QRnet: optimal regulator design with LQR-augmented neural networks

TL;DR

This paper introduces QRnet, a physics-informed neural network approach that augments LQR with neural networks to efficiently design optimal regulators for high-dimensional nonlinear systems, demonstrated on Burgers' equation.

Contribution

The paper presents a novel neural network-based method that directly solves high-dimensional Hamilton-Jacobi-Bellman equations without state space discretization.

Findings

Improved robustness over existing neural network formulations.

Effective handling of high-dimensional nonlinear control problems.

Successful application to unstable Burgers' equation.

Abstract

In this paper we propose a new computational method for designing optimal regulators for high-dimensional nonlinear systems. The proposed approach leverages physics-informed machine learning to solve high-dimensional Hamilton-Jacobi-Bellman equations arising in optimal feedback control. Concretely, we augment linear quadratic regulators with neural networks to handle nonlinearities. We train the augmented models on data generated without discretizing the state space, enabling application to high-dimensional problems. We use the proposed method to design a candidate optimal regulator for an unstable Burgers' equation, and through this example, demonstrate improved robustness and accuracy compared to existing neural network formulations.