# Neutrosophic Compound Orthogonal Neural Network and Its Applications in Neutrosophic Function Approximation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts and Operations of NsNs

^{−}, I

^{+}] are combined. Hence, it can depict and express the certain and/or uncertain information in indeterminate problems.

^{−}, I

^{+}] is possibly specified in actual applications to satisfy some applied requirement. For instance, the indeterminacy I is specified as such a possible interval I ∈ [0, 2]. Thus, it is equivalent to N = [5, 11]. If I ∈ [1, 3], then there is N = [8, 14]. It is obvious that it is a changeable interval depending on the specified indeterminate range of I ∈ [I

^{−}, I

^{+}], which is also denoted by N = [c + uI

^{−}, c + uI

^{+}].

_{1}= c

_{1}+ u

_{1}I and N

_{2}= c

_{2}+ u

_{2}I for N

_{1}, N

_{2}∈ U and I ∈ [I

^{−}, I

^{+}], then their operational laws are introduced as follows [21]:

**x, I**): U

^{n}→ U for

**x**= [x

_{1}, x

_{2}, …, x

_{n}]

^{T}∈ U

^{n}and I ∈ [I

^{−}, I

^{+}], which is then a neutrosophic nonlinear or linear function.

^{−}, I

^{+}] is a neutrosophic nonlinear function, while ${y}_{2}(x,I)={N}_{1}{x}_{1}^{}+{N}_{2}{x}_{2}^{}+{N}_{3}=({c}_{1}+{u}_{1}I){x}_{1}^{}+({c}_{2}+{u}_{2}I){x}_{2}^{}+({c}_{3}+{u}_{3}I)$ for

**x**= [x

_{1}, x

_{2}]

^{T}∈ U

^{2}and I ∈ [I

^{−}, I

^{+}] is a neutrosophic linear function.

**x**and y(

**x**) are NsNs (usually, but not always).

## 3. NCONN with NsNs

_{j}(j = 1, 2, …, p); x

_{k}(k = 1, 2, …, n) is the kth NsN input signal; y

_{k}is the kth NsN output signal; and p is the number of the hidden layer neutrosophic neurons.

^{−}, I

^{+}], the actual output value is given as:

^{−}, I

^{+}] is given as follows:

^{−}, I

^{+}], the learning algorithm of NCONN permits changeable interval operations, which are different from existing neural network algorithms and show its advantage of approximating neutrosophic nonlinear functions/NsN data in an uncertain/NsN setting.

## 4. NsN Nonlinear Function Approximation Applied by the Proposed NCONN

**Example 1**. Supposing there is a neutrosophic nonlinear function:

**Example 2**. Considering a neutrosophic nonlinear function:

## 5. Actual Example on the Approximation of the JRC NsNs Based on the Proposed NCONN

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A three-layer feedforward neutrosophic compound orthogonal neural network (NCONN structure).

**Figure 2.**The desired output ${y}_{1d}=[{y}_{1d}^{-},{y}_{1d}^{+}]$ and actual output ${y}_{1}=[{y}_{1}^{-},{y}_{1}^{+}]$ of the proposed NCONN.

**Figure 3.**The desired output ${\mathrm{y}}_{2d}=[{y}_{2d}^{-},{y}_{2d}^{+}]$ and actual output ${y}_{2}=[{y}_{2}^{-},{y}_{2}^{+}]$ of the proposed NCONN.

**Figure 4.**The proposed NCONN approximation results of the JRC NsN data regarding different sampling lengths L.

NCONN Structure | α | λ | The Number of the Specified Learning Iteration | $\tilde{E}$ |
---|---|---|---|---|

1 × 8 × 1 | 2.5 | 0.25 | 20 | [3.2941, 8.5088] |

NCONN Structure | α | λ | The Number of the Specified Learning Iteration | $\tilde{E}$ |
---|---|---|---|---|

1 × 8 × 1 | 8 | 0.3 | 20 | [0.5525, 1.1261] |

**Table 3.**NsN data of rock joint roughness coefficient (JRC) regarding different sampling lengths for I ∈ [0, 1].

Sample Length L (cm) | x_{k} | JRC | y_{k} |
---|---|---|---|

9.8 + 0.4I | [9.8, 10.2] | 8.321 + 6.231I | [8.321, 14.552] |

19.8 + 0.4I | [19.8, 20.2] | 7.970 + 6.419I | [7.970, 14.389] |

29.8 + 0.4I | [29.8, 30.2] | 7.765 + 6.529I | [7.765, 14.294] |

39.8 + 0.4I | [39.8, 40.2] | 7.762 + 6.464I | [7.762, 14.226] |

49.8 + 0.4I | [49.8, 50.2] | 7.507 + 6.64I | [7.507, 14.147] |

59.8 + 0.4I | [59.8, 60.2] | 7.417 + 6.714I | [7.417, 14.131] |

69.8 + 0.4I | [69.8, 70.2] | 7.337 + 6.758I | [7.337, 14.095] |

79.8 + 0.4I | [79.8, 80.2] | 7.269 + 6.794I | [7.269, 14.063] |

89.8 + 0.4I | [89.8, 90.2] | 7.210 + 6.826I | [7.210, 14.036] |

99.8 + 0.4I | [99.8, 100.2] | 7.156 + 6.855I | [7.156, 14.011] |

NCONN Structure | α | λ | The Number of the Specified Learning Iteration | $\tilde{E}$ |
---|---|---|---|---|

1 × 8 × 1 | 8 | 0.11 | 5 | [3.2715, 22.3275] |

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Ye, J.; Cui, W.
Neutrosophic Compound Orthogonal Neural Network and Its Applications in Neutrosophic Function Approximation. *Symmetry* **2019**, *11*, 147.
https://doi.org/10.3390/sym11020147

**AMA Style**

Ye J, Cui W.
Neutrosophic Compound Orthogonal Neural Network and Its Applications in Neutrosophic Function Approximation. *Symmetry*. 2019; 11(2):147.
https://doi.org/10.3390/sym11020147

**Chicago/Turabian Style**

Ye, Jun, and Wenhua Cui.
2019. "Neutrosophic Compound Orthogonal Neural Network and Its Applications in Neutrosophic Function Approximation" *Symmetry* 11, no. 2: 147.
https://doi.org/10.3390/sym11020147