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

arXiv: https://arxiv.org/abs/2005.05409 · HTML: https://mlsanity.com/paper/2005.05409/

## Key facts

- Cited by 82 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 May 2020.

## Built on by (most cited first, 30 of 82)

- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
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- [Sequential Controlled Langevin Diffusions](https://mlsanity.com/paper/2412.07081/) (2024)
- [DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transform](https://mlsanity.com/paper/2406.01781/) (2024)
- [Stochastic Optimal Control Matching](https://mlsanity.com/paper/2312.02027/) (2023)
- [LEAPS: A discrete neural sampler via locally equivariant networks](https://mlsanity.com/paper/2502.10843/) (2025)
- [Underdamped Diffusion Bridges with Applications to Sampling](https://mlsanity.com/paper/2503.01006/) (2025)
- [Trajectory Balance with Asynchrony: Decoupling Exploration and Learning for Fast, Scalable LLM Post-Training](https://mlsanity.com/paper/2503.18929/) (2025)
- [No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplers](https://mlsanity.com/paper/2502.06685/) (2025)
- [Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inference](https://mlsanity.com/paper/2508.12511/) (2025)
- [Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems](https://mlsanity.com/paper/2112.03749/) (2021)
- [Transport meets Variational Inference: Controlled Monte Carlo Diffusions](https://mlsanity.com/paper/2307.01050/) (2023)
- [MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Control](https://mlsanity.com/paper/2508.10684/) (2025)
- [Dynamical Measure Transport and Neural PDE Solvers for Sampling](https://mlsanity.com/paper/2407.07873/) (2024)
- [Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs](https://mlsanity.com/paper/2311.15283/) (2023)
- [From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster training](https://mlsanity.com/paper/2501.06148/) (2025)
- [Solving high-dimensional parabolic PDEs using the tensor train format](https://mlsanity.com/paper/2102.11830/) (2021)
- [Outsourced diffusion sampling: Efficient posterior inference in latent spaces of generative models](https://mlsanity.com/paper/2502.06999/) (2025)
- [Stochastic Optimal Control for Diffusion Bridges in Function Spaces](https://mlsanity.com/paper/2405.20630/) (2024)
- [Proximal Diffusion Neural Sampler](https://mlsanity.com/paper/2510.03824/) (2025)
- [Stein Variational Gradient Descent: many-particle and long-time asymptotics](https://mlsanity.com/paper/2102.12956/) (2021)
- [Deep learning probability flows and entropy production rates in active matter](https://mlsanity.com/paper/2309.12991/) (2023)
- [Conditioning Diffusions Using Malliavin Calculus](https://mlsanity.com/paper/2504.03461/) (2025)
- [TR2-D2: Tree Search Guided Trajectory-Aware Fine-Tuning for Discrete Diffusion](https://mlsanity.com/paper/2509.25171/) (2025)
- [Value Gradient Guidance for Flow Matching Alignment](https://mlsanity.com/paper/2512.05116/) (2025)
- [Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning](https://mlsanity.com/paper/2206.10588/) (2022)
- [A Taxonomy of Loss Functions for Stochastic Optimal Control](https://mlsanity.com/paper/2410.00345/) (2024)

## 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.