Attractor reconstruction with reservoir computers: The effect of the reservoir's conditional Lyapunov exponents on faithful attractor reconstruction

Reservoir computing is a machine learning technique which has been shown to be able to replicate the chaotic attractor, including the fractal dimension and the entire Lyapunov spectrum, of the dynamical system on which it is trained. We quantitatively relate the generalized synchronization dynamics of a driven reservoir computer during the training stage to the performance of the autonomous reservoir computer at the attractor reconstruction task. We show that, for successful attractor reconstruction and Lyapunov exponent estimation, the largest conditional Lyapunov exponent of the driven reservoir must be significantly smaller (more negative) than the smallest (most negative) Lyapunov exponent of the true system. We find that the maximal conditional Lyapunov exponent of the reservoir depends strongly on the spectral radius of the reservoir adjacency matrix, and therefore, for attractor reconstruction and Lyapunov exponent estimation, small spectral radius reservoir computers perform better in general. Our arguments are supported by numerical examples on well-known chaotic systems.