Data-driven optimal prediction with control

TL;DR

This paper extends data-driven optimal prediction methods to controlled dynamical systems, using dynamic mode decomposition to efficiently predict averaged trajectories of unresolved variables, demonstrated on Hamiltonian systems.

Contribution

It introduces a novel data-driven approach combining optimal prediction and dynamic mode decomposition for controlled systems with unresolved variables.

Findings

Method is significantly faster than Monte Carlo simulations.

Approach reliably predicts averaged trajectories.

Effective on Hamiltonian dynamical systems.

Abstract

This study presents the extension of the data-driven optimal prediction approach to the dynamical system with control. The optimal prediction is used to analyze dynamical systems in which the states consist of resolved and unresolved variables. The latter variables can not be measured explicitly. They may have smaller amplitudes and affect the resolved variables that can be measured. The optimal prediction approach recovers the averaged trajectories of the resolved variables by computing conditional expectations, while the distribution of the unresolved variables is assumed to be known. We consider such dynamical systems and introduce their additional control functions. To predict the targeted trajectories numerically, we develop a data-driven method based on the dynamic mode decomposition. The proposed approach takes the $\mathit{measured}$ trajectories of the resolved variables, constructs an approximate linear operator from the Mori-Zwanzig decomposition, and reconstructs the $\mathit{averaged}$ trajectories of the same variables. It is demonstrated that the method is much faster than the Monte Carlo simulations and it provides a reliable prediction. We experimentally confirm the efficacy of the proposed method for two Hamiltonian dynamical systems.