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

TL;DR

This work addresses non-intrusive reduced-order modeling by employing regularized kernel interpolation in an RKHS to learn reduced dynamics from data. By leveraging feature-map kernels, the approach can embed known structure from the full-order model into the ROM, yielding interpretable dynamics and the option to incorporate closure terms via hybrid kernels. The authors derive a computable a posteriori error bound that couples kernel interpolation error with intrusive ROM error, and demonstrate competitive performance against operator inference and intrusive methods on advection-diffusion, Burgers', and Euler--Riemann problems. The framework supports POD and quadratic-manifold reductions, provides flexible kernel design strategies, and offers a path toward scalable, structure-preserving, data-driven ROMs with certifiable error behavior.

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.

Figures (11)

• Figure 1: Solutions of the full-order advection diffusion model \ref{['eq:advection_diffusion_fom']} with initial conditions \ref{['eq:advection_diffusion_initial_condition']} for various choices of ${\boldsymbol{\mu}}$.
• Figure 2: Relative ROM error at the test parameter $\bar{{\boldsymbol{\mu}}} = (0.3,0.1)$ as a function of number of basis vectors in linear POD (left) and quadratic manifold (right) reduced state approximations for the advection-diffusion problem \ref{['eq:advection_diffusion_pde']}.
• Figure 3: Effect of the quadratic manifold regularization parameter $\rho$ on the projection error and intrusive ROM error for two different reduced state dimensions $r$, using data from the advection-diffusion problem \ref{['eq:advection_diffusion_pde']} at the test parameter $\bar{{\boldsymbol{\mu}}} = (0.3, 0.1)$.
• Figure 4: Approximate error bounds for POD and QM Kernel ROMs for the advection-diffusion problem \ref{['eq:advection_diffusion_pde']}.
• Figure 5: Solutions of the full-order Burgers' model \ref{['eq:burgers_fom']} with initial conditions \ref{['eq:burgers_initial_condition']} for various choices of ${\boldsymbol{\mu}}$.

Theorems & Definitions (23)

• Definition 2.1: Positive-definite kernels
• Definition 2.2: RKHS
• Definition 2.3: Regularized kernel interpolant
• Theorem 2.1: Representer Theorem
• Theorem 2.2: Power function error bound
• Corollary 2.1: Vector Representer Theorem
• Corollary 2.2
• proof
• Remark 2.1: Input normalization
• Remark 3.1: Kronecker redundancy