2.1

Chaotic Mapping Based On Hyper Space Dimensional

Hyper dimensional chaos is a common phenomenon in nonlinear systems. It is a random motion state obtained from the deterministic equations[7].hyperspace dimensional chaotic motion is characterized by randomness, universality and sensitivity to the initial value. It is a kind of random and regular motion model and state structure[8]. Therefore, the application of hyperspace dimensional chaotic mapping to artificial neural network can overcome the defects of artificial neural network into local convergence. In this paper, we propose a concept of hyperspace dimensional and infinite folding inversion iterative chaos mapping, and discuss its chaos strictly from the mathematical point of view. By calculating the maximum Lyapunov exponent, we prove that the system has chaotic features[9]. The hyperspace dimensional chaotic system has a strong adaptive control ability, and the hyperspace dimensional infinite folding inversion iterative chaotic mapping is described as follows:

First, generate a sequence of variables that are distributed between [-1, 1]

Wherex_k is the k-th component, a is control parameter, and generally a can take 5.65. By taking N samples of intelligent frogs with typical characteristic differences, we can get the chaotic variables of N trajectories with self-similar features. In the artificial intelligence control application, the variable interval of each variable is different, and the hyperspace dimensional chaotic variable needs to be controlled by the interval (-1,1) carrier to the ultra-pure interval of the particle swarm. Set x_lmaxval and x_lminval to be the upper and lower bounds of the l-th dimension variable, respectively. According to the following formula, we get the chaotic variable characteristic solution mapped to (-1,1).

According to the above formula to update the hyper space dimensional chaotic variable cx_l, according to the following formula into a decision variable x_l:

By the above equation, we obtain hyperspace dimensional infinite folding iterative chaotic mapping iterative variable, and adaptive control of artificial BP neural network under infinite folding inversion iteration. The learning process is guided by the hyperspace dimensional chaotic map, which is divided into is divided into two processes: forward spread and backward spread. The literatures have proved that a 3-layer BP neural network can close in on the nonlinear function very precisely. Next, take the 3-layer neural network as an example. Assume that the input layer has an L node, the hidden layer has m nodes, and the output layer has n nodes. BP artificial neural network mean square error function is:

Where q is the number of input sample of the artificial neural network system, ε_kis the output error of the k-th node in the output layer of the artificial neural network system,d_k is the expected output value of the k-th node,c_k is the actual output value of the k-th node.

2.2

Particle Swarm Optimization

In this paper, based on the concept of hyperspace dimensional infinite folding inversion iterative chaotic mapping, a PSO algorithm is introduced, and the early maturing judgment is carried out. For this algorithm, a group of particles is randomly initialized. Through its infinite evolution, the position of the particle is updated and evolved, and the optimal generalization solution P_b is obtained by adaptive adjustment. This optimal solution is called the individual extremum. With the progression of the particle swarm in layers of the neural network system, the global optimal solution is obtained. In the hyperspace dimensional chaotic mapping system, it is assumed that in the D-dimensional searching space, there are m particles to form a population. The position of the particle i in the D-dimensional space can be expressed as X_i = (x_i1,x_i2, … x_iD), The i-th group of individuals through the traversal, to achieve the optimal location logo is P_i = (p_i1, p_i2, … p_iD), As a result of the intelligent frog group network, the jump speed of each individual group is V_i = (v_i1,v_i2, …,v_iD),i = 1,2, … m, In the whole leap group, the optimal traversal position experienced by the individual in the state space is P_g = (P_g1, P_g2, … P_gD), For each generation of frogs, through tracking the state variables, update the single extremes and global state optimization values, the new position is

Where k is the number of iterations,c₁ and c₂ are learning factors. By tracking individual extremes and global extremes to update their own speed and position, it is:

Set rand() to a random number between [0,1], and ωas a linear decreasing inertia weight. Where the value of w is :

Where w_maxval is the upper limit of the inertial control weight of the infinite folding inversion iterative chaotic map, w_minval is the lower limit, T_maxval is the maximum of iterations, and t is the current iteration number. With the infinite folding inversion of iterative chaotic mapping artificial neural network control population evolution, the emergence of mapping state results clustering leads to artificial neural network system local precocity. In order to suppress the early maturity of this population, we propose to use the value of the fitness of the frog leaping swarm jumping transformation to adjust the state and make the decision.

Assuming the global particle number of the hyperspace dimensional chaotic frog leaping swarm of artificial neural network system is N, the fitness f_i of the frog leaping swarm jumping transformation of the i-th particle is defined, and the mean value is defined as f_avg. Then, the shuffled frog leaping swarm evolution average mean square error σ²is defined as:

Where f is the normalized factor, the effect is to limit the size of σ². The value f is: when max|f_i – f_avg| > 1,;when max|f_i – f_avg| ≤ 1, f =1.

Through the above formula, we can obtain the mean square root error σ² of the artificial shuffled frog leaping swarm neural network control system. The response of σ² to the physical properties is expressed as the degree of “aggregation” of the adaptive shuffled frog leaping swarm. The smaller the value σ² is, the greater the degree of “aggregation” is. With the increase of population evolution and convergence iteration, the individual fitness of the population tends to be self-similar, resulting in convergence characteristics, which will become smaller and smaller. When σ² > C, and the fitness does not meet the end conditions, it is necessary for early maturity treatment, that is, chaotic transformation operation.

3.1

Shuffled Frog Leaping Swarm Algorithm

In recent years, many scholars have used the bionic genetic algorithm such as bee colony, ant colony and frog leaping swarm to train the neural network as an artificial intelligence method, and achieved good results. The shuffled frog leaping swarm algorithm can be well combined with the evolution of the module group, and can effectively overcome the shortcomings such as the slow convergence speed and the local extreme value of the feedforward neural network in the intelligent control design training, and the dependence on the initial weight. The shuffled frog swam algorithm is widely combined with artificial neural network(ANN) system with fast convergence speed, simple modeling and easy implementation. Its realization mechanism is to complete an intelligent solution through the simulation of the information interaction behavior of frog foraging information in the natural environment. The basic step of the algorithm implementation is described as follows:

3.2

Realization of Frog Leaping Swarm Neural Network Algorithm

In this paper, the hyperspace dimensional infinite folding inversion iterative chaotic map is combined with the gradient descent method based on error back transfer. The shuffled frog swam algorithm has a strong search ability for global optimal solution, but the search speed is slow near the optimal solution and prone to early maturity phenomenon. In order to solve the shortcomings of the traditional PSO algorithm, we introduce the hyperspace dimensional infinite folding inversion iterative chaotic mapping and the gradient descent algorithm based on error back transfer to enhance the performance of particle swarm optimization algorithm. When the PSO algorithm is caught in the local optimal solution, the chaotic map is introduced to adjust the weight parameter of the shuffled frog leaping swarm in the network distribution system, so that each individual of the frog leaping swarm can escape the local convergence point and realize the global search, so as to avoid the occurrence of precocious population. According to the convergence, the algorithm can automatically switch between global and local searches.

According to the PSO algorithm early maturity judgment mechanism, when the fitness variance σ²is less than a given threshold, it shows that the population appears precocious. At this time, the current global optimal position is taken as the initial point, and the chaos map is carried out according to a certain probability, leading the frog to escape the local extreme value region. In order to speed up the local search process, call the BP algorithm and re-enter the PSO optimization process. The iteration is done until the algorithm terminates. This algorithm can acquire the diversity of frog populations by using the chaotic map, and the individual frog can search the whole space under the premise of rapid local search. The algorithm implementation steps are as follows:

Through the above analysis and improvement, the algorithm is an artificial neural network control algorithm combined with the advantages of shuffled frog leaping swarm algorithm, particle swarm algorithm, chaos map and BP algorithm. The shuffled frog leaping swarm algorithm and the chaotic mapping algorithm are simpler and the training speed is faster than the simple BP algorithm. In terms of complexity, it is equivalent to the traditional PSO-BP and GA-BP algorithms. In terms of convergence, due to the introduction of hyperspace dimensional infinite folding inversion iterative chaotic mapping mechanism, the population particles have a strong global search ability; Due to the acceleration of local optimization of particles for BP, PSO performance has been a good play. Therefore, the convergence of the algorithm is superior, and the speed of convergence is faster than the traditional PSO-BP and GA-BP algorithm.