Data-driven prediction and control of extreme events in a chaotic flow

Abstract

An extreme event is a sudden and violent change in the state of a nonlinear system. In fluid dynamics, extreme events can have adverse effects on the system's optimal design and operability, which calls for accurate methods for their prediction and control. In this paper, we propose a data-driven methodology for the prediction and control of extreme events in a chaotic shear flow.The approach is based on echo state networks, which are a type of reservoir computing that learn temporal correlations within a time-dependent dataset. The objective is five-fold. First, we exploit ad-hoc metrics from binary classification to analyse (i) how many of the extreme events predicted by the network actually occur in the test set (precision), and (ii) how many extreme events are missed by the network (recall). We apply a principled strategy for optimal hyperparameter selection, which is key to the networks' performance. Second, we focus on the time-accurate prediction of extreme events. We show that echo state networks are able to predict extreme events well beyond the predictability time, i.e., up to more than five Lyapunov times. Third, we focus on the longterm prediction of extreme events from a statistical point of view. By training the networks with datasets that contain non-converged statistics, we show that the networks are able to learn and extrapolate the flow's long-term statistics. In other words, the networks are able to extrapolate in time from relatively short time series. Fourth, we design a simple and effective control strategy to prevent extreme events from occurring. The control strategy decreases the occurrence of extreme events up to one order of magnitude with respect to the uncontrolled system. Finally, we analyse the robustness of the results for a range of Reynolds numbers. We show that the networks perform well across a wide range of regimes. This work opens up new possibilities for the data-driven prediction and control of extreme events in chaotic systems.

Stability analysis of chaotic systems from data

“…The echo state property enforces the independence of the reservoir state on the initial conditions, which is satisfied by rescaling W by a multiplication factor, such that the absolute value of the largest eigenvalue [37], i.e., the spectral radius, is smaller than unity. Following [28,62,35,63], we add a bias in the input and output layers to break the inherent symmetry of the basic ESN architecture. Specifically, the input bias, b in is a hyperparameter, selected in order to have the same order of magnitude as the normalized inputs, ŷin .…”

“…In general, instead of Eq. ( 16), other types of error functions can be used for the hyperparamer tuning, such as the maximization of the prediction horizon [28,33,35] or the minimization of the kinetic energy differences [63]. Here the input scaling, σ in , the spectral radius, ρ, and the Tikhonov parameter, β , are the ESN hyperparameters that are being tuned [37,63].…”

“…( 16), other types of error functions can be used for the hyperparamer tuning, such as the maximization of the prediction horizon [28,33,35] or the minimization of the kinetic energy differences [63]. Here the input scaling, σ in , the spectral radius, ρ, and the Tikhonov parameter, β , are the ESN hyperparameters that are being tuned [37,63]. In order to select the optimal hyperparameters, σ in and ρ, we employ a Bayesian Optimization, which is a strategy for finding the extrema of objective functions that are expensive to evaluate [65,63].…”

“…Here the input scaling, σ in , the spectral radius, ρ, and the Tikhonov parameter, β , are the ESN hyperparameters that are being tuned [37,63]. In order to select the optimal hyperparameters, σ in and ρ, we employ a Bayesian Optimization, which is a strategy for finding the extrema of objective functions that are expensive to evaluate [65,63]. Within the optimal [σ in , ρ] we perform a grid search to select β [63].…”

“…In order to select the optimal hyperparameters, σ in and ρ, we employ a Bayesian Optimization, which is a strategy for finding the extrema of objective functions that are expensive to evaluate [65,63]. Within the optimal [σ in , ρ] we perform a grid search to select β [63]. In particular, [σ in , ρ] are searched in the hyperparameter space [0.1, 5] × [0.1, 1] in logarithmic scale, while for β we test {10 −6 , 10 −8 , 10 −10 , 10 −12 }.…”