mlsanityData to 14 Oct 2026

An optimal control perspective on diffusion-based generative modeling

arXiv 2211.01364 ↗First posted 2 Nov 2022cs.LG math.OC stat.MLcited by 147

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

We establish a connection between stochastic optimal control and generative models based on stochastic differential equations (SDEs), such as recently developed diffusion probabilistic models. In particular, we derive a Hamilton-Jacobi-Bellman equation that governs the evolution of the log-densities of the underlying SDE marginals. This perspective allows to transfer methods from optimal control theory to generative modeling. First, we show that the evidence lower bound is a direct consequence of the well-known verification theorem from control theory. Further, we can formulate diffusion-based generative modeling as a minimization of the Kullback-Leibler divergence between suitable measures in path space. Finally, we develop a novel diffusion-based method for sampling from unnormalized densities -- a problem frequently occurring in statistics and computational sciences. We demonstrate that our time-reversed diffusion sampler (DIS) can outperform other diffusion-based sampling approaches on multiple numerical examples.

Key facts. Cited by 147 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 52 of its 99 references that are in the map. First posted on arXiv 2 Nov 2022.

Builds on

52 of 99 references.

Closest ancestors · 13

Same direction, newest first

  1. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  2. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  3. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  4. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  5. 2022Neural Lagrangian Schr"odinger Bridge: Diffusion Modeling for Population Dynamics11 Apr
  6. 2021Variational Diffusion Models1 Jul
  7. 2021Score-based Generative Modeling in Latent Space10 Jun
  8. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matching5 Jun
  9. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  10. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  11. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  12. 2020Denoising Diffusion Probabilistic Models19 Jun
1 more
  1. 2020Improved Techniques for Training Score-Based Generative Models16 Jun

Other references · 39

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  4. 2016Densely Connected Convolutional Networkscited by 6,720
  5. 2016Gaussian Error Linear Units (GELUs)cited by 3,502
  6. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,316
  7. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  8. 2015Variational Inference with Normalizing Flowscited by 2,170
  9. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,354
  10. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,323
  11. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  12. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
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  1. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 384
  2. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 373
  3. 2022DiffDock: Diffusion Steps, Twists, and Turns for Molecular Dockingcited by 313
  4. 2020Scalable Gradients for Stochastic Differential Equationscited by 271
  5. 2022Torsional Diffusion for Molecular Conformer Generationcited by 263
  6. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 219
  7. 2021Path Integral Sampler: a stochastic control approach for samplingcited by 181
  8. 2020Stochastic Normalizing Flowscited by 157
  9. 2022Flow Annealed Importance Sampling Bootstrapcited by 127
  10. 2017Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inferencecited by 98
  11. 2019Theoretical guarantees for sampling and inference in generative models with latent diffusionscited by 97
  12. 2021Annealed Flow Transport Monte Carlocited by 90
  13. 2020Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path spacecited by 82
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  15. 2021Efficient and Accurate Gradients for Neural SDEscited by 71
  16. 2021Bayesian Learning via Neural Schr"odinger-F"ollmer Flowscited by 56
  17. 2022Score-Based Diffusion meets Annealed Importance Samplingcited by 51
  18. 2018Solving the Kolmogorov PDE by means of deep learningcited by 41
  19. 2021Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networkscited by 29
  20. 2020Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learningcited by 22
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 1; 2023: 10; 2024: 45; 2025: 54; 2026: 37.

147 papers.

