Greedy Kernel Methods for Center Manifold Approximation

Abstract

For certain dynamical systems it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to a non-hyperbolic equilibrium point, and to obtain meaningful predictions of its behavior by analyzing a reduced dimensional problem. Since the manifold is usually not known, approximation methods are of great interest to obtain qualitative estimates. In this work, we use a data-based greedy kernel method to construct a suitable approximation of the manifold close to the equilibrium. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used to construct a surrogate model of the manifold. The method is tested on different examples which show promising performance and good accuracy.

Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem

“…First we prove that if x = 0 is an equilibrium of (9) then (x, y) = (0, 0) is an equilibrium of the full order system. Indeed, since h(0) = 0 by definition, we have that (12) implies that ĥ(0) = 0 whatever is the value of q > 0. Thus, since x = 0 is an equilibrium of (9), we have…”

“…We then turn our attention to concrete procedures to construct approximants that provide that kind of error control, and to this end we revise and refine a kernel-based approximation method that we introduced in [12]. This approximation can be constructed based solely on the knowledge of points on the trajectories of the system close to the equilibrium point, that are obtained in practice by the numerical integration of the system.…”

“…This method is formulated in the Reproducing Kernel Hilbert Spaces (RKHS) of a strictly positive kernel (see [2]), which have provided strong mathematical foundations for analyzing dynamical systems in other settings (see e.g. [1,3,4,5,6,7,9,11,12,13,18,19]). Our method explicitly incorporates into the approximant some prior knowledge on the structure of the manifold.…”

Learning dynamical systems from data: A simple cross-validation perspective, part I: Parametric kernel flows

“…On the other hand, RNN-LSTM were observed to be accurate for estimating Lyapunov exponents but not as good as reservoir computing for predictions (see [14] for a survey). Although Reproducing Kernel Hilbert Spaces (RKHS) [16] have provided strong mathematical foundations for analyzing dynamical systems [5,6,8,20,24,7,18,4,22,23,2], the accuracy of these emulators depends on the kernel and the problem of selecting a good kernel has received less attention.…”

Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series

“…Amongst various learning-based approaches, kernel-based methods hold potential for considerable advantages in terms of theoretical analysis, numerical implementation, regularization, guaranteed convergence, automatization, and interpretability [11,32]. Indeed, reproducing kernel Hilbert spaces (RKHS) [14] have provided strong mathematical foundations for analyzing dynamical systems [6,21,19,20,4,24,25,1,26,7,8,9] and surrogate modeling (we refer the reader to [38] for a survey). Yet, the accuracy of these emulators depends on the kernel, and the problem of selecting a good kernel has received less attention.…”