Adaptive Deep Learning for High-dimensional Hamilton-Jacobi-Bellman Equations
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Authors
Nakamura-Zimmerer, Tenavi
Gong, Qi
Kang, Wei
Subjects
Hamilton--Jacobi--Bellman equations||optimal feedback control||deep learning||neural networks||nonlinear dynamical systems||optimization
Advisors
Date of Issue
2021
Date
2021
Publisher
SIAM
Language
Abstract
Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton--Jacobi--Bellman (HJB) equations, which are notoriously difficult when the state dimension is large. Existing strategies for high-dimensional problems often 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 semiglobal solutions to HJB equations for general high-dimensional nonlinear systems and compute candidate optimal feedback controls in real-time. To accomplish this, we model solutions to HJB equations with neural networks (NNs) trained on data generated without discretizing the state space. Training is made more effective and data-efficient by leveraging the known physics of the problem and using the partially trained NN to aid in adaptive data generation. We demonstrate the effectiveness of our method by learning solutions to HJB equations corresponding to the attitude control of a six-dimensional nonlinear rigid body and nonlinear systems of dimension up to 30 arising from the stabilization of a Burgers'-type partial differential equation. The trained NNs are then used for real-time feedback control of these systems.
Type
Preprint
Description
The article of record as published may be found at http://dx.doi.org/10.1137/19M1288802
Series/Report No
Department
Applied Mathematics
Organization
Identifiers
NPS Report Number
Sponsors
Defense Advanced Research Projects Agency (DARPA)
Funder
The work of the first and second authors was partially supported with funding from the Defense Advanced Research Projects Agency (DARPA) grant FA8650-18-1-7842.
Format
27 p.
Citation
Nakamura-Zimmerer, Tenavi, Qi Gong, and Wei Kang. "Adaptive Deep Learning for High-Dimensional Hamilton--Jacobi--Bellman Equations." SIAM Journal on Scientific Computing 43.2 (2021): A1221-A1247.
Distribution Statement
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States