Non-linear reduced modeling of dynamical systems using kernel methods and low-rank approximation

TL;DR

This paper introduces a novel kernel-based reduced modeling approach for non-linear dynamical systems, embedding trajectories in RKHS to improve approximation accuracy and computational efficiency.

Contribution

It proposes an efficient algorithm leveraging low-rank approximation in RKHS for data-driven reduced modeling of non-linear dynamics, outperforming existing methods.

Findings

Enhanced approximation accuracy demonstrated by numerical simulations

Reduced computational complexity compared to prior approaches

Effective linearization in RKHS for non-linear systems

Abstract

Reduced modeling of a computationally demanding dynamical system aims at approximating its trajectories, while optimizing the trade-off between accuracy and computational complexity. In this work, we propose to achieve such an approximation by first embedding the trajectories in a reproducing kernel Hilbert space (RKHS), which exhibits appealing approximation and computational capabilities, and then solving the associated reduced model problem. More specifically, we propose a new efficient algorithm for data-driven reduced modeling of non-linear dynamics based on linear approximations in a RKHS. This algorithm takes advantage of the closed-form solution of a low-rank constraint optimization problem while exploiting advantageously kernel-based computations. Reduced modeling with this algorithm reveals a gain in approximation accuracy, as shown by numerical simulations, and in complexity with respect to existing approaches.