Learning dynamical systems from data: a simple cross-validation perspective

The prediction of chaotic systems from time-series (initially investigated in [13]) has been investigated from the regression perspectives of support vector machines [29, 28], reservoir computing [35, 25], deep feed-forward artificial neural networks (ANN), and recurrent neural networks with long short-term memory (RNN-LSTM) [11, 12, 10, 37]. Reservoir computing was observed to be efficient for predictions but not very accurate for estimating Lyapunov exponents. 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, 7, 18, 4, 23, 24, 22, 2], the accuracy of these emulators depends on the kernel and the problem of selecting a good kernel has received less attention. We investigate Kernel Flows [31] (KF) as a generic tool for selecting the kernel used to learn chaotic dynamical systems.