  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.
