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

Abstract

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

“…A reminder on kernel methods for regularly sampled time series. The simplest approach to forecasting the time series (employed in [22]) is to assume that x 1 , x 2 , . .…”

“…, x k−τ † +1 ) to predict future state. Given τ ∈ N * (see [22] for how τ can be learned in practice), the approximation of the dynamical system can then be recast as that of interpolating f † from pointwise measurements…”

“…Yet, the accuracy of these emulators depends on the kernel, and the problem of selecting a good kernel has received less attention. Recently, the experiments by Hamzi and Owhadi [22] show that when the time series is regularly sampled, Kernel Flows (KF) [34] (an RKHS technique) can successfully reconstruct the dynamics of some prototypical chaotic dynamical systems. KFs have subsequently been applied to complex large-scale systems, including climate data [30,41].…”

Classification of regular and chaotic motions in Hamiltonian systems with deep learning

“…The paper will demonstrate also the generalization capabilities of InceptionTime deep Convolutional Neural Networks (CNN) that would allow the cartographic study of analogous albeit not identical dynamical systems. A review of recent machine learning (ML) methods used in dynamical astronomy can be found in 1 , while some advanced techniques to learn dynamical systems can be found in 2 . Our work differentiates from the recent literature on ML for dynamical systems and dynamical astronomy in that we provide some evidence of the ability of DL at classifying types of motion from dynamic indicators associated to time series.…”

Data-driven modelling of nonlinear dynamics by barycentric coordinates and memory

“…Despite current success of these methods, tackling high-dimensional nonlinear dynamics is still a challenge. This is, because neural networks require an abundance of parameters whose estimation is costly (e.g., due to a non-convex optimization problem), kernel-based methods require a suitable choice of the kernel [HO20], regression methods need the right choice of transformations, making a suitable basis desirable (for instance by restricting the parametrization to collective variables or reaction coordinates [BKK + 17]), and can result in ill-conditioned problems [FWM10]. Lastly, probabilistic models are more stable and do not require strong intuition about the problem, however often suffer from the curse of dimensionality.…”