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

TL;DR

It is found that the maximal conditional Lyapunov exponent of the reservoir depends strongly on the spectral radius of the reservoir adjacency matrix; therefore, for attractor reconstruction and Lyapunov spectrum estimation, small spectral radius reservoir computers perform better in general.

Abstract

Reservoir computing is a machine learning framework that 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 during the training stage to the performance of the trained reservoir computer at the attractor reconstruction task. We show that, in order to obtain successful attractor reconstruction and Lyapunov spectrum estimation, the largest conditional Lyapunov exponent of the driven reservoir must be significantly more negative than the most negative Lyapunov exponent of the target system. We also 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 spectrum estimation, small spectral radius reservoir computers perform better in general. Our arguments are supported by numerical examples on well-known chaotic systems.

Figures (5)

• Figure 1: Lorenz system with standard parameters: $\sigma=10$, $\beta=8/3$ and $\rho=28$. (a) Three largest Lyapunov exponents of the trained autonomous RC $\lambda_i^{(RC)}$ vs the maximal CLE of the driven reservoir $\lambda_{max}^{(r)}$. The Lyapunov exponents of the target Lorenz system are indicated as solid gray lines. The line $\lambda=\lambda_{max}^{(r)}$ is shown as a dashed gray line. (b) Maximal CLE of the driven reservoir $\lambda_{max}^{(r)}$ vs the reservoir spectral radius. (c) Three largest Lyapunov exponents of the trained autonomous RC $\lambda^{(RC)}$ vs the reservoir spectral radius.
• Figure 2: Lorenz system with $\sigma=20$, $\beta=4$, and $\rho=45.92$. (a) Three largest Lyapunov exponents of the trained autonomous RC $\lambda_i^{(RC)}$ vs the maximal CLE of the driven reservoir $\lambda_{max}^{(r)}$. The Lyapunov exponents of the target Lorenz system are indicated as solid gray lines. The line $\lambda=\lambda_{max}^{(r)}$ is shown as a dashed gray line. (b) Maximal CLE of the driven reservoir $\lambda_{max}^{(r)}$ vs the reservoir spectral radius. (c) Three largest Lyapunov exponents of the trained autonomous RC $\lambda^{(RC)}$ vs the reservoir spectral radius.
• Figure 3: Kaplan-Yorke dimension of the trained autonomous RC vs maximal CLE of the driven reservoir before training for the Lorenz system with (a) $\sigma=10$, $\beta=8/3$ and $\rho=28$ and (b) $\sigma=20$, $\beta=4$, and $\rho=45.92$. In both cases, the Kaplan-Yorke dimension of the RC increases sharply as the maximal CLE of the reservoir $\lambda_{max}^{(r)}$ approaches zero (i.e., as $\rho$ increases), indicating that the autonomous RC does not accurately replicate the Lorenz attractor geometry when $\rho$ is near unity.
• Figure 4: Qi system with $p_1=35$, $p_2=10$, $p_3=1$ and $p_4=10$. (a) Four largest Lyapunov exponents of the trained autonomous RC $\lambda_i^{(RC)}$ vs the maximal CLE of the driven reservoir $\lambda_{max}^{(r)}$. The Lyapunov exponents of the target Qi system are indicated as solid gray lines. The line $\lambda=\lambda_{max}^{(r)}$ is shown as a dashed gray line. (b) Maximal CLE of the driven reservoir $\lambda_{max}^{(r)}$ vs the reservoir spectral radius. (c) Four largest Lyapunov exponents of the trained autonomous RC $\lambda_i^{(RC)}$ vs the reservoir spectral radius.
• Figure 5: Kaplan-Yorke dimension of the trained autonomous RC vs largest CLE $\lambda_{max}^{(r)}$ of the driven reservoir for the Qi system. For the Qi system, the Kaplan-Yorke dimension of the RC does not depend much on the largest CLE because it is determined solely by the three largest Lyapunov exponents, which are all relatively small in absolute value.