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A mathematical framework for time-delay reservoir computing analysis

Anh-Tuan Clabaut, Jean Auriol, Islam Boussaada, Guilherme Mazanti

Abstract

Reservoir computing is a well-established approach for processing data with a much lower complexity compared to traditional neural networks. Despite two decades of experimental progress, the core properties of reservoir computing (namely separation, robustness, and fading memory) still lack rigorous mathematical foundations. This paper addresses this gap by providing a control-theoretic framework for the analysis of time-delay-based reservoir computers. We introduce formal definitions of the separation property and fading memory in terms of functional norms, and establish their connection to well-known stability notions for time-delay systems as incremental input-to-state stability. For a class of linear reservoirs, we derive an explicit lower bound for the separation distance via Fourier analysis, offering a computable criterion for reservoir design. Numerical results on the NARMA10 benchmark and continuous-time system prediction validate the approach with a minimal digital implementation.

A mathematical framework for time-delay reservoir computing analysis

Abstract

Reservoir computing is a well-established approach for processing data with a much lower complexity compared to traditional neural networks. Despite two decades of experimental progress, the core properties of reservoir computing (namely separation, robustness, and fading memory) still lack rigorous mathematical foundations. This paper addresses this gap by providing a control-theoretic framework for the analysis of time-delay-based reservoir computers. We introduce formal definitions of the separation property and fading memory in terms of functional norms, and establish their connection to well-known stability notions for time-delay systems as incremental input-to-state stability. For a class of linear reservoirs, we derive an explicit lower bound for the separation distance via Fourier analysis, offering a computable criterion for reservoir design. Numerical results on the NARMA10 benchmark and continuous-time system prediction validate the approach with a minimal digital implementation.
Paper Structure (14 sections, 4 theorems, 24 equations, 1 figure)

This paper contains 14 sections, 4 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Desired properties for a reservoir
  3. Time-delay reservoir computing
  4. Time-delay systems as reservoirs
  5. Input driving
  6. Timescales
  7. Framework for reservoir analysis
  8. Stability
  9. Separation property
  10. Robustness to noise
  11. Fading memory
  12. Case of a linear reservoir
  13. Simulations and results
  14. Discussion and perspectives

Key Result

Proposition 1

System eq gen is ISS if and only if there exist a functional $V \colon C \rightarrow \mathbb{R}$ Lipschitz on bounded sets, $\alpha_1, \alpha_2, \alpha_3, \gamma \in \mathcal{K}_\infty$ such that for every $\phi \in C$ and every $u \in U$, the corresponding solution $x$ satisfies

Figures (1)

  • Figure 1: NARMA10 predictions for System \ref{['eq lin']} ($a_0=-1$, $a_1=0.9e^{-0.1}$, $\tau=1$, $\mathrm{NRMSE}=0.38$).

Theorems & Definitions (10)

  • Proposition 1
  • Definition 2: Incremental Input-to-State Stability
  • Theorem 3
  • proof
  • Proposition 4
  • Proposition 5
  • proof
  • Definition 6: Class $\mathcal{K}$, $\mathcal{K}_\infty$ function
  • Definition 7: Class $\mathcal{KL}$ function
  • Definition 8: Driver derivative