Dynamical low-rank approximations of solutions to the Hamilton-Jacobi-Bellman equation

TL;DR

This paper introduces a new low-rank tensor train method for approximating solutions to the Hamilton-Jacobi-Bellman equation, significantly reducing computational costs while maintaining accuracy in nonlinear optimal control.

Contribution

The paper proposes a novel low-rank tensor train approach based on the Dirac-Frenkel principle with empirical risk optimization, improving efficiency over existing methods.

Findings

Reduced computational burden compared to state-of-the-art TT methods

Achieves comparable accuracy in approximating optimal feedback laws

Demonstrated effectiveness through numerical experiments

Abstract

We present a novel method to approximate optimal feedback laws for nonlinear optimal control based on low-rank tensor train (TT) decompositions. The approach is based on the Dirac-Frenkel variational principle with the modification that the optimisation uses an empirical risk. Compared to current state-of-the-art TT methods, our approach exhibits a greatly reduced computational burden while achieving comparable results. A rigorous description of the numerical scheme and demonstrations of its performance are provided.