🤖 AI Summary
Existing non-intrusive reduced-order modeling (ROM) methods suffer from limited interpretability and structural flexibility. To address this, we propose a regularized kernel interpolation framework grounded in reproducing kernel Hilbert spaces (RKHS). Our method explicitly embeds physics-informed feature maps and nonlinear closure terms into the ROM structure, enabling optimal low-dimensional dynamical approximation without requiring access to full-order model operators. This work is the first to integrate kernel interpolation with ROM structural constraints, thereby unifying model interpretability and representational flexibility. We further derive a computable a posteriori error bound that jointly accounts for projection error and kernel approximation error. Numerical experiments demonstrate that, compared to classical operator inference approaches, our method achieves higher predictive accuracy, superior generalization across diverse problems, and enhanced robustness—while preserving explicit correspondence between the ROM structure and the underlying physical system.
📝 Abstract
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.