# Reservoir Computing with Delayed Input for Fast and Easy Optimisation

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Reservoir Computing

#### Error Measure

#### 2.2. Reservoir Model

#### 2.3. Input and Mask

#### 2.4. Time Series Prediction Tasks

#### 2.4.1. Mackey–Glass

#### 2.4.2. NARMA10

#### 2.4.3. Lorenz

#### 2.5. Simulation Conditions

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

RC | Reservoir computing |

NVAR | Nonlinear vector autoregression |

NRMSE | Normalised root mean squared error |

## Appendix A. Optimised Input Parameters

**Figure A1.**Values for the optimised input parameters ${G}_{2}$ (orange) and d (blue), corresponding to the Mackey–Glass $s=10$ results depicted in Figure 5, as a function of (

**a**) the virtual node coupling strength K and (

**b**) the coupling delay N. All remaining parameters are as stated in Section 2.5.

## Appendix B. Stuart-Landau Delay-Based Reservoir Computer

Parameter | Value | Parameter | Value |
---|---|---|---|

${\lambda}_{SL}$ | −0.02 | $\omega $ | 0 |

$\gamma $ | −0.1 | K | 0.1 |

$\tau $ | 105 | $\varphi $ | 0 |

${N}_{v}$ | 30 | ${R}_{\mathrm{noise}}$ | 1 × 10${}^{-7}$ |

${G}_{1}$ | 0.01 | ${J}_{0}$ | 0 |

T | 80 |

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**Figure 1.**(

**a**) Sketch of the reservoir computing concept. The vector ${\underline{\mathbf{x}}}_{\mathrm{out}}^{{k}^{\prime}}$ is the responses of the reservoir to an input ${I}_{{k}^{\prime}}$ and the corresponding output ${o}_{{k}^{\prime}}$ is generated by a weighted sum of these responses. The read-out weights $\underline{\mathbf{W}}$ are trained. (

**b**) Sketch of the memory cell reservoir described in Section 2.2, where a time-multiplexed input $J\left(k\right)$ is fed into the reservoir ($k={N}_{v}{k}^{\prime}+{k}^{\u2033}$, please see Figure 2 for the construction of the time-multiplexed input). The index n labels the memory cells (in total N) that are addressed via the coupling matrices ${\mathbf{K}}_{\mathbf{in}}$ and ${\mathbf{K}}_{\mathbf{out}}$. K labels the feedback strength at which the output of the memory cells ${x}_{\mathrm{out}}\left(k\right)$ (given by Equation (5)) is fed back into the nonlinearity $\mathcal{G}$. The elements of the state matrix $\underline{\underline{\mathbf{S}}}$ are given by ${S}^{{k}^{\prime},{k}^{\u2033}}={x}_{\mathrm{out}}\left({N}_{v}{k}^{\prime}+{k}^{\u2033}\right)$, with ${k}^{\prime}\in (1\cdots {K}_{tr})$ and ${k}^{\u2033}\in (1\cdots {N}_{v})$, i.e., one row of $\underline{\underline{\mathbf{S}}}$ corresponds to the vector of responses ${\underline{\mathbf{x}}}_{\mathrm{out}}^{{k}^{\prime}}$ in (

**a**).

**Figure 2.**Sketch of the generation of the final time-multiplexed input sequence $J\left(k\right)$ using the task-dependent input $I\left({k}^{\prime}\right)$, a delayed version of this input $I\left({k}^{\prime}-d\right)$, and the masks ${M}_{1}\left({k}^{\u2033}\right)$ and ${M}_{2}\left({k}^{\u2033}\right)$, as described by Equation (8).

**Figure 3.**NRMSE as a function of the delayed input parameters d and ${G}_{2}$ for Mackey–Glass (

**a**) one, (

**b**) three, and (

**c**) ten steps ahead prediction, (

**d**) NARMA10, (

**c**) Lorenz x one step ahead prediction, and (

**f**) Lorenz z one step ahead cross-prediction. Parameters are as stated in Section 2.5, except for (

**a**,

**e**) where ${K}_{\mathrm{te}}=30,000$.

**Figure 4.**NRMSE for optimised input delay d, as a function of the delayed-input scaling ${G}_{2}$ for Mackey–Glass (

**a**) one, (

**b**) three, and (

**c**) ten steps ahead prediction, (

**d**) NARMA10, (

**c**) Lorenz x one step ahead prediction, and (

**f**) Lorenz z one step ahead cross-prediction. The error bars indicate the standard deviation. The optimal input delays are (

**a**) $d=14$, (

**b**) $d=13$, (

**c**) $d=9$, (

**d**) $d=9$, (

**e**) $d=1$, and (

**f**) $d=15$. The remaining parameters are as stated in Section 2.5, except for (

**a**,

**e**) where ${K}_{\mathrm{te}}=30,000$.

**Figure 5.**NRMSE for Mackey–Glass 10 step ahead prediction as a function of (

**a**) the virtual node coupling strength K and (

**b**) the coupling delay N. The orange dotted (blue dashed) lines show the results without (with) delayed input. Along the blue curve the delayed input parameters d and ${G}_{2}$ have been optimised (see Figure A1 in Appendix A for their values). The error bars indicate the standard deviation. All remaining parameters are as stated in Section 2.5.

**Figure 6.**NRMSE for the NARMA10 task as a function of the delayed input parameters d and ${G}_{2}$ using the Stuart–Landau delay-based reservoir computer described in Appendix B.

Parameter | Value | Parameter | Value |
---|---|---|---|

${g}_{0}$ | 40 | ${P}_{\mathrm{sat}}$ | 1 |

K | 0.02 | N | 31 |

${N}_{v}$ | 30 | $\lambda $ | 5 × 10${}^{-6}$ |

${G}_{1}$ (Mackey–Glass) | 1 | ${J}_{0}$ (Mackey–Glass) | 0 |

${G}_{1}$ (Lorenz) | 0.03 | ${J}_{0}$ (Lorenz) | 0.85 |

${G}_{1}$ (NARMA10) | 1.8 | ${J}_{0}$ (NARMA10) | 0.4 |

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**MDPI and ACS Style**

Jaurigue, L.; Robertson, E.; Wolters, J.; Lüdge, K. Reservoir Computing with Delayed Input for Fast and Easy Optimisation. *Entropy* **2021**, *23*, 1560.
https://doi.org/10.3390/e23121560

**AMA Style**

Jaurigue L, Robertson E, Wolters J, Lüdge K. Reservoir Computing with Delayed Input for Fast and Easy Optimisation. *Entropy*. 2021; 23(12):1560.
https://doi.org/10.3390/e23121560

**Chicago/Turabian Style**

Jaurigue, Lina, Elizabeth Robertson, Janik Wolters, and Kathy Lüdge. 2021. "Reservoir Computing with Delayed Input for Fast and Easy Optimisation" *Entropy* 23, no. 12: 1560.
https://doi.org/10.3390/e23121560