Mapping topological characteristics of dynamical systems into neural networks: A reservoir computing approach

Significant advances have recently been made in modeling chaotic systems with the reservoir computing approach, especially for prediction. We find that although state prediction of the trained reservoir computer will gradually deviate from the actual trajectory of the original system, the associated geometric features remain invariant. Specifically, we show that the typical geometric metrics including the correlation dimension, the multiscale entropy, and the memory effect are nearly identical between the trained reservoir computer and its learned chaotic systems. We further demonstrate this fact on a broad range of chaotic systems ranging from discrete and continuous chaotic systems to hyperchaotic systems. Our findings suggest that the successfully reservoir computer may be topologically conjugate to an observed dynamical system.