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

Pradeep Singh, Sutirtha Ghosh, Ashutosh Kumar, Hrishit B P, Balasubramanian Raman

TL;DR

HypER introduces a hyperbolic reservoir for ESNs by embedding neurons in the Poincaré ball and connecting them with an exponential kernel in hyperbolic distance, embedding the chaos-stretching geometry into the latent space. The approach preserves standard ESN components (sparsity, leaky integration, spectral-radius control) while training only a regularized readout, and it provides a state-divergence bound that mirrors Lyapunov growth. Empirical evaluations across canonical chaotic systems (Lorenz-63, Rössler, Chen-Ueta, Chua, Mackey–Glass) and real-world datasets (Sunspots, Santa Fe laser, MIT–BIH ECG) show longer valid-prediction horizons, improved NRMSE, and better phase-space fidelity, with results validated over 30 seeds. Ablation analyses confirm the necessity of hyperbolic geometry, sampling strategies, and kernel width, establishing a principled connection between negative curvature and long-horizon forecasting in chaotic time series. The work highlights hyperbolic geometry as a practical inductive bias for robust, long-range prediction in complex dynamical systems.

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.

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

TL;DR

HypER introduces a hyperbolic reservoir for ESNs by embedding neurons in the Poincaré ball and connecting them with an exponential kernel in hyperbolic distance, embedding the chaos-stretching geometry into the latent space. The approach preserves standard ESN components (sparsity, leaky integration, spectral-radius control) while training only a regularized readout, and it provides a state-divergence bound that mirrors Lyapunov growth. Empirical evaluations across canonical chaotic systems (Lorenz-63, Rössler, Chen-Ueta, Chua, Mackey–Glass) and real-world datasets (Sunspots, Santa Fe laser, MIT–BIH ECG) show longer valid-prediction horizons, improved NRMSE, and better phase-space fidelity, with results validated over 30 seeds. Ablation analyses confirm the necessity of hyperbolic geometry, sampling strategies, and kernel width, establishing a principled connection between negative curvature and long-horizon forecasting in chaotic time series. The work highlights hyperbolic geometry as a practical inductive bias for robust, long-range prediction in complex dynamical systems.

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.

Paper Structure

This paper contains 43 sections, 5 theorems, 23 equations, 18 figures, 13 tables, 1 algorithm.

Table of Contents

  1. Summary of Key Contributions.
  2. Background and Related Works
  3. Chaotic Dynamics and the Limits of Predictability.
  4. Reservoir Computing for Chaotic Time Series.
  5. Hyperbolic Geometry in Machine Learning.
  6. Methodology
  7. Problem Statement.
  8. Hyperbolic Embeddings and Adjacency
  9. Poincaré Ball Model.
  10. Node Placement.
  11. Connectivity Kernel.
  12. Sparsity & Spectral Normalization.
  13. Reservoir Configuration
  14. Leaky Echo State Update.
  15. Polynomial Readout.
  16. ...and 28 more sections

Key Result

Lemma 1

Let $\{\mathbf{p}_i\}_{i=1}^N\subset\mathcal{B}^d$ be distinct points. Set $\delta=\min_{i\neq j}d_{\mathbb H}(\mathbf{p}_i,\mathbf{p}_j).$ Then $\lambda_{\min}(\mathbf{\widetilde{W}})\;\ge\;1-(N-1)e^{-\delta/\sigma}, \rho(\mathbf{\widetilde{W}})\;\le\;1+(N-1)e^{-\delta/\sigma}$. (Proof in supplem

Figures (18)

  • Figure 1: Two‑dimensional PCA projections of high-dimensional reservoir states (no read-out training applied) for (a) a standard ESN and (b) the proposed HypER, when both networks are driven by the Lorenz system.
  • Figure 2: HypER embedded in the Poincaré disk: nodes placed evenly in hyperbolic volume, links decay with distance.
  • Figure 3: Predicted trajectories by HypER alongside ground truth for the test segment of the Lorenz system under autoregressive forecasting.
  • Figure 4: NRMSE for autoregressive predictions across multiple horizons.
  • Figure 5: Boxplots of NRMSE for autoregressive predictions at a 1000-step prediction horizon.
  • ...and 13 more figures

Theorems & Definitions (8)

  • Lemma 1: Spectrum of the hyperbolic kernel
  • Lemma 2: Linearized forward sensitivity
  • Theorem 3: State–divergence lower bound
  • proof
  • Lemma 4: Spectrum of the hyperbolic kernel
  • proof
  • Lemma 5: Linearized forward sensitivity
  • proof