# The Effect of Blue Noise on the Optimization Ability of Hopfield Neural Network
## 4 Results and Discussion
### Chaotic Neural Network Model Based on Blue Noise  
Energy function selection is the same as before, \(\varepsilon_{0}=0.6\), \(\varepsilon_{1}=125\) remains unchanged, and _coef_ is equal to 0 and \(1/250\), respectively. Let the simulation program take the 30 cities after classical normalization. Let the emulator take the classic 30 cities [48], and the coordinates are as follows: (41,94); (37,84); (54,67); (25,62); (7,64); (2,99); (68,58); (71,44); (54,62); (83,69); (64,60); (18,54); (22,60); (83,46); (91,38); (25,38); (24,42); (58,69); (71,71); (74,78); (87,76); (18,40); (13,40); (82,7); (62,32); (58,35); (45,21); (41,26); (44,35); and (4,50). The shortest path of the 30-city TSP problem is 423.7406, and its shortest path is shown in Figure 8. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multirow{2}{*}{\(SNR\)} & **Legal** & **Optimal** & **Legal** & **Optimal** \\  & & **Path** & **Path** & **Ratio** & **Ratio** \\ \hline  & Noiseless & 163 & 118 & 81.5\% & 59.0\% \\ This paper & 40 & 40 & 7 & 20.0\% & 3.5\% \\ Reference [41] & 40 & 84 & 13 & 42.0\% & 6.5\% \\ This paper & 50 & 118 & 54 & 59.0\% & 27.0\% \\ Reference [41] & 126 & 64 & 63.0\% & 32.0\% \\ This paper & 60 & 156 & 111 & 78.0\% & 55.5\% \\ Reference [41] & 60 & 144 & 92 & 72.5\% & 47.0\% \\ This paper & 162 & 116 & 81.0\% & 58.0\% \\ Reference [41] & 70 & 145 & 94 & 80.0\% & 56.0\% \\ This paper & 163 & 117 & 81.5\% & 58.5\% \\ Reference [41] & 80 & 160 & 112 & 75.5\% & 53.0\% \\ This paper & 90 & 169 & 128 & 84.5\% & 64.0\% \\ Reference [41] & 151 & 106 & 74.5\% & 51.5\% \\ This paper & 161 & 109 & 80.5\% & 54.5\% \\ Reference [41] & 149 & 103 & 74.5\% & 51.5\% \\ This paper & 164 & 113 & 82.0\% & 55.5\% \\ Reference [41] & 149 & 104 & 74.5\% & 52.0\% \\ This paper & 166 & 114 & 83.0\% & 57.0\% \\ Reference [41] & 180 & 152 & 103 & 76.0\% & 51.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental simulation results for different values of the parameter \(SNR\). When using the Hopfield neural network model with blue noise perturbation to solve the TSP problem, a self-feedback term \(z_{i}(t)\) with a piece-wise simulated annealing mechanism is introduced, where the formula is as follows:  
\[z_{i}(t+1)= \begin{cases}(1-\beta_{1})z_{i}(t),&z_{i}(t)>0.4\\ (1-\beta_{2})z_{i}(t),&\text{\emph{otherwise}}\end{cases} , \tag{26}\]  
Let \(\beta_{1}=0.01\), \(\beta_{2}=0.02\), and \(z_{i}(1)=0.1\). This enables the neural network not only to give full play to the chaotic search ability but also to maintain a reasonable convergence rate. In the simulation experiment, different _SNR_ and excitation function coefficients _coef_ are used, and the experimental results are compared and analyzed. Table 3 shows the simulation test data of 100 times randomly assigned initial values when different values of colored noise parameters are taken. The program simulation results are shown in the table below:  
As can be seen from the above table, the improved neural network model performs perfectly when solving the legal path, which is significantly improved compared with the previous neural network model. As noise will affect network optimization within a certain range, it can be seen that when _SNR_ = 30, the noise interference is too strong and the performance of the neural network on the optimal path is not as good as that without noise interference. However, when _SNR_ = 60, the noise disturbance enhances the chaos of the  
Figure 8: The optimal 30-city path. \begin{table}
\begin{tabular}{c c c c c c} \hline _coef_ & _SNR_ & **Legal Path** & **Optimal Path** & **Legal Ratio** & **Optimal Ratio** \\ \hline  & 0 & 91 & 21 & 91.0\% & 21.0\% \\ _coef_ = 0 & 30 & 87 & 12 & 87.0\% & 12.0\% \\  & 60 & 99 & 26 & 99.0\% & 26.0\% \\  & 90 & 93 & 22 & 93.0\% & 22.0\% \\  & 0 & 100 & 27 & 100\% & 27.0\% \\ _coef_ = & 30 & 100 & 21 & 100\% & 21.0\% \\  & 60 & 100 & 31 & 100\% & 31.0\% \\  & 90 & 100 & 27 & 100\% & 27.0\% \\ \hline \end{tabular}
\end{table} 
Table 3: Simulation results of the solved 30-city travel salesman problem.