Be greedy and learn: efficient and certified algorithms for parametrized optimal control problems

TL;DR

This paper introduces an efficient approach combining greedy reduced basis methods and machine learning to solve parametrized linear-quadratic optimal control problems with certified error bounds.

Contribution

It extends greedy control algorithms to include penalty terms and integrates machine learning surrogates for faster online evaluations with error certification.

Findings

Significant reduction in computational costs demonstrated

Effective error bounds provided for the combined approach

Numerical examples show high potential of the methodology

Abstract

We consider parametrized linear-quadratic optimal control problems and provide their online-efficient solutions by combining greedy reduced basis methods and machine learning algorithms. To this end, we first extend the greedy control algorithm, which builds a reduced basis for the manifold of optimal final time adjoint states, to the setting where the objective functional consists of a penalty term measuring the deviation from a desired state and a term describing the control energy. Afterwards, we apply machine learning surrogates to accelerate the online evaluation of the reduced model. The error estimates proven for the greedy procedure are further transferred to the machine learning models and thus allow for efficient a posteriori error certification. We discuss the computational costs of all considered methods in detail and show by means of two numerical examples the tremendous potential of the proposed methodology.