, yet many real-world problems necessitate operations in infinite-dimensional function spaces for more natural and interpretable formulations. In this paper, we present a theory of stochastic optimal control (SOC) tailored to infinite-dimensional spaces, aiming to extend diffusion-based algorithms to function spaces. Specifically, we demonstrate how Doob's h-transform, the fundamental tool for constructing diffusion bridges, can be derived from the SOC perspective and expanded to infinite dimensions. This expansion presents a challenge, as infinite-dimensional spaces typically lack closed-form densities. Leveraging our theory, we establish that solving the optimal control problem with a specific objective function choice is equivalent to learning diffusion-based generative models. We propose two applications: (1) learning bridges between two infinite-dimensional distributions and (2) generative models for sampling from an infinite-dimensional distribution. Our approach proves effective for diverse problems involving continuous function space representations, such as resolution-free images, time-series data, and probability density functions.

Key facts. Cited by 16 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 47 of its 76 references that are in the map. First posted on arXiv 31 May 2024.

Builds on

47 of 76 references.

Closest ancestors · 17

Same direction, newest first

  1. 2024Conditioning non-linear and infinite-dimensional diffusion processes2 Feb
  2. 2023Non-Denoising Forward-Time Diffusions22 Dec
  3. 2023Generative Modeling with Phase Stochastic Bridges11 Oct
  4. 2023Denoising Diffusion Bridge Models29 Sep
  5. 2023Diffusion Schr"odinger Bridge Matching29 Mar
  6. 2023Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation8 Mar
  7. 2023Continuous-Time Functional Diffusion Processes1 Mar
  8. 2023Diffusion Probabilistic Fields1 Mar
  9. 2023Score-based Diffusion Models in Function Space14 Feb
  10. 2023I2SB: Image-to-Image Schr"odinger Bridge12 Feb
  11. 2022Scalable Diffusion Models with Transformers19 Dec
  12. 2022An optimal control perspective on diffusion-based generative modeling2 Nov
5 more
  1. 2022Spectral Diffusion Processes28 Sep
  2. 2022First Hitting Diffusion Models for Generating Manifold, Graph and Categorical Data2 Sep
  3. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  4. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  5. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 30

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,214
  3. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,170
  4. 2019StarGAN v2: Diverse Image Synthesis for Multiple Domainscited by 1,095
  5. 2023Improving and generalizing flow-based generative models with minibatch optimal transportcited by 835
  6. 2021Perceiver IO: A General Architecture for Structured Inputs & Outputscited by 616
  7. 2021CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputationcited by 577
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  9. 2018Conditional Neural Processescited by 421
  10. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
  11. 2021COIN: COmpression with Implicit Neural representationscited by 232
  12. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 216
18 more
  1. 2021Path Integral Sampler: a stochastic control approach for samplingcited by 175
  2. 2022Generative Modelling With Inverse Heat Dissipationcited by 164
  3. 2023Improved sampling via learned diffusionscited by 103
  4. 2019Theoretical guarantees for sampling and inference in generative models with latent diffusionscited by 97
  5. 2020Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path spacecited by 80
  6. 2023Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modelingcited by 80
  7. 2021Bayesian Learning via Neural Schr"odinger-F"ollmer Flowscited by 56
  8. 2022Neural Diffusion Processescited by 50
  9. 2022Deep Generalized Schr"odinger Bridgecited by 47
  10. 2019Function Space Particle Optimization for Bayesian Neural Networkscited by 41
  11. 2020Bootstrapping Neural Processescited by 40
  12. 2023Generalized Schr"odinger Bridge Matchingcited by 40
  13. 2022Modeling Temporal Data as Continuous Functions with Stochastic Process Diffusioncited by 39
  14. 2023Stochastic Optimal Control Matchingcited by 39
  15. 2023Conditional score-based diffusion models for Bayesian inference in infinite dimensionscited by 37
  16. 2017Deep Relaxation: partial differential equations for optimizing deep neural networkscited by 27
  17. 2020Generalization bound of globally optimal non-convex neural network training: Transportation map estimation by infinite dimensional Langevin dynamicscited by 14
  18. 2019Variational approach to rare event simulation using least-squares regressioncited by 12

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 4; 2026: 10.

16 papers.

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  1. 2025UniDB: A Unified Diffusion Bridge Framework via Stochastic Optimal Controlcited by 18
  2. 2024Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Seriescited by 9
  3. 2025Functional Adjoint Sampler: Scalable Sampling on Infinite Dimensional Spacescited by 4
  4. 2026Discrete Adjoint Schr"odinger Bridge Samplercited by 4
  5. 2024Conditional Lagrangian Wasserstein Flow for Time Series Imputationcited by 3
  6. 2025Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysiscited by 3
  7. 2026Supervised Guidance Training for Infinite-Dimensional Diffusion Modelscited by 3
  8. 2026Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Controlcited by 2
  9. 2026Inference-Time Attribute Distribution Alignment for Unconditional Diffusioncited by 2
  10. 2025A Unified and Fast-Sampling Diffusion Bridge Framework via Stochastic Optimal Controlcited by 1
  11. 2026Energy-oriented Diffusion Bridge for Image Restoration with Foundational Diffusion Modelscited by 1
  12. 2026Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inferencecited by 1
4 more
  1. 2026LatentFlow: A General Framework for Conditioning Stochastic Processescited by 1
  2. 2026Stochastic Interpolants in Hilbert Spacescited by 0
  3. 2026Infinite-dimensional generative diffusions via Doob's h-transformcited by 0
  4. 2026Sparsely-Supervised Data Assimilation via Physics-Informed Schr"odinger Bridgecited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2025Control Consistency Losses for Diffusion Bridges
  2. 2024Stochastic Control for Fine-tuning Diffusion Models: Optimality, Regularity, and Convergence
  3. 2023Infinite-Dimensional Diffusion Models
  4. 2022Diffusion Generative Models in Infinite Dimensions
  5. 2024Soft-constrained Schrodinger Bridge: a Stochastic Control Approach
  6. 2026Stochastic Optimal Control Sampling for Diffusion Inverse Problems
  7. 2025Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction
  8. 2025Underdamped Diffusion Bridges with Applications to Sampling