Doria Rosales Elizabeth, Carbone Vincenzo, Lepreti Fabio
Department of Physics, University of Trento, Via Sommarive, Povo, 38123, Trento, Italy.
Physics Department, University of Calabria, Ponte P. Bucci Cubo 31C, Rende, 87036, Cosenza, Italy.
Sci Rep. 2025 Jul 2;15(1):23412. doi: 10.1038/s41598-025-03607-2.
The dynamics of non-integrable Hamiltonian systems, described by area-preserving mappings, are regulated by the KAM theorem. This states that the phase space of the system is made up of interwoven sets of regular and chaotic dynamics, whose extent depends on a chaoticity parameter k. The chaoticity parameter measures the degree of non-integrability of the Hamiltonian; the extent of regular orbits decreases as the non-integrable contribution increases. Deep learning is proving increasingly useful in predicting natural phenomena from a set of data, even for chaotic time series forecasting. In this paper we investigate numerical simulations of the standard map with different values of k, as a learning process of a typical non-integrable Hamiltonian system. Our aim is to investigate to what extent a deep learning process is able to recognize the degree of non-integrability of a Hamiltonian system, namely to forecast the actual value of the chaoticity parameter, using data obtained from the same system used in the learning process. Results show that, in general, forecasting the chaoticity parameter is far from being guaranteed, because the KAM theorem is at work. However, the accuracy of the forecasting process depends on both the number of initial conditions and the length of the trajectories used in the learning process. The maximum of the forecasting accuracy is obtained for intermediate values of k, when the phase space is formed by roughly equally spaced regular and irregular trajectories. On the contrary, both relatively low values of k (prevalence of regular orbits) and high values of k (prevalence of irregular orbits), are more difficult to predict. According to our results, a standard deep learning process has difficulty distinguishing between regular and slightly irregular dynamics, and between a purely stochastic system and a system with residual regular orbits.
由保面积映射描述的不可积哈密顿系统的动力学受KAM定理支配。该定理指出，系统的相空间由规则动力学和混沌动力学相互交织的集合组成，其范围取决于混沌参数k。混沌参数衡量哈密顿量的不可积程度；随着不可积贡献的增加，规则轨道的范围减小。深度学习在从一组数据预测自然现象方面越来越有用，甚至对于混沌时间序列预测也是如此。在本文中，我们研究了具有不同k值的标准映射的数值模拟，作为一个典型不可积哈密顿系统的学习过程。我们的目的是研究深度学习过程在多大程度上能够识别哈密顿系统的不可积程度，即使用从学习过程中使用的同一系统获得的数据来预测混沌参数的实际值。结果表明，一般来说，预测混沌参数远不能得到保证，因为KAM定理在起作用。然而，预测过程的准确性取决于学习过程中使用的初始条件数量和轨迹长度。当相空间由大致等间距的规则和不规则轨迹组成时，对于k的中间值可获得预测准确性的最大值。相反，k的相对较低值（规则轨道占主导）和较高值（不规则轨道占主导）都更难预测。根据我们的结果，一个标准的深度学习过程难以区分规则和轻微不规则的动力学，以及区分纯随机系统和具有残余规则轨道的系统。
