# Adaptive Deep Learning for High Dimensional Hamilton-Jacobi-Bellman Equations

@article{NakamuraZimmerer2021AdaptiveDL, title={Adaptive Deep Learning for High Dimensional Hamilton-Jacobi-Bellman Equations}, author={Tenavi Nakamura-Zimmerer and Qi Gong and Wei Kang}, journal={SIAM J. Sci. Comput.}, year={2021}, volume={43}, pages={A1221-A1247} }

Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton-Jacobi-Bellman (HJB) equations, which, in high dimensions, are notoriously difficult. Existing strategies for high dimensional problems generally rely on specific, restrictive problem structures, or are valid only locally around some nominal trajectory. In this paper, we propose a data-driven method to approximate semi-global solutions to HJB equations for general high dimensional nonlinear systems and…

## 48 Citations

Deep neural network approximations for the stable manifolds of the Hamilton-Jacobi equations

- Mathematics, Computer Science
- 2020

This paper rigorously proves that if an approximation is sufficiently close to the exact stable manifold of the HJB equation, then the corresponding control derived from this approximation is near optimal, and proposes a deep learning method to approximate the stable manifolds.

Actor-Critic Method for High Dimensional Static Hamilton-Jacobi-Bellman Partial Differential Equations based on Neural Networks

- Computer Science, MathematicsSIAM Journal on Scientific Computing
- 2021

A novel numerical method for high dimensional Hamilton–Jacobi– Bellman (HJB) type elliptic partial differential equations (PDEs) based on neural network parametrization of the value and control functions using stochastic calculus is proposed.

A Neural Network Approach for High-Dimensional Optimal Control

- Computer Science
- 2021

A neural network approach for solving high-dimensional optimal control problems arising in real-time applications by fusing the Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman approaches and parameterizing the value function with a neural network is proposed.

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

- Computer Science, MathematicsArXiv
- 2021

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.

A Neural Network Approach for Real-Time High-Dimensional Optimal Control

- Computer Science
- 2021

A neural network approach for solving high-dimensional optimal control problems arising in real-time applications that fuse the HamiltonJacobi-Bellman (HJB) and Pontryagin Maximum Principle approaches by parameterizing the value function with an NN, and empirically observe that the number of parameters in the approach scales linearly with the dimension of the control problem, thereby mitigating the curse of dimensionality.

Approximating optimal feedback controllers of finite horizon control problems using hierarchical tensor formats

- Mathematics, Computer ScienceArXiv
- 2021

A linear error propagation with respect to the time discretization is proved and numerical evidence is given by controlling a diffusion equation with unstable reaction term and an Allen-Kahn equation.

Data-Driven Recovery of Optimal Feedback Laws through Optimality Conditions and Sparse Polynomial Regression

- Computer Science
- 2020

An extended set of low and high-dimensional numerical tests in nonlinear optimal control reveal that enriching the dataset with gradient information reduces the number of training samples, and that the sparse polynomial regression consistently yields a feedback law of lower complexity.

Neural Network Optimal Feedback Control with Guaranteed Local Stability

- Computer ScienceArXiv
- 2022

Several novel NN architectures are proposed, which show guarantee local stability while retaining the semi-global approximation capacity to learn the optimal feedback policy, and are found to be near-optimal in testing.

A Tensor Decomposition Approach for High-Dimensional Hamilton-Jacobi-Bellman Equations

- MathematicsArXiv
- 2019

The proposed method combines a tensor train approximation for the value function together with a Newton-like iterative method for the solution of the resulting nonlinear system, solving Hamilton-Jacobi equations with more than 100 dimensions at modest cost.

Data-Driven Computational Methods for the Domain of Attraction and Zubov's Equation

- Computer Science, MathematicsArXiv
- 2021

It is proved that a neural network approximation exists for the Lyapunov function of power systems such that the approximation error is a cubic polynomial of the number of generators.

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