mlsanityData to 7 Oct 2026

Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space

arXiv 2005.05409 ↗First posted 11 May 2020math.OC cs.LG cs.NAcited by 80

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Cited by 23 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

Optimal control of diffusion processes is intimately connected to the problem of solving certain Hamilton-Jacobi-Bellman equations. Building on recent machine learning inspired approaches towards high-dimensional PDEs, we investigate the potential of textititerative diffusion optimisation techniques, in particular considering applications in importance sampling and rare event simulation, and focusing on problems without diffusion control, with linearly controlled drift and running costs that depend quadratically on the control. More generally, our methods apply to nonlinear parabolic PDEs with a certain shift invariance. The choice of an appropriate loss function being a central element in the algorithmic design, we develop a principled framework based on divergences between path measures, encompassing various existing methods. Motivated by connections to forward-backward SDEs, we propose and study the novel textitlog-variance divergence, showing favourable properties of corresponding Monte Carlo estimators. The promise of the developed approach is exemplified by a range of high-dimensional and metastable numerical examples.

Key facts. Cited by 80 papers in the map (data through 7 Oct 2026). First posted on arXiv 11 May 2020.

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Citing papers per year: 2021: 5; 2022: 4; 2023: 11; 2024: 17; 2025: 28; 2026: 15.

80 papers.

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  1. 2022An optimal control perspective on diffusion-based generative modelingcited by 145
  2. 2023Improved sampling via learned diffusionscited by 103
  3. 2024Amortizing intractable inference in diffusion models for vision, language, and controlcited by 70
  4. 2021Bayesian Learning via Neural Schr"odinger-F"ollmer Flowscited by 56
  5. 2024Improved off-policy training of diffusion samplerscited by 47
  6. 2024Sequential Controlled Langevin Diffusionscited by 47
  7. 2024DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transformcited by 44
  8. 2023Stochastic Optimal Control Matchingcited by 39
  9. 2025LEAPS: A discrete neural sampler via locally equivariant networkscited by 30
  10. 2025Underdamped Diffusion Bridges with Applications to Samplingcited by 28
  11. 2025Trajectory Balance with Asynchrony: Decoupling Exploration and Learning for Fast, Scalable LLM Post-Trainingcited by 27
  12. 2025No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplerscited by 25
38 more
  1. 2025Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inferencecited by 22
  2. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  3. 2023Transport meets Variational Inference: Controlled Monte Carlo Diffusionscited by 21
  4. 2025MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Controlcited by 20
  5. 2023Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEscited by 19
  6. 2024Dynamical Measure Transport and Neural PDE Solvers for Samplingcited by 19
  7. 2021Solving high-dimensional parabolic PDEs using the tensor train formatcited by 17
  8. 2025From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster trainingcited by 17
  9. 2024Stochastic Optimal Control for Diffusion Bridges in Function Spacescited by 16
  10. 2025Outsourced diffusion sampling: Efficient posterior inference in latent spaces of generative modelscited by 16
  11. 2025Proximal Diffusion Neural Samplercited by 16
  12. 2021Stein Variational Gradient Descent: many-particle and long-time asymptoticscited by 15
  13. 2023Deep learning probability flows and entropy production rates in active mattercited by 15
  14. 2025Conditioning Diffusions Using Malliavin Calculuscited by 15
  15. 2025Value Gradient Guidance for Flow Matching Alignmentcited by 14
  16. 2022Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learningcited by 13
  17. 2025TR2-D2: Tree Search Guided Trajectory-Aware Fine-Tuning for Discrete Diffusioncited by 13
  18. 2024Solving Poisson Equations using Neural Walk-on-Spherescited by 11
  19. 2024A Taxonomy of Loss Functions for Stochastic Optimal Controlcited by 11
  20. 2025Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyondcited by 11
  21. 2024Transition Path Sampling with Improved Off-Policy Training of Diffusion Path Samplerscited by 10
  22. 2024Learned Reference-based Diffusion Sampling for multi-modal distributionscited by 9
  23. 2025Non-equilibrium Annealed Adjoint Samplercited by 8
  24. 2026Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matchingcited by 7
  25. 2021Gradient-augmented Supervised Learning of Optimal Feedback Laws Using State-dependent Riccati Equationscited by 6
  26. 2025Iterative Importance Fine-tuning of Diffusion Modelscited by 6
  27. 2025Deep Learning for Continuous-Time Stochastic Control with Jumpscited by 6
  28. 2025Adaptive Destruction Processes for Diffusion Samplerscited by 6
  29. 2025Control Consistency Losses for Diffusion Bridgescited by 6
  30. 2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited by 5
  31. 2023From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEscited by 5
  32. 2024Scalable Simulation-free Entropic Unbalanced Optimal Transportcited by 5
  33. 2025A brief review of the Deep BSDE method for solving high-dimensional partial differential equationscited by 5
  34. 2025Reinforced sequential Monte Carlo for amortised samplingcited by 5
  35. 2022Computational Doob's h-transforms for Online Filtering of Discretely Observed Diffusionscited by 4
  36. 2025Neural Implicit Solution Formula for Efficiently Solving Hamilton-Jacobi Equationscited by 4
  37. 2026Scalable Maximum Entropy Reinforcement Learning for Diffusion Policies via Adjoint Matchingcited by 4
  38. 2024Generative Modelling with Tensor Train approximations of Hamilton--Jacobi--Bellman equationscited by 3

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