# Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations

arXiv: https://arxiv.org/abs/2103.05744 · HTML: https://mlsanity.com/paper/2103.05744/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 21 references that are in the map.
- First posted on arXiv 9 Mar 2021.

## Builds on (most cited first, 9 of 9)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations](https://mlsanity.com/paper/1809.03062/) (2018)
- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions](https://mlsanity.com/paper/2007.05384/) (2020)
- [Space-time error estimates for deep neural network approximations for differential equations](https://mlsanity.com/paper/1908.03833/) (2019)
- [Using Neural Networks to Compute Approximate and Guaranteed Feasible Hamilton-Jacobi-Bellman PDE Solutions](https://mlsanity.com/paper/1611.03158/) (2016)

## Built on by (most cited first, 5 of 5)

- [Deep neural network expressivity for optimal stopping problems](https://mlsanity.com/paper/2210.10443/) (2022)
- [Is RL fine-tuning harder than regression? A PDE learning approach for diffusion models](https://mlsanity.com/paper/2509.02528/) (2025)
- [Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions](https://mlsanity.com/paper/2508.07559/) (2025)
- [DeepMartingale: Duality of the Optimal Stopping Problem with Expressivity and High-Dimensional Hedging](https://mlsanity.com/paper/2510.13868/) (2025)
- [Neural operators approximate strongly continuous convex monotone semigroups](https://mlsanity.com/paper/2609.02727/) (2026)

## Abstract (arXiv)

The approximation of solutions to second order Hamilton--Jacobi--Bellman (HJB) equations by deep neural networks is investigated. It is shown that for HJB equations that arise in the context of the optimal control of certain Markov processes the solution can be approximated by deep neural networks without incurring the curse of dimension. The dynamics is assumed to depend affinely on the controls and the cost depends quadratically on the controls. The admissible controls take values in a bounded set.