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  1. 2024Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Controlcited by 233
  2. 2024Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Controlcited by 143
  3. 2024Iterated Denoising Energy Matching for Sampling from Boltzmann Densitiescited by 116
  4. 2023Improved sampling via learned diffusionscited by 104
  5. 2025Test-time Alignment of Diffusion Models without Reward Over-optimizationcited by 93
  6. 2024Physics-Informed Diffusion Modelscited by 90
  7. 2025Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matchingcited by 85
  8. 2023Spontaneous Symmetry Breaking in Generative Diffusion Modelscited by 80
  9. 2024Amortizing intractable inference in diffusion models for vision, language, and controlcited by 72
  10. 2024NETS: A Non-Equilibrium Transport Samplercited by 69
  11. 2024Particle Denoising Diffusion Samplercited by 65
  12. 2024A Diffusion Model Framework for Unsupervised Neural Combinatorial Optimizationcited by 59
38 more
  1. 2025DIME:Diffusion-Based Maximum Entropy Reinforcement Learningcited by 58
  2. 2023Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimizationcited by 57
  3. 2024Beyond ELBOs: A Large-Scale Evaluation of Variational Methods for Samplingcited by 56
  4. 2024Symbolic Music Generation with Non-Differentiable Rule Guided Diffusioncited by 54
  5. 2025Any-Order Flexible Length Masked Diffusioncited by 51
  6. 2024Improved off-policy training of diffusion samplerscited by 48
  7. 2024Sequential Controlled Langevin Diffusionscited by 48
  8. 2024DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transformcited by 46
  9. 2023Reverse Diffusion Monte Carlocited by 41
  10. 2023Stochastic Optimal Control Matchingcited by 40
  11. 2025Scalable Equilibrium Sampling with Sequential Boltzmann Generatorscited by 38
  12. 2024Model-Based Diffusion for Trajectory Optimizationcited by 32
  13. 2025Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distancecited by 29
  14. 2025Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physicscited by 29
  15. 2025Underdamped Diffusion Bridges with Applications to Samplingcited by 29
  16. 2023A mean-field games laboratory for generative modelingcited by 28
  17. 2024Stochastic Localization via Iterative Posterior Samplingcited by 28
  18. 2025No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplerscited by 28
  19. 2025Progressive Inference-Time Annealing of Diffusion Models for Sampling from Boltzmann Densitiescited by 26
  20. 2025Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inferencecited by 23
  21. 2025Variational Control for Guidance in Diffusion Modelscited by 22
  22. 2023Transport meets Variational Inference: Controlled Monte Carlo Diffusionscited by 21
  23. 2025Adjoint Schr"odinger Bridge Samplercited by 21
  24. 2025MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Controlcited by 21
  25. 2024Dynamical Measure Transport and Neural PDE Solvers for Samplingcited by 20
  26. 2025UniDB: A Unified Diffusion Bridge Framework via Stochastic Optimal Controlcited by 20
  27. 2024Stochastic Control for Fine-tuning Diffusion Models: Optimality, Regularity, and Convergencecited by 19
  28. 2025From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster trainingcited by 19
  29. 2024Stochastic Sampling from Deterministic Flow Modelscited by 18
  30. 2025Outsourced diffusion sampling: Efficient posterior inference in latent spaces of generative modelscited by 17
  31. 2025Accelerated Parallel Tempering via Neural Transportscited by 17
  32. 2024Stochastic Optimal Control for Diffusion Bridges in Function Spacescited by 16
  33. 2024Hamilton-Jacobi Reachability in Reinforcement Learning: A Surveycited by 16
  34. 2022DriftRec: Adapting diffusion models to blind JPEG restorationcited by 15
  35. 2024BNEM: A Boltzmann Sampler Based on Bootstrapped Noised Energy Matchingcited by 15
  36. 2025Temperature-Annealed Boltzmann Generatorscited by 15
  37. 2025Conditioning Diffusions Using Malliavin Calculuscited by 15
  38. 2025Value Gradient Guidance for Flow Matching Alignmentcited by 14

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  3. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
  4. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models
  5. 2026A Stochastic Analysis Perspective on Diffusion Models
  6. 2023A Geometric Perspective on Diffusion Models
  7. 2023Exploring the Optimal Choice for Generative Processes in Diffusion Models: Ordinary vs Stochastic Differential Equations
  8. 2026Formalizing the Sampling Design Space of Diffusion-Based Generative Models via Adaptive Solvers and Wasserstein-Bounded Timesteps