mlsanityData to 7 Oct 2026

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

arXiv 2103.05744 ↗First posted 9 Mar 2021math.NA cs.LG cs.NAcited by 5

Not on the map. Why?

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.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 9 of its 21 references that are in the map. First posted on arXiv 9 Mar 2021.

Builds on

9 of 21 references.

References · 9

  1. 2016Error bounds for approximations with deep ReLU networkscited by 567
  2. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  3. 2019Deep Neural Network Approximation Theorycited by 92
  4. 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
  5. 2018Analysis 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 Equationscited by 44
  6. 2019Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equationscited by 40
  7. 2020Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditionscited by 25
  8. 2019Space-time error estimates for deep neural network approximations for differential equationscited by 6
  9. 2016Using Neural Networks to Compute Approximate and Guaranteed Feasible Hamilton-Jacobi-Bellman PDE Solutionscited by 5

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 0; 2024: 0; 2025: 3; 2026: 1.

5 papers.

  1. 2022Deep neural network expressivity for optimal stopping problemscited by 9
  2. 2025Is RL fine-tuning harder than regression? A PDE learning approach for diffusion modelscited by 3
  3. 2025Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditionscited by 1
  4. 2025DeepMartingale: Duality of the Optimal Stopping Problem with Expressivity and High-Dimensional Hedgingcited by 0
  5. 2026Neural operators approximate strongly continuous convex monotone semigroupscited by 0