Classification of chaotic time series with deep learning

We use deep neural networks to classify time series generated by discrete and continuous dynamical systems based on their chaotic behaviour. Our approach to circumvent the lack of precise models for some of the most challenging real-life applications is to train different neural networks on a data set from a dynamical system with a basic or low-dimensional phase space and then use these networks to classify time series of a dynamical system with more intricate or high-dimensional phase space. We illustrate this extrapolation approach using the logistic map, the sine-circle map, the Lorenz system, and the Kuramoto-Sivashinsky equation. We observe that the proposed convolutional neural network with large kernel size outperforms state-of-the-art neural networks for time series classification and is able to classify time series as chaotic or non-chaotic with high accuracy. Introduction Data and in particular time series are generated from numerous observations and experiments across different scientific fields such as atmospheric and oceanic sciences for climate predictions, nuclear fusion for control and safety, biology and medicine for diagnosis. Fourier transforms, radial basis functions approximation and standard numerical techniques have been extensively applied to perform short and long term predictions of chaotic time series [1, 2, 3, 4]. On the other hand, the spectacular success of machine learning and deep learning techniques to image classification [5, 6], which have recently surpassed human-level performance on the ImageNet data set [7], has inspired the development of neural network techniques for time series forecasting [8, 9] and classification [10]. Recently, deep learning approaches have been used to solve partial differential equations in high dimensions [11, 12, 13] and identify hidden physics models from experimental data [14, 15, 16, 17].