We present novel model reduction methods for rapid solution of parametrized nonlinear partial differential equations (PDEs) in real-time or many-query contexts. Our approach combines reduced basis (RB) space for rapidly convergent approximation of the parametric solution manifold, Galerkin projection of the underlying PDEs onto the RB space for dimensionality reduction, and high-order empirical interpolation for efficient treatment of the nonlinear terms. We propose a class of high-order empirical interpolation methods to derive basis functions and interpolation points by using high-order partial derivatives of the nonlinear terms. As these methods can generate high-quality basis functions and interpolation points from a snapshot set of full-order model (FOM) solutions, they significantly improve the approximation accuracy. We develop effective a posteriori estimator to quantify the interpolation errors and construct a parameter sample via greedy sampling. Furthermore, we implement two hyperreduction schemes to construct efficient reduced-order models: one that applies the empirical interpolation before Newton's method and another after. The latter scheme shows flexibility in controlling hyperreduction errors. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed methods.

翻译：本文提出了一种新颖的模型降阶方法，用于在实时或多查询场景中快速求解参数化非线性偏微分方程（PDEs）。该方法结合了用于快速逼近参数化解流形的降阶基（RB）空间、将原始偏微分方程通过伽辽金投影映射到降阶基空间以实现维度约简，以及利用高阶经验插值高效处理非线性项。我们提出了一类高阶经验插值方法，通过利用非线性项的高阶偏导数来构造基函数和插值点。由于这些方法能够从全阶模型（FOM）解的样本集中生成高质量的基函数和插值点，从而显著提高了逼近精度。我们开发了有效的后验误差估计器来量化插值误差，并通过贪婪采样构建参数样本集。此外，我们实现了两种超降阶方案来构建高效的降阶模型：一种在牛顿法之前应用经验插值，另一种在之后应用。后一种方案在控制超降阶误差方面表现出更高的灵活性。数值结果验证了所提方法的精度和效率。