# A Ferroelectric Memristor-Based Transient Chaotic Neural Network for Solving Combinatorial Optimization Problems

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## Abstract

**:**

## 1. Introduction

## 2. Ferroelectric Memristor Model

_{3}/Nb:SrTiO

_{3}FTJ [21]), while S is a value between 0 and 1 depending on the amplitude and duration of the applied pulse.

## 3. The FM-TCNN

## 4. FM-TCNN for TSP

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Some examples of the Lorentzian distribution of switching time at different applied voltages.

**Figure 2.**Area fractions of downward domains as a function of pulse time at different pulse amplitude. (

**a**) The applied pulse voltage is positive, and all the domains are initially aligned upward. (

**b**) The applied pulse voltage is negative, and all the domains are initially aligned downward.

**Figure 3.**Area fractions of downward domains as a function of pulse amplitude at different pulse durations. (

**a**) Schematic illustration of the applied pulse train. Multiple (

**b**) $S-V$ and (

**c**) $G-V$ hysteresis loops obtained with the application of pulse trains, as schematically shown in (

**a**), at different pulse durations.

**Figure 5.**Dynamic characteristics of a single neuron of TCNN. Here $\epsilon =1/\phantom{1250,}\phantom{\rule{0.0pt}{0ex}}250,\phantom{\rule{3.33333pt}{0ex}}\alpha =0.015,$$k=0.9,{I}_{0}=0.65,\phantom{\rule{3.33333pt}{0ex}}y\left(1\right)=0.08,\phantom{\rule{3.33333pt}{0ex}}z\left(1\right)=0.08$.

**Figure 6.**The potential circuit implementation of Equation (10).

**Figure 7.**Dynamic characteristics of a single neuron of FM-TCNN. (

**a**) The iteration of $x\left(t\right)$. (

**b**) The iteration of $z\left(t\right)$.

**Figure 9.**Iterations of neuron outputs with steady-state values equal to 1 for the 10-city TSP. Here, ${W}_{1}=2,\phantom{\rule{3.33333pt}{0ex}}{W}_{2}=1.1,\phantom{\rule{3.33333pt}{0ex}}b=0.01,\phantom{\rule{3.33333pt}{0ex}}c=-2,\phantom{\rule{3.33333pt}{0ex}}\gamma =50,\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\Delta t=1\times {10}^{-7}$.

b | −0.05 | −0.02 | −0.01 | 0 | 0.01 | 0.02 | 0.05 |
---|---|---|---|---|---|---|---|

Global optimum | 861 | 899 | 931 | 938 | 966 | 956 | 946 |

Local optimum | 139 | 101 | 69 | 62 | 34 | 44 | 54 |

Infeasible solution | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Average iterations for convergence | 432.994 | 498.198 | 526.895 | 548.036 | 570.581 | 595.305 | 666.763 |

Average distance | 2.7199 | 2.721 | 2.7041 | 2.696 | 2.6934 | 2.6972 | 2.7007 |

TCNN | HPM-TCNN | FM-TCNN | |
---|---|---|---|

$\mathit{\beta}=\mathbf{0.001}$ | $\mathit{b}=\mathbf{100},$ $\mathit{c}=\mathbf{1.25}\times {\mathbf{10}}^{-\mathbf{5}}$ | $\mathit{b}=\mathbf{0.01},\mathit{c}=-\mathbf{2}$ | |

Global optimum | 943 | 853 | 966 |

Local optimum | 57 | 144 | 34 |

Infeasible solution | 0 | 3 | 0 |

Average iterations for convergence | 757.822 | 1423.019 | 570.581 |

Average distance | 2.6961 | 2.7121 | 2.6934 |

City Scale | b | The Best Path Length | The Average Path Length | Feasible Solution Rate |
---|---|---|---|---|

10-city TSP (Global optimum path length: 2.6907) | 0.43 | 2.6907 | 2.6907 | 100% |

0.44 | 2.6907 | 2.6907 | 100% | |

0.45 | 2.6907 | 2.6907 | 100% | |

0.46 | 2.6907 | 2.6907 | 100% | |

0.47 | 2.6907 | 2.6907 | 100% | |

30-city TSP (Global optimum path length: 423.74) | 0.43 | 429.38 | 467.8 | 94% |

0.44 | 429.38 | 460.58 | 92% | |

0.45 | 435.00 | 469.53 | 98% | |

0.46 | 429.53 | 468.31 | 94% | |

0.47 | 429.38 | 463.27 | 96% | |

50-city TSP (Global optimum path length: 428.10) | 0.43 | 457.61 | 514.48 | 68% |

0.44 | 473.43 | 514.83 | 80% | |

0.45 | 482.99 | 538.85 | 74% | |

0.46 | 469.88 | 569.32 | 80% | |

0.47 | 496.34 | 595.38 | 66% | |

75-city TSP (Global optimum path length: 543.45) | 0.43 | 575.1 | 615.96 | 68% |

0.44 | 587.82 | 616.47 | 66% | |

0.45 | 578.30 | 624.37 | 62% | |

0.46 | 583.26 | 616.91 | 68% | |

0.47 | 580.84 | 613.07 | 82% |

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**MDPI and ACS Style**

Lin, Z.; Fan, Z. A Ferroelectric Memristor-Based Transient Chaotic Neural Network for Solving Combinatorial Optimization Problems. *Symmetry* **2023**, *15*, 59.
https://doi.org/10.3390/sym15010059

**AMA Style**

Lin Z, Fan Z. A Ferroelectric Memristor-Based Transient Chaotic Neural Network for Solving Combinatorial Optimization Problems. *Symmetry*. 2023; 15(1):59.
https://doi.org/10.3390/sym15010059

**Chicago/Turabian Style**

Lin, Zhuosheng, and Zhen Fan. 2023. "A Ferroelectric Memristor-Based Transient Chaotic Neural Network for Solving Combinatorial Optimization Problems" *Symmetry* 15, no. 1: 59.
https://doi.org/10.3390/sym15010059