# Physical Reservoir Computing Enabled by Solitary Waves and Biologically Inspired Nonlinear Transformation of Input Data

## Abstract

**:**

## 1. Introduction

## 2. State-of-the-Art

#### 2.1. Traditional Reservoir Computing Algorithm

- 1
- Create a vector ${\mathbf{u}}_{n}$ of ${N}_{u}$ input values;
- 2
- Use a random number generator to define an input matrix ${\mathbf{W}}^{in}$ consisting of ${N}_{x}\times {N}_{u}$ elements and a recurrent weight matrix $\mathbf{W}$ containing ${N}_{x}\times {N}_{x}$ elements;
- 3
- 4
- Compute a vector ${\mathbf{x}}_{n}$ of ${N}_{x}$ neural activations as$$\begin{array}{c}\hfill {\mathbf{x}}_{n}=(1-\alpha ){\mathbf{x}}_{n-1}+\alpha tanh({\mathbf{W}}^{in}{\mathbf{u}}_{n}+\mathbf{W}{\mathbf{x}}_{n-1})\phantom{\rule{0.166667em}{0ex}},\end{array}$$
- 5
- Construct the state matrix $\mathbf{X}$ using the values of ${\mathbf{x}}_{n}$;
- 6
- Train the output as ${\mathbf{W}}^{out}={\mathbf{Y}}^{target}{\mathbf{X}}^{\top}{(\mathbf{X}{\mathbf{X}}^{\top}+\beta \mathbf{I})}^{-1}$, where $\mathbf{I}$ is the identity matrix, $\beta $ is a regularisation coefficient, ${\mathbf{X}}^{\top}$ is the transpose of $\mathbf{X}$, and ${\mathbf{Y}}^{target}$ is a matrix composed of target outputs ${\mathbf{y}}_{n}^{target}$ for each time instant ${t}_{n}$;
- 7
- Solve Equation (1) with a new set of target data ${\mathbf{u}}_{n}$, and compute the output vector as ${\mathbf{y}}_{n}={\mathbf{W}}^{out}[1;{\mathbf{u}}_{n};{\mathbf{x}}_{n}]$.

#### 2.2. Next Generation Algorithmic Reservoir Computing and Adjacent Techniques

#### 2.3. Physical Reservoir Computing Systems Based on Solitary Waves

## 3. Experimental Setup

## 4. Results

#### 4.1. Formation of the Nonlinear Functional

#### 4.2. Advantages for Generative Mode Operation

#### 4.3. Free-Running Forecast of Chaotic Time Series

## 5. Discussion

#### 5.1. Energy Efficiency, Power Consumption, and Cost

^{TM}PCIe Board with a BrainChip neuromorphic processor is USD 499, but an assembled ‘Development Kit’ system based on a personal computer costs USD 9995 and consumes 180 W [94]. Generally speaking, in light of an exponential increase in the computing power demand seen in the last decade [95], the liquid-based unconventional computing system appears to be a plausible alternative to conventional microelectronics [96,97]. Unconventional liquid-state computational systems can also outperform emergent photonics-based computers [98] in terms of energy efficiency since the latter may require high-intensity laser light to induce the nonlinear effects needed for a physical implementation of a neural network [70], but nonlinear processes in liquids can be obtained virtually effortlessly [82].

#### 5.2. Potential Applications

## 6. Conclusions

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AI | artificial intelligence |

DC | direct current |

ESN | Echo State Network |

FLOPS | floating point operations per second |

KdV | Korteweg–de Vries |

LSM | Liquid State Machine |

MGTS | Mackey–Glass time series |

RC | reservoir computing |

RAM | random access memory |

SL | solitary-like |

UV | ultraviolet |

USD | United States Dollar |

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**Figure 1.**Schematic representation of (

**a**) a traditional algorithmic RC system and (

**b**) an RC system with a reservoir of random connections substituted by a nonlinear functional of the input data.

**Figure 2.**(

**a**) Sketch and (

**b**) top view fluorescence photograph of the experimental setup used to validate the proposed architecture of the physical RC system. The fluorescent dye, UV light, and digital camera play an auxiliary role and can be removed from the setup without compromising its operation. The remaining components of the setup are controlled by an Arduino microcontroller, which is also used to process the raw data traces. Note that the curvature of the wavefront of SL waves due to the boundary effect does not affect the operation of the RC system since all measurements are taken on the centreline.

**Figure 3.**(

**a**) Input sinusoidal signal (the dotted curve) and the SL waves excited by it (the solid curve). (

**b**) Fourier spectra of the signals in Panel (

**a**). (

**c**) Free-running forecast of the future evolution of the sinusoidal waves made by the RC system based on the SL waves. Note that the timescale in Panel (

**c**) is unrelated to that in Panel (

**a**).

**Figure 4.**Input MGTS signal (the dotted curve) and the SL waves excited by it (the solid curve). Unlike in Figure 3a, since each variation of the MGTS results in the generation of SL waves with different amplitudes and propagation speeds, the SL waves collide and form more complex wave profiles.

**Figure 5.**Generative mode operation (free-running forecast) of (

**a**) physical RC system based on SL waves and (

**b**) traditional algorithmic RC system (the solid curve) compared with the target MGTS (the dotted curve). (

**c**) Modulus of the absolute error of the forecasts produced by the physical and traditional RC algorithmic systems. Note that, for the sake of comparison, the error of the traditional RC system is plotted with the negative sign.

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## Share and Cite

**MDPI and ACS Style**

Maksymov, I.S.
Physical Reservoir Computing Enabled by Solitary Waves and Biologically Inspired Nonlinear Transformation of Input Data. *Dynamics* **2024**, *4*, 119-134.
https://doi.org/10.3390/dynamics4010007

**AMA Style**

Maksymov IS.
Physical Reservoir Computing Enabled by Solitary Waves and Biologically Inspired Nonlinear Transformation of Input Data. *Dynamics*. 2024; 4(1):119-134.
https://doi.org/10.3390/dynamics4010007

**Chicago/Turabian Style**

Maksymov, Ivan S.
2024. "Physical Reservoir Computing Enabled by Solitary Waves and Biologically Inspired Nonlinear Transformation of Input Data" *Dynamics* 4, no. 1: 119-134.
https://doi.org/10.3390/dynamics4010007