# The Effect of Blue Noise on the Optimization Ability of Hopfield Neural Network
## 4 Results and Discussion
### Blue Noise Hopfield Neural Networks in Optimization Problems
#### 4.1.1 The Effect of Blue Noise on the Ability of Neural Networks to Optimize Continuous Functions  
The statistical analysis of the Hopfield neural network's ability to optimize Equation (24) [39, 40] under blue noise perturbations with different signal-to-noise ratios. \[f(x_{1},x_{2})=(x_{1}-0.7)^{2}[(x_{2}+0.6)^{2}+0.1]+(x_{2  
Figure 3 shows the noise-free scenario, and the solution process of the neural network is as follows: the network reaches an unstable state after the search, at which time the network starts gradient descent and soon reaches a stable state, searching for the global minimum point (0.7, 0.5) with a minimum value of 0.  
It can be seen from Figure 4 that when the _SNR_ is 30, it has a great impact on the neural network and never reaches a stable state, and after 1000 iterations, the global minimum point (0.6917, 0.5043) is searched for at this time, with a minimum value of 1.1537 \(\times\) 10\({}^{-4}\) and a minimum value of 1.12 \(\times\) 10\({}^{-2}\) in the literature [41].  
Figure 5 shows the solution of the neural network when the _SNR_ is 50. Although after 500 iterations, a stable state is still not reached, the floating range is very close to the correct solution, and at this point, the global minimum point (0.7014, 0.5002) is searched  
Figure 5: Changes of \(\times 1\) and \(\times 2\) and energy evolution in a noisy environment (_SNR_ = 50).  
Figure 6: Changes of \(\times 1\) and \(\times 2\) and energy evolution in a noisy environment (_SNR_ = 90).  
for, with a minimum value of \(2.5668\times 10^{-6}\) and the minimum value of \(5.3560\times 10^{-6}\) in the literature [41].  
It can be seen from Figure 6 that when the \(SNR\) is 90, the interference of noise for the neural network is basically negligible and achieves the same effect as when there is no noise, when the global minimum point (0.7000, 0.5000) is searched for with a minimum value of \(6.0226\times 10^{-10}\) and a minimum value of \(6.2693\times 10^{-10}\) in the literature [41].  
From the above experiments, it can be seen that the presence of noise has an impact on the optimization ability of Hopfield neural network; in particular, when the signal-to-noise ratio is less than 50, the convergence speed of the energy function is greatly affected, and the impact on the solution ability is also great, but when the signal-to-noise ratio is greater than 90, the noise has little effect on the network model, and the optimization performance of this network model reaches the same degree as the ideal noise-free environment. The comparison with the optimization results conducted in the literature [41] also shows the superiority of this model, and the range of the anti-interference ability of the Hopfield neural network under the influence of blue noise is sought.  
1.2 The Effect of Blue Noise on the Optimization Power of Neural Networks for Combinatorial Problems  
In this paper, the colored noise Hopfield neural network is applied to the 10-city traveling salesman problem (TSP) [42; 43; 44]. The traveling salesman problem is described as the shortest loop required to visit each city once and return to the starting city, given the distance between two cities.  
The energy function to take the shortest path and satisfy all constraints [45; 46] is shown in Equation (25). Where \(V_{xi}\) denotes that the \(x\)th city is visited on the \(i\)th order and \(d_{xy}\) is the distance between cities \(x\) and \(y\). Therefore, the global minimum \(E\) value reached after several iterations represents the shortest effective path.  
\[E=\frac{A}{2}\sum_{x=1}^{n}\,(\sum_{i=1}^{n}V_{xi}-1)^{2}+\frac{B}{2}\sum_{i=1 }^{n}\,(\sum_{x=1}^{n}\,V_{xi}-1)^{2}+\frac{D}{2}\sum_{x=1}^{n}\,\sum_{y=1}^{ n}\,d_{xy}V_{xi}V_{y,i+1} \tag{25}\]  
In this paper, we use 10 cities after classical normalization, with the following coordinates: (0.4, 0.4439); (0.2439, 0.1463); (0.1707, 0.2293); (0.2293, 0.716); (0.5171, 0.9414); (0.8732, 0.6536); (0.6878, 0.5219); (0.8488, 0.3609); (0.6683, 0.2536); and (0.6195, 0.2634).