# 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 |

## References

- Nakajima, K.; Fischer, I. Reservoir Computing: Theory, Physical Implementations, and Applications; Springer: New York, NY, USA, 2021. [Google Scholar]
- Jaeger, H. The ’Echo State’ Approach to Analysing and Training Recurrent Neural Networks; GMD Report 148; GMD—German National Research Institute for Computer Science: Darmstadt, Germany, 2001. [Google Scholar]
- Dutoit, X.; Schrauwen, B.; Van Campenhout, J.; Stroobandt, D.; Van Brussel, H.; Nuttin, M. Pruning and regularization in reservoir computing. Neurocomputing
**2009**, 72, 1534–1546. [Google Scholar] [CrossRef] - Rodan, A.; Tiňo, P. Minimum Complexity Echo State Network. IEEE Trans. Neural Netw.
**2011**, 22, 131–144. [Google Scholar] [CrossRef] [PubMed] - Grigoryeva, L.; Henriques, J.; Larger, L.; Ortega, J.P. Stochastic nonlinear time series forecasting using time-delay reservoir computers: Performance and universality. Neural Netw.
**2014**, 55, 59. [Google Scholar] [CrossRef] [PubMed] - Nguimdo, R.M.; Verschaffelt, G.; Danckaert, J.; Van der Sande, G. Simultaneous Computation of Two Independent Tasks Using Reservoir Computing Based on a Single Photonic Nonlinear Node With Optical Feedback. IEEE Trans. Neural Netw. Learn. Syst.
**2015**, 26, 3301–3307. [Google Scholar] [CrossRef] [PubMed] - Griffith, A.; Pomerance, A.; Gauthier, D.J. Forecasting chaotic systems with very low connectivity reservoir computers. Chaos
**2019**, 29, 123108. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Carroll, T.L. Path length statistics in reservoir computers. Chaos
**2020**, 30, 083130. [Google Scholar] [CrossRef] - Zheng, T.Y.; Yang, W.H.; Sun, J.; Xiong, X.Y.; Li, Z.T.; Zou, X.D. Parameters optimization method for the time-delayed reservoir computing with a nonlinear duffing mechanical oscillator. Sci. Rep.
**2021**, 11, 997. [Google Scholar] [CrossRef] - Ortín, S.; Pesquera, L. Reservoir Computing with an Ensemble of Time-Delay Reservoirs. Cogn. Comput.
**2017**, 9, 327–336. [Google Scholar] [CrossRef] - Röhm, A.; Lüdge, K. Multiplexed networks: Reservoir computing with virtual and real nodes. J. Phys. Commun.
**2018**, 2, 085007. [Google Scholar] [CrossRef] - Brunner, D. Photonic Reservoir Computing, Optical Recurrent Neural Networks; De Gruyter: Berlin, Germany, 2019. [Google Scholar]
- Gauthier, D.J.; Bollt, E.M.; Griffith, A.; Barbosa, W.A.S. Next generation reservoir computing. Nat. Commun.
**2021**, 12, 5564. [Google Scholar] [CrossRef] [PubMed] - Vandoorne, K.; Dambre, J.; Verstraeten, D.; Schrauwen, B.; Bienstman, P. Parallel reservoir computing using optical amplifiers. IEEE Trans. Neural Netw.
**2011**, 22, 1469–1481. [Google Scholar] [CrossRef] [PubMed] - Duport, F.; Schneider, B.; Smerieri, A.; Haelterman, M.; Massar, S. All-optical reservoir computing. Opt. Express
**2012**, 20, 22783–22795. [Google Scholar] [CrossRef] - Tanaka, G.; Yamane, T.; Héroux, J.B.; Nakane, R.; Kanazawa, N.; Takeda, S.; Numata, H.; Nakano, D.; Hirose, A. Recent advances in physical reservoir computing: A review. Neural Netw.
