Research And Optimization Of The Four-color Map Problem Based On Chaotic Neural Network

Artificial neural network, what has rapid development based on the research of human cognitive process, has evolved into many subjects, but it is still the central issue of machine learning methods and the construction of machine learning problems. It is the simulation of the human brain information processing disciplines which is premise of human intelligent behavior. Chaos and Quantum and Relativity as three of the last century of the physic discovery have rapid development. While the feature of human brain's memory is chaotic phenomenon, so it is natural that contact the chaos with artificial neural network to generate a new subject—chaotic neural network. Chaotic neural network by its unique complex dynamic has made brilliant achievements both in information processing and optimization calculation. And it is widely regarded as one of the best way to solve NP-complete problems. Different from conventional neural network, it has richer types of attractors instead of down search methods for optimizing search. This paper describes a more comprehensive definition and concept of chaos and the development of its history, the feature and measure of chaos, chaotic neural network model and so on.Four-color map problem is one of the three world's mathematical problems and has not been mathematical proof, after K.Appel and W.Haken use order approximation method to solve this problem by computer-aided witch consumed about 1200 hours, Dahldl is a pioneer who use Hopfield neural network (HNN) to solve four-color map problem first, and achieved good results, but it has two drawbacks in his model: Firstly many valid solutions: secondly it is easy to fall into local minima. The chaotic neural network model, which I used to solve this problem in this paper, can both avoid the local minimum and search effectively, On this basis , I take a new Gaussian wavelet as the activation function for further improve search accuracy. Wavelet analysis as a wonderful work in mathematics has a reputation called a mathematical microscope. It has an advantage that analyze signal in multi- scale analysis which different from the Fourier transform. It integrated into the conventional chaotic neural network which has only monotonous Sigmoid function as an activation function. It can develop search properties and make more advantages on the global search. Finally, I split monotonous annealing parameters of the self-feedback connection weights. The traditional chaotic neural network only use a single annealing parameters, and it cause the search process was rather monotonous, This paper adopt sub-annealing parameters, I take smaller parameter values in early chaotic, so It can make the network fully chaotic, than I take a larger value of the annealing parameters for speeding up convergence process in chaos post , It make all the network more abundantly. The simulation results proved my model is better than HNN both search efficiency and the ability of function approximation.