Interpretable and flexible non-intrusive reduced-order models using reproducing kernel Hilbert spaces

TL;DR

This paper introduces an interpretable, flexible non-intrusive reduced-order modeling method using regularized kernel interpolation within reproducing kernel Hilbert spaces, enabling structure-mirroring and error estimation.

Contribution

It proposes a novel kernel-based approach for non-intrusive ROMs that enhances interpretability and flexibility over existing methods by embedding feature maps and deriving error bounds.

Findings

Kernel interpolation yields optimal ROM dynamics approximation.

The method produces interpretable ROMs mirroring full-order models.

Numerical experiments demonstrate competitive accuracy and interpretability.

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.