HypER: Hyperbolic Echo State Networks for Capturing Stretch-and-Fold Dynamics in Chaotic Flows

TL;DR

HypER introduces a hyperbolic reservoir for Echo State Networks that better captures chaotic dynamics, significantly extending prediction horizons in both synthetic and real-world chaotic systems.

Contribution

It proposes a novel hyperbolic embedding reservoir that aligns with chaos geometry, improving long-term prediction in chaotic systems over traditional ESNs.

Findings

HypER outperforms Euclidean ESNs in predicting chaotic systems.

HypER achieves statistically significant longer prediction horizons.

Results are validated on synthetic and real-world datasets.

Abstract

Forecasting chaotic dynamics beyond a few Lyapunov times is difficult because infinitesimal errors grow exponentially. Existing Echo State Networks (ESNs) mitigate this growth but employ reservoirs whose Euclidean geometry is mismatched to the stretch-and-fold structure of chaos. We introduce the Hyperbolic Embedding Reservoir (HypER), an ESN whose neurons are sampled in the Poincare ball and whose connections decay exponentially with hyperbolic distance. This negative-curvature construction embeds an exponential metric directly into the latent space, aligning the reservoir's local expansion-contraction spectrum with the system's Lyapunov directions while preserving standard ESN features such as sparsity, leaky integration, and spectral-radius control. Training is limited to a Tikhonov-regularized readout. On the chaotic Lorenz-63 and Roessler systems, and the hyperchaotic Chen-Ueta attractor, HypER consistently lengthens the mean valid-prediction horizon beyond Euclidean and graph-structured ESN baselines, with statistically significant gains confirmed over 30 independent runs; parallel results on real-world benchmarks, including heart-rate variability from the Santa Fe and MIT-BIH datasets and international sunspot numbers, corroborate its advantage. We further establish a lower bound on the rate of state divergence for HypER, mirroring Lyapunov growth.