Reservoir computing with large valid prediction time for the Lorenz system

TL;DR

This paper investigates how hyperparameters affect the Valid Prediction Time of Reservoir Computers when modeling the Lorenz system, achieving high prediction times and proposing efficient evaluation methods.

Contribution

It identifies hyperparameter regimes that maximize VPT, relates VPT to Lyapunov exponents, and emphasizes the importance of solver choice and the concept of Valid Ground Truth Time.

Findings

High VPT (>30 Lyapunov times) achievable under certain conditions.

VPT can be predicted from early prediction errors using Lyapunov exponents.

Two spectral radius regimes yield large VPT: near zero and at the edge of chaos.

Abstract

We study the dependence of the Valid Prediction Time (VPT) of Reservoir Computers (RCs) on hyperparameters including the regularization coefficient, reservoir size, and spectral radius. Under carefully chosen conditions, the RC can achieve approximately 70% of a benchmark performance, based on the output of a single prediction step used as initial conditions for the Lorenz equations. We report high VPT values (>30 Lyapunov times), as we are predicting a noiseless system where overfitting can be beneficial. While these conditions may not hold for noisy systems, they could still be useful for real-world applications with limited noise. Furthermore, utilizing knowledge of the Lyapunov exponent, we find that the VPT can be predicted by the error in the first few prediction steps, offering a computationally efficient evaluation method. We emphasize the importance of the numerical solver used to generate the Lorenz dataset and define a Valid Ground Truth Time (VGTT), during which the outputs of several common solvers agree. A VPT exceeding the VGTT is not meaningful, as a different solver could produce a different result. Lastly, we identify two spectral radius regimes that achieve large VPT: a small radius near zero, resulting in simple but stable operation, and a larger radius operating at the "edge of chaos."