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.

翻译：精确误差估计在模型降阶中至关重要，既有助于获得小型化降阶模型，又能确保其在数字孪生等下游应用中的精度。现有后验误差估计方法（如为降基法提出的残差型误差估计器）需强制已知时间积分格式，这给使用常微分方程求解器库进行时间积分带来挑战。为此，我们提出一种数据增强的后验误差估计方法。新公式使残差型误差估计器与时间积分方法无关。通过引入包含数据驱动闭合项的修正降阶模型提升精度，该闭合项在温和假设下与对应时间积分格式的局部截断误差相关。我们提出仅需适度训练数据即可高效近似闭合项的计算方案，并将新误差估计器纳入贪婪算法框架以获取参数化降阶模型。对三个不同系统的数值结果表明，所提误差估计方法具有高精度，且能生成泛化性能良好的降阶模型。