Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data.

By Jaideep Pathak, Zhixin Lu, Brian R. Hunt, Michelle Girvan, Edward Ott. Chaos, 2017

674 citations recorded by the source corpus.

We use recent advances in the machine learning area known as "reservoir computing" to formulate a method for model-free estimation from data of the Lyapunov exponents of a chaotic process. The technique uses a limited time series of measurements as input to a high-dimensional dynamical system called a "reservoir." After the reservoir's response to the data is recorded, linear regression is used to learn a large set of parameters, called the "output weights." The learned output weights are then used to form a modified autonomous reservoir designed to be capable of producing an arbitrarily long time series whose ergodic properties approximate those of the input signal. When successful, we say that the autonomous reservoir reproduces the attractor's "climate." Since the reservoir equations and output weights are known, we can compute the derivatives needed to determine the Lyapunov exponents of the autonomous reservoir, which we then use as estimates of the Lyapunov exponents for the original input generating system. We illustrate the effectiveness of our technique with two examples, the Lorenz system and the Kuramoto-Sivashinsky (KS) equation. In the case of the KS equation, we note that the high dimensional nature of the system and the large number of Lyapunov exponents yield a challenging test of our method, which we find the method successfully passes.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f6cb47a3b5f27833cfc3f7b7cebffea1e0dfb91b

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In Artificial Intelligence

In Statistical and Nonlinear Physics

Citations in the corpus

Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks (2019, methodology): “[66] reservoir computing was used to predict the K-S system, however the model is trained on the past history of a specific state.”
Backpropagation algorithms and Reservoir Computing in Recurrent Neural Networks for the forecasting of complex spatiotemporal dynamics (2019, methodology): “In Pathak et al. (2017), Pathak, Wikner et al. (2018) RC is utilized to build surrogate models for chaotic systems and compute their Lyapunov exponents based solely on data.”
Machine learning and the physical sciences (2019, background): “For instance (Pathak et al., 2018, 2017) used recurrent neural networks called an echo state networks or reservoir computers (Jaeger and Haas, 2004) to predict the trajectories of a chaotic dynamical system and of models used for weather prediction.”