Optimizing Reservoir Computing for Reconstructing Ergodic Properties

Reservoir computing offers an appealing framework for modeling dynamical systems from data due to its universality and computational efficiency. However, a major challenge is achieving a stable long-term forecast, or climate, from a trained network, which is essential for inferring ergodic properties such as Lyapunov exponents. A common approach is to optimize the reservoir’s macroscopic parameters, such as the spectral radius of its adjacency matrix, by maximizing prediction time. However, we show that even accurate predictions over multiple Lyapunov times do not guarantee a correct climate. Instead, we propose optimizing macroscopic properties by minimizing the error in the reconstructed invariant distribution (or its projections), which is easily available from data. Our approach reproduces the Lyapunov exponents of model dynamical systems, including the double pendulum, even with partial observations. We apply our method to the posture dynamics of the nematode worm C. elegans across development, and find chaotic Lyapunov spectra with an attractor dimension that decreases with developmental stage.