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

## TL;DR

This paper introduces a neural network-based, data-driven approach to approximate solutions of high-dimensional Hamilton-Jacobi-Bellman equations, enabling real-time feedback control for complex nonlinear systems without discretizing the state space.

## Contribution

It presents a novel method that leverages physics-informed neural networks and adaptive data generation to solve high-dimensional HJB equations efficiently.

## Key findings

- Successfully applied to 6D rigid body attitude control
- Extended to systems of dimension up to 30
- Achieved real-time feedback control capabilities

## 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 semi-global 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.