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

Abstract

This paper demonstrates the capabilities of convolutional neural networks (CNNs) at classifying types of motion starting from time series, without any prior knowledge of the underlying dynamics. The paper applies different forms of deep learning to problems of increasing complexity with the goal of testing the ability of different deep learning architectures at predicting the character of the dynamics by simply observing a time-ordered set of data. We will demonstrate that a properly trained CNN can correctly classify the types of motion on a given data set. We also demonstrate effective generalisation capabilities by using a CNN trained on one dynamic model to predict the character of the motion governed by another dynamic model. The ability to predict types of motion from observations is then verified on a model problem known as the forced pendulum and on a relevant problem in Celestial Mechanics where observational data can be used to predict the long-term evolution of the system.

Datasets for learning of unknown characteristics of dynamical systems

“…The method can estimate the largest Lyapunov exponent from noisy data, based on training deep learning models on synthetically generated trajectories 22 . The deep learning techniques were trained with different indexing methods based on chaos indicators, like Fast Lyapunov Indicators and the frequency map analysis 23 . The goal was to classify types of motion (series corresponds to a chaotic, librational, or rotational motion) by observing samples taken from time series coming from simple Hamiltonian systems.…”

Machine learning tuning by diversity oriented firefly metaheuristics for Industry 4.0

“…While many ML approaches exist, per the no free lunch theorem (NFL) Wolpert and Macready (1997), no single approach works best for all given problems Celletti et al (2020). Nevertheless, researchers strive to improve models and their methodologies to achieve the best possible general performance.…”

Perturbed Keplerian Hamiltonian Systems

“…Tese types of perturbation systems have been the focus of interest by a number of authors and the references therein [1][2][3][4][5]. We mention in particular the works of Poincaré, regarding the three-body problem (these orbits were called "frst-view sort solutions") [6], and the studies by Ambrosetti et al [7] and Celletti et al [8] which showed the existence of a skew-T/2 periodic solution of the following problem:…”

“…Theorem 1 (see [7,8,12]). Tere exists ε 0 > 0 such that ∀|ε| < ε 0 , the perturbed system (6), has at least one noncollision symmetric (skew-T/2 periodic) orbit near the circular orbit of (6).…”