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

2025-08-25

AI Summary

Chaotic systems are fundamentally limited in long-term predictability due to exponential error divergence (Lyapunov timescale), and conventional Euclidean echo state networks (ESNs) fail to capture their intrinsic stretch-and-fold dynamics. To address this, we propose Hyperbolic Embedding Reservoir (HypER), the first reservoir computing framework embedding hidden states in the negatively curved Poincaré ball geometry. This enables alignment between the reservoir’s expansion–contraction spectrum and the system’s Lyapunov directions. HypER defines connection decay via hyperbolic distance and incorporates sparsity, leaky integration, and spectral radius constraints—only the Tikhonov-regularized readout layer is trained. Evaluated on Lorenz-63, Rössler, Chen–Ueta systems, and real-world heart rate and sunspot time series, HypER significantly outperforms both Euclidean and graph-structured ESNs, with statistically validated extension of prediction horizons. Moreover, it establishes, for the first time, a theoretical lower bound on state divergence under chaotic dynamics.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Forecasting chaotic dynamics beyond Lyapunov times
• Mismatch between Euclidean geometry and chaotic structure
• Improving prediction horizon in chaotic systems

Innovation

• Hyperbolic embedding reservoir with exponential distance decay
• Aligns reservoir spectrum with Lyapunov directions
• Preserves sparsity and spectral-radius control features