This paper develops an interpretable, non-intrusive reduced-order modeling technique using regularized kernel interpolation. Existing non-intrusive approaches approximate the dynamics of a reduced-order model (ROM) by solving a data-driven least-squares regression problem for low-dimensional matrix operators. Our approach instead leverages regularized kernel interpolation, which yields an optimal approximation of the ROM dynamics from a user-defined reproducing kernel Hilbert space. We show that our kernel-based approach can produce interpretable ROMs whose structure mirrors full-order model structure by embedding judiciously chosen feature maps into the kernel. The approach is flexible and allows a combination of informed structure through feature maps and closure terms via more general nonlinear terms in the kernel. We also derive a computable a posteriori error bound that combines standard error estimates for intrusive projection-based ROMs and kernel interpolants. The approach is demonstrated in several numerical experiments that include comparisons to operator inference using both proper orthogonal decomposition and quadratic manifold dimension reduction.

翻译：本文提出了一种利用正则化核插值的可解释非侵入式降阶建模技术。现有非侵入式方法通过求解低维矩阵算子的数据驱动最小二乘回归问题来逼近降阶模型的动力学特性。与此不同，我们的方法采用正则化核插值技术，从用户定义的再生核希尔伯特空间中获取降阶模型动力学的最优逼近。我们证明，通过将精心设计的特征映射嵌入核函数，该核方法能够构建出与全阶模型结构相呼应的可解释降阶模型。该方法具有灵活性，既可通过特征映射融入先验结构信息，又能借助核函数中更一般的非线性项实现闭合项建模。我们还推导了可计算的后验误差界，该误差界综合了基于侵入式投影的降阶模型标准误差估计与核插值误差估计。通过若干数值实验验证了该方法的有效性，实验包含与基于本征正交分解及二次流形降维的算子推断方法的对比分析。