Predicting phase and sensing phase coherence in chaotic systems with machine learning

Abstract

Recent interest in exploiting machine learning for model-free prediction of chaotic systems focused on the time evolution of the dynamical variables of the system as a whole, which include both amplitude and phase. In particular, in the framework based on reservoir computing, the prediction horizon as determined by the largest Lyapunov exponent is often short, typically about five or six Lyapunov times that contain approximately equal number of oscillation cycles of the system. There are situations in the real world where the phase information is important, such as the ups and downs of species populations in ecology, the polarity of a voltage variable in an electronic circuit, and the concentration of certain chemical above or below the average. Using classic chaotic oscillators and a chaotic food-web system from ecology as examples, we demonstrate that reservoir computing can be exploited for long-term prediction of the phase of chaotic oscillators. The typical prediction horizon can be orders of magnitude longer than that with predicting the entire variable, for which we provide a physical understanding. We also demonstrate that a properly designed reservoir computing machine can reliably sense phase synchronization between a pair of coupled chaotic oscillators with implications to the design of the parallel reservoir scheme for predicting large chaotic systems.

Anticipating synchronization with machine learning

“…The idea and principle of exploiting reservoir computing for predicting the state evolution of chaotic systems were first laid out about two decades ago [11,12]. In recent years, modelfree predication of chaotic systems using reservoir computing has gained considerable momentum [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. The neural architecture of reservoir computing consists of a single hidden layer of a complex dynamical network that receives input data and generates output data.…”

“…In the training phase, the whole system is set to the "open-loop" mode, where it receives input data and optimizes its parameters to match its output with the true output corresponding to the input. In the standard reservoir computing setting [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48], the adjustable parameters are those associated with the output matrix that maps the internal dynamical state of the hidden layer to the output layer, while other parameters, such as those defining the network and the input matrix, are fixed (the hyperparameters). In the prediction phase, the neural machine operates in the "closeloop" mode where the output variables are fed directly into the input, so that the whole system becomes a self-evolving dynamical system.…”

Optimized ensemble deep learning framework for scalable forecasting of dynamics containing extreme events

“…In the last few years, machine learning techniques are used extensively to explore several emergent phenomena in nonlinear dynamical systems. In this context, reservoir computing (RC) [29][30][31], a version of recurrent neural network model, is effective for inference of unmeasured variables in chaotic systems using values of a known variable [32], forecasting dynamics of chaotic oscillators [33,34], predicting the evolution of the phase of chaotic dynamics [35], and prediction of critical transition in dynamical systems [36]. Also, RC is used to detect synchronization [37][38][39], spiking-bursting phenomena [40], inferring network links [41] in coupled systems.…”

Role of assortativity in predicting burst synchronization using echo state network

“…For instance, Pathak et al [33] showed that a reservoir computation based ESN approach can indeed predict a large spatiotemporal chaotic data and the prediction efficiency is very good up to few multiples of Lyapunov time scale. Since then, ESN based prediction has received growing attention from the researchers and has been used in several aspects of nonlinear dynamics, for example, prediction of a chaotic time * arindammishra@gmail.com series data [33][34][35][36][37], spatiotemporal dynamics [38], determining Lyapunov exponents [39,40], dynamics of multiscale systems [35], predicting critical transition [41] and critical range for the efficiency of the reservoir [42], to name a few. Another interesting aspect namely the collective or macroscopic behavior of ensemble of interacting oscillators, such as, synchronization, quenching of oscillations, chimera states etc.…”