**2019**, 115, 100–123. [Google Scholar] [CrossRef] [PubMed] - Canaday, D.; Griffith, A.; Gauthier, D.J. Rapid time series prediction with a hardware-based reservoir computer. Chaos
**2018**, 28, 123119. [Google Scholar] [CrossRef] [Green Version] - Harkhoe, K.; Verschaffelt, G.; Katumba, A.; Bienstman, P.; Van der Sande, G. Demonstrating delay-based reservoir computing using a compact photonic integrated chip. Opt. Express
**2020**, 28, 3086. [Google Scholar] [CrossRef] - Freiberger, M.; Sackesyn, S.; Ma, C.; Katumba, A.; Bienstman, P.; Dambre, J. Improving Time Series Recognition and Prediction With Networks and Ensembles of Passive Photonic Reservoirs. IEEE J. Sel. Top. Quantum Electron.
**2020**, 26, 7700611. [Google Scholar] [CrossRef] [Green Version] - Waibel, A.; Hanazawa, T.; Hinton, G.E.; Shikano, K.; Lang, K.J. Phoneme recognition using time-delay neural networks. IEEE Trans. Signal Process.
**1989**, 37, 328–339. [Google Scholar] [CrossRef] - Karamouz, M.; Razavi, S.; Araghinejad, S. Long-lead seasonal rainfall forecasting using time-delay recurrent neural networks: A case study. Hydrol. Process.
**2008**, 22, 229–241. [Google Scholar] [CrossRef] - Han, B.; Han, M. An Adaptive Algorithm of Universal Learning Network for Time Delay System. In Proceedings of the 2005 International Conference on Neural Networks and Brain, Beijing, China, 13–15 October 2005; Volume 3, pp. 1739–1744. [Google Scholar] [CrossRef]
- Ranzini, S.M.; Da Ros, F.; Bülow, H.; Zibar, D. Tunable Optoelectronic Chromatic Dispersion Compensation Based on Machine Learning for Short-Reach Transmission. Appl. Sci.
**2019**, 9, 4332. [Google Scholar] [CrossRef] [Green Version] - Bardella, P.; Drzewietzki, L.; Krakowski, M.; Krestnikov, I.; Breuer, S. Mode locking in a tapered two-section quantum dot laser: Design and experiment. Opt. Lett.
**2018**, 43, 2827–2830. [Google Scholar] [CrossRef] [PubMed] - Takano, K.; Sugano, C.; Inubushi, M.; Yoshimura, K.; Sunada, S.; Kanno, K.; Uchida, A. Compact reservoir computing with a photonic integrated circuit. Opt. Express
**2018**, 26, 29424–29439. [Google Scholar] [CrossRef] [PubMed] - Appeltant, L.; Soriano, M.C.; Van der Sande, G.; Danckaert, J.; Massar, S.; Dambre, J.; Schrauwen, B.; Mirasso, C.R.; Fischer, I. Information processing using a single dynamical node as complex system. Nat. Commun.
**2011**, 2, 468. [Google Scholar] [CrossRef] [Green Version] - Paquot, Y.; Duport, F.; Smerieri, A.; Dambre, J.; Schrauwen, B.; Haelterman, M.; Massar, S. Optoelectronic Reservoir Computing. Sci. Rep.
**2012**, 2, 1–6. [Google Scholar] [CrossRef] [PubMed] - Brunner, D.; Penkovsky, B.; Marquez, B.A.; Jacquot, M.; Fischer, I.; Larger, L. Tutorial: Photonic neural networks in delay systems. J. Appl. Phys.
**2018**, 124, 152004. [Google Scholar] [CrossRef] - Brunner, D.; Soriano, M.C.; Mirasso, C.R.; Fischer, I. Parallel photonic information processing at gigabyte per second data rates using transient states. Nat. Commun.
**2013**, 4, 1364. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wolters, J.; Buser, G.; Horsley, A.; Béguin, L.; Jöckel, A.; Jahn, J.P.; Warburton, R.J.; Treutlein, P. Simple Atomic Quantum Memory Suitable for Semiconductor Quantum Dot Single Photons. Phys. Rev. Lett.
**2017**, 119, 060502. [Google Scholar] [CrossRef] [Green Version] - Jiang, N.; Pu, Y.F.; Chang, W.; Li, C.; Zhang, S.; Duan, L.M. Experimental realization of 105-qubit random access quantum memory. NPJ Quantum Inf.
