Data-driven prediction of Jacobians and Covariant Lyapunov Vectors in chaotic flows

In a chaotic flow, infinitesimal perturbations grow exponentially in time. Their evolution is governed by the tangent dynamics, which, in turn, is governed by the Jacobian. From the Jacobian, the perturbations' growth rates (Lyapunov exponents, LEs), and directions (covariant Lyapunov vectors, CLVs) can be computed, which are key to computing the stability and the gradient for design optimization. The derivation and numerical solution of the tangent equation and its adjoint, however, may be time consuming and cumbersome. To overcome this, we propose a method to infer the tangent dynamics from data. We develop and train Echo State Networks (ESNs) on data of prototypical chaotic dynamical systems. We compute the Networks' Jacobian, from which we extract its spectrum of LEs, and CLVs. We find that (i) the long-term statistics, and (ii) the finite time variations in physical space of both LEs and CLVs agree with the target sets. Hence, we show that (iii) ESNs are able to accurately learn the stability properties of chaotic attractors. This work opens opportunities to physically interpret the stability and compute the gradient of chaotic and turbulent flows from experimental data.