Accurate error estimation for model reduction of nonlinear dynamical systems via data-enhanced error closure

TL;DR

This paper introduces a data-enhanced, residual-based a posteriori error estimator for nonlinear dynamical systems that is independent of the time integration scheme, improving accuracy and robustness of reduced-order models.

Contribution

It proposes a novel data-driven correction to residual-based error estimators, enabling scheme independence and better error control in model reduction.

Findings

The new error estimator is accurate across different systems.

The approach produces reduced models that generalize well.

Numerical results validate the effectiveness of the method.

Abstract

Accurate error estimation is crucial in model order reduction, both to obtain small reduced-order models and to certify their accuracy when deployed in downstream applications such as digital twins. In existing a posteriori error estimation approaches, knowledge about the time integration scheme is mandatory, e.g., the residual-based error estimators proposed for the reduced basis method. This poses a challenge when automatic ordinary differential equation solver libraries are used to perform the time integration. To address this, we present a data-enhanced approach for a posteriori error estimation. Our new formulation enables residual-based error estimators to be independent of any time integration method. To achieve this, we introduce a corrected reduced-order model which takes into account a data-driven closure term for improved accuracy. The closure term, subject to mild assumptions, is related to the local truncation error of the corresponding time integration scheme. We propose efficient computational schemes for approximating the closure term, at the cost of a modest amount of training data. Furthermore, the new error estimator is incorporated within a greedy process to obtain parametric reduced-order models. Numerical results on three different systems show the accuracy of the proposed error estimation approach and its ability to produce ROMs that generalize well.