# The Effect of Blue Noise on the Optimization Ability of Hopfield Neural Network
## 5 Conclusions  
In this paper, the color noise generation scheme with custom spectral slope is improved so that the improved generator can meet the needs of neural networks. A novel method combining colored noise and an original signal sequence according to signal-to-noise ratio is proposed, and the effect of noise on the Hopfield neural network is systematically analyzed by introducing blue noise into a simulation experiment. Moreover, the paper improves the original neural network by changing the excitation function and introducing the piece-wise simulated annealing mechanism to improve the performance of the neural network in the face of more complex combinatorial optimization problems. In the simulation results, applying the Hopfield neural network based on colored noise in function optimization and combinatorial optimization, it can be seen that the Hopfield neural network has very good robustness in noise resistance ability, and the appropriate addition of noise can also help the neural network improve optimization ability. In conclusion, this study presents a new approach to training neural networks that can actively simulate real-world noise environments. This allows models to be trained in laboratory settings to account for noise disturbances in specific scenarios, which can prevent models from performing well in the laboratory but failing in real-world applications, ultimately improving the overall accuracy of the neural network. However, to establish the reliability and effectiveness of this method, further research is needed to evaluate its performance in real-world environments and to expand its training range and application.  
The research results indicate that it is possible to consider the noise conditions in specific environments during training for fusion in order to avoid a significant amount of repetitive sampling work. A training dataset can be created for models in various scenarios, provided that different noise characteristics are incorporated. Furthermore, adding noise perturbations can also be beneficial for improving the accuracy of neural networks. While the study evaluated the accuracy of the model under noise perturbations, further research is required to explore how to construct a more realistic spectral noise scenario and how to combine this approach with other neural network techniques, such as cyclic and graph neural networks. Our next research focus will be the examination of the corresponding noise perturbation of neural networks in different application fields by considering the noise characteristics specific to each scenario.  
Conceptualization, Y.Z.; methodology, Y.X.; formal analysis, Y.X.; investigation, L.L.; data curation, S.W. and Y.W.; writing--original draft, B.C.; writing--review and editing, Y.Z.; supervision, Y.X. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Nature Science Foundation of Heilongjiang Province, grant number LH2022F036.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest.