Non-linear reduced modeling of dynamical systems using kernel methods and low-rank approximation
This work addresses the challenge of efficient reduced modeling for non-linear dynamical systems, which is incremental as it builds on kernel methods and low-rank techniques.
The paper tackles the problem of approximating trajectories of computationally demanding dynamical systems by proposing a new algorithm that embeds trajectories in a reproducing kernel Hilbert space and uses low-rank optimization, resulting in improved approximation accuracy and reduced complexity compared to existing methods.
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.