**2019**, 5, 28. [Google Scholar] [CrossRef] - Katz, O.; Firstenberg, O. Light storage for one second in room-temperature alkali vapor. Nat. Commun.
**2018**, 9, 2074. [Google Scholar] [CrossRef] [Green Version] - Arecchi, F.T.; Giacomelli, G.; Lapucci, A.; Meucci, R. Two-dimensional representation of a delayed dynamical system. Phys. Rev. A
**1992**, 45, R4225. [Google Scholar] [CrossRef] [PubMed] - Zajnulina, M.; Lingnau, B.; Lüdge, K. Four-wave Mixing in Quantum Dot Semiconductor Optical Amplifiers: A Detailed Analysis of the Nonlinear Effects. IEEE J. Sel. Top. Quantum Electron.
**2017**, 23, 3000112. [Google Scholar] [CrossRef] - Lingnau, B.; Lüdge, K. Quantum-Dot Semiconductor Optical Amplifiers. In Handbook of Optoelectronic Device Modeling and Simulation; Series in Optics and Optoelectronics; Piprek, J., Ed.; CRC Press: Boca Raton, FL, USA, 2017; Volume 1, Chapter 23. [Google Scholar] [CrossRef]
- Mackey, M.C.; Glass, L. Oscillation and chaos in physiological control systems. Science
**1977**, 197, 287. [Google Scholar] [CrossRef] - Atiya, A.F.; Parlos, A.G. New results on recurrent network training: Unifying the algorithms and accelerating convergence. IEEE Trans. Neural Netw.
**2000**, 11, 697–709. [Google Scholar] [CrossRef] [Green Version] - Lorenz, E.N. Deterministic nonperiodic flow. J. Atmos. Sci.
**1963**, 20, 130. [Google Scholar] [CrossRef] [Green Version] - Goldmann, M.; Mirasso, C.R.; Fischer, I.; Soriano, M.C. Exploiting transient dynamics of a time-multiplexed reservoir to boost the system performance. In Proceedings of the 2021 International Joint Conference on Neural Networks (IJCNN), Shenzhen, China, 18–22 July 2021; pp. 1–8. [Google Scholar] [CrossRef]
- Ortín, S.; Soriano, M.C.; Pesquera, L.; Brunner, D.; San-Martín, D.; Fischer, I.; Mirasso, C.R.; Gutierrez, J.M. A Unified Framework for Reservoir Computing and Extreme Learning Machines based on a Single Time-delayed Neuron. Sci. Rep.
**2015**, 5, 14945. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Köster, F.; Yanchuk, S.; Lüdge, K. Insight into delay based reservoir computing via eigenvalue analysis. J. Phys. Photonics
**2021**, 3, 024011. [Google Scholar] [CrossRef] - Köster, F.; Ehlert, D.; Lüdge, K. Limitations of the recall capabilities in delay based reservoir computing systems. Cogn. Comput.
**2020**, 2020, 1–8. [Google Scholar] [CrossRef] - Röhm, A.; Jaurigue, L.C.; Lüdge, K. Reservoir Computing Using Laser Networks. IEEE J. Sel. Top. Quantum Electron.
**2019**, 26, 7700108. [Google Scholar] [CrossRef] - Manneschi, L.; Ellis, M.O.A.; Gigante, G.; Lin, A.C.; Del Giudice, P.; Vasilaki, E. Exploiting Multiple Timescales in Hierarchical Echo State Networks. Front. Appl. Math. Stat.
**2021**, 6, 76. [Google Scholar] [CrossRef] - Stelzer, F.; Röhm, A.; Lüdge, K.; Yanchuk, S. Performance boost of time-delay reservoir computing by non-resonant clock cycle. Neural Netw.
**2020**, 124, 158–169. [Google Scholar] [CrossRef] - Nooteboom, P.D.; Feng, Q.Y.; López, C.; Hernández-García, E.; Dijkstra, H.A. Using network theory and machine learning to predict El Niño. Earth Syst. Dyn.
**2018**, 9, 969–983. [Google Scholar] [CrossRef] [Green Version]

**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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## Share and Cite

**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