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

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.