On the attractor in high-dimensional neural network dynamics of reservoir computing: A Lyapunov analysis viewpoint

Recent theoretical studies on reservoir computing have shown that when the spectral radius of the adjacency matrix is sufficiently small, the dynamics of a reference system can be embedded in the reservoir space, enabling the reconstruction of dynamical invariants. However, reservoir models often reproduce time series accurately even when the spectral radius is relatively large, where the underlying mechanism is not well understood. In this study, we investigate the reconstruction of dynamical structures from the perspective of Lyapunov analysis using reservoir computing applied to the Hénon map. By comparing the Lyapunov spectrum of the reservoir dynamics with that of the reference system, we show that the reference dynamics are embedded in a low-dimensional inertial manifold in the reservoir space. We further demonstrate that the full Lyapunov spectrum of the reference system can be recovered by restricting the analysis to the tangent space of this manifold, even when the spectral radius is relatively large. These results clarify the geometric mechanism underlying the successful reconstruction of chaotic dynamics by reservoir computing beyond the regime where theoretical guarantees currently exist.