论文标题

监督机器学习以估计混乱系统中的不稳定性:估计本地莱普诺诺夫指数

Supervised machine learning to estimate instabilities in chaotic systems: estimation of local Lyapunov exponents

论文作者

Ayers, Daniel, Lau, Jack, Amezcua, Javier, Carrassi, Alberto, Ojha, Varun

论文摘要

在混乱的动力系统(例如天气)中,预测错误在某些情况下的增长速度比其他情况更快。关于误差增长的实时知识可以使策略能够随时调整建模和预测基础架构,以提高准确性或减少计算时间。例如,可以更改整体大小或目标观测的分布和类型。局部Lyapunov指数是已知的指标,即在有限的时间间隔内非常小的预测误差的速率。但是,它们的计算非常昂贵:它需要维护和进化的切线线性模型,正交化算法和存储大型矩阵。 在这项可行性研究中,我们从系统轨迹的当前和最新时间​​步骤的输入中研究了监督机器学习在估计当前局部Lyapunov指数方面的准确性,这是经典方法的替代方法。因此,这里的机器学习不是在此不用于模仿物理模型或其某些组件,而是作为互补工具而不是内在的。我们测试了四种受欢迎的监督学习算法:回归树,多层感知,卷积神经网络和长期短期记忆网络。实验是在普通微分方程的两个低维混沌系统上进行的,Rössler和Lorenz 63型号。我们发现,平均而言,机器学习算法准确地预测了稳定的本地Lyapunov指数,不稳定的指数准确地预测了中性指数,并且中性指数仅准确。我们表明,更高的预测准确性与系统吸引子上局部Lyapunov指数的局部同质性有关。重要的是,(预测)错误生长最快的情况不一定与使用机器学习预测本地Lyapunov指数更难的情况相同。

In chaotic dynamical systems such as the weather, prediction errors grow faster in some situations than in others. Real-time knowledge about the error growth could enable strategies to adjust the modelling and forecasting infrastructure on-the-fly to increase accuracy or reduce computation time. One could, e.g., change the ensemble size, or the distribution and type of target observations. Local Lyapunov exponents are known indicators of the rate at which very small prediction errors grow over a finite time interval. However, their computation is very expensive: it requires maintaining and evolving a tangent linear model, orthogonalisation algorithms and storing large matrices. In this feasibility study, we investigate the accuracy of supervised machine learning in estimating the current local Lyapunov exponents, from input of current and recent time steps of the system trajectory, as an alternative to the classical method. Thus machine learning is not used here to emulate a physical model or some of its components, but non intrusively as a complementary tool. We test four popular supervised learning algorithms: regression trees, multilayer perceptrons, convolutional neural networks and long short-term memory networks. Experiments are conducted on two low-dimensional chaotic systems of ordinary differential equations, the Rössler and the Lorenz 63 models. We find that on average the machine learning algorithms predict the stable local Lyapunov exponent accurately, the unstable exponent reasonably accurately, and the neutral exponent only somewhat accurately. We show that greater prediction accuracy is associated with local homogeneity of the local Lyapunov exponents on the system attractor. Importantly, the situations in which (forecast) errors grow fastest are not necessarily the same as those where it is more difficult to predict local Lyapunov exponents with machine learning.

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