Long-term prediction of chaotic systems with machine learning

Fan, Huawei; Jiang, Junjie; Chun, Zhang; Wang, Xingang; Lai, Ying Cheng

doi:10.1103/physrevresearch.2.012080

Cited by 135 publications

(66 citation statements)

References 36 publications

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“…As we aim to simulate such systems with increasing levels of fidelity, we need to increase the numerical resolutions and/or incorporate more physical processes from a wide range of spatiotemporal scales into the models. For example, in atmospheric modeling for predicting the weather and climate systems, we need to account for the nonlinear interactions across the scales of cloud microphysics processes, gravity waves, convection, baroclinic waves, synoptic eddies, and large-scale circulation, just to name a few (not to mention the fast/slow processes involved in feedbacks from the ocean, land, and cryosphere; Collins et al, 2006Collins et al, , 2011Flato, 2011;Bauer et al, 2015;Jeevanjee et al, 2017).…”

Section: Introductionmentioning

confidence: 99%

Data-driven predictions of a multiscale Lorenz 96 chaotic system using machine-learning methods: reservoir computing, artificial neural network, and long short-term memory network

Chattopadhyay

Hassanzadeh

Subramanian

2020

Nonlin. Processes Geophys.

161

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Abstract. In this paper, the performance of three machine-learning methods for predicting short-term evolution and for reproducing the long-term statistics of a multiscale spatiotemporal Lorenz 96 system is examined. The methods are an echo state network (ESN, which is a type of reservoir computing; hereafter RC–ESN), a deep feed-forward artificial neural network (ANN), and a recurrent neural network (RNN) with long short-term memory (LSTM; hereafter RNN–LSTM). This Lorenz 96 system has three tiers of nonlinearly interacting variables representing slow/large-scale (X), intermediate (Y), and fast/small-scale (Z) processes. For training or testing, only X is available; Y and Z are never known or used. We show that RC–ESN substantially outperforms ANN and RNN–LSTM for short-term predictions, e.g., accurately forecasting the chaotic trajectories for hundreds of numerical solver's time steps equivalent to several Lyapunov timescales. The RNN–LSTM outperforms ANN, and both methods show some prediction skills too. Furthermore, even after losing the trajectory, data predicted by RC–ESN and RNN–LSTM have probability density functions (pdf's) that closely match the true pdf – even at the tails. The pdf of the data predicted using ANN, however, deviates from the true pdf. Implications, caveats, and applications to data-driven and data-assisted surrogate modeling of complex nonlinear dynamical systems, such as weather and climate, are discussed.

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Section: Introductionmentioning

confidence: 99%

Data-driven predictions of a multiscale Lorenz 96 chaotic system using machine-learning methods: reservoir computing, artificial neural network, and long short-term memory network

Chattopadhyay

Hassanzadeh

Subramanian

2020

Nonlin. Processes Geophys.

161

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“…Reservoir computing using ESNs for predicting chaotic dynamics has already been demonstrated in 2004 by Jaeger and Haas [37]. Since then many studies appeared analyzing and optimizing this approach (see, for example [38][39][40][41][42][43][44], and references cited therein). In particular, it has been pointed out how reservoir computing exploits generalized synchronization of uni-directionally coupled systems [45,46].…”

Section: Echo State Networkmentioning

confidence: 99%

Reconstructing Complex Cardiac Excitation Waves From Incomplete Data Using Echo State Networks and Convolutional Autoencoders

Herzog

Zimmermann

Abele

et al. 2021

Front. Appl. Math. Stat.

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The mechanical contraction of the pumping heart is driven by electrical excitation waves running across the heart muscle due to the excitable electrophysiology of heart cells. With cardiac arrhythmias these waves turn into stable or chaotic spiral waves (also called rotors) whose observation in the heart is very challenging. While mechanical motion can be measured in 3D using ultrasound, electrical activity can (so far) not be measured directly within the muscle and with limited resolution on the heart surface, only. To bridge the gap between measurable and not measurable quantities we use two approaches from machine learning, echo state networks and convolutional autoencoders, to solve two relevant data modelling tasks in cardiac dynamics: Recovering excitation patterns from noisy, blurred or undersampled observations and reconstructing complex electrical excitation waves from mechanical deformation. For the synthetic data sets used to evaluate both methods we obtained satisfying solutions with echo state networks and good results with convolutional autoencoders, both clearly indicating that the data reconstruction tasks can in principle be solved by means of machine learning.

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“…The field of complex systems and nonlinear dynamics has also witnessed a recent spurt in the use of machine learning techniques, particularly in characterization or identification of a variety of system properties or phenomena. For instance, the machine learning algorithms have been successfully implemented in community detection in networks [50], finding fixed points attractors [51], spatiotemporal chaotic systems [52], detecting phase transition [53], prediction of chaotic systems [54], and identification of chimera states [55].…”

Section: Introductionmentioning

confidence: 99%

Machine Learning Assisted Chimera and Solitary States in Networks

et al. 2021

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Chimera and Solitary states have captivated scientists and engineers due to their peculiar dynamical states corresponding to co-existence of coherent and incoherent dynamical evolution in coupled units in various natural and artificial systems. It has been further demonstrated that such states can be engineered in systems of coupled oscillators by suitable implementation of communication delays. Here, using supervised machine learning, we predict (a) the precise value of delay which is sufficient for engineering chimera and solitary states for a given set of system's parameters, as well as (b) the intensity of incoherence for such engineered states. Ergo, using few initial data points we generate a machine learning model which can then create a more refined phase plot as well as by including new parameter values. We demonstrate our results for two different examples consisting of single layer and multi layer networks. First, the chimera states (solitary states) are engineered by establishing delays in the neighboring links of a node (the interlayer links) in a 2-D lattice (multiplex network) of oscillators. Then, different machine learning classifiers, K-nearest neighbors (KNN), support vector machine (SVM) and multi-layer perceptron neural network (MLP-NN) are employed by feeding the data obtained from the network models. Once a machine learning model is trained using the limited amount of data, it predicts the precise value of critical delay as well as the intensity of incoherence for a given unknown systems parameters values. Testing accuracy, sensitivity, and specificity analysis reveal that MLP-NN classifier is better suited than Knn or SVM classifier for the predictions of parameters values for engineered chimera and solitary states. The technique provides an easy methodology to predict critical delay values as well as intensity of incoherence for that delay value for designing an experimental setup to create solitary and chimera states.

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Long-term prediction of chaotic systems with machine learning

Cited by 135 publications

References 36 publications

Data-driven predictions of a multiscale Lorenz 96 chaotic system using machine-learning methods: reservoir computing, artificial neural network, and long short-term memory network

Data-driven predictions of a multiscale Lorenz 96 chaotic system using machine-learning methods: reservoir computing, artificial neural network, and long short-term memory network

Reconstructing Complex Cardiac Excitation Waves From Incomplete Data Using Echo State Networks and Convolutional Autoencoders

Machine Learning Assisted Chimera and Solitary States in Networks

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