The model and algorithm of BP neural network optimized by expanded multichain quantum optimization algorithm with super parallel and ultrahigh speed are proposed based on the analysis of the research status quo and defects of BP neural network to overcome the defects of overfitting, the random initial weights, and the oscillation of the fitting and generalization ability along with subtle changes of the network parameters. The method optimizes the structure of the neural network effectively and can overcome a series of problems existing in the BP neural network optimized by basic genetic algorithm such as slow convergence speed, premature convergence, and bad computational stability. The performance of the BP neural network controller is further improved. The simulation experimental results show that the model is with good stability, high precision of the extracted parameters, and good realtime performance and adaptability in the actual parameter extraction.
Artificial neural network (ANN) is a new information processing and computer system which is based on the modern neuroscience research and is formed by abstracting, simplifying, and simulating of biological structure. The features of ANN are as follows. ANN can fully approximate to any complex nonlinear relations; all quantitative or qualitative information keeps in storage equipotentially in each neuron of network so it is with very strong robustness and fault tolerance; ANN is a kind of system which can emulate and adapt to the unknown system and is able to deal with the quantitative and qualitative knowledge at the same time [
BP neural network is currently the most popular neural network model in application [
At present, the research based on ANN to carry out quantitative structureactivity relationship research has gradually been applied to various fields. Wang et al. [
Although BP algorithm has become the most widely used artificial neural network, the BP neural network has the following defects.
In order to further improve the efficiency of the BP neural network and overcome the shortage of it, a lot of research has been conducted. Sun et al. established improved BP neural network prediction model and quantitatively researched related parameters [
Genetic algorithm imitates evolutionary and genetic rule of biology and is a mathematical algorithm which can solve problems to find the global optimization. Due to the strong macroscopic search ability and good global optimization performance of genetic algorithm, many scholars try to use genetic algorithm to optimize the connection weights, structure, learning rules, and so forth of BP neural network. Xiao et al. [
In order to further overcome the shortage of the model based on genetic algorithm to optimize the BP neural network, the model and algorithm of BP neural network based on expanded multichain quantum optimization are proposed. The structure of neural network is effectively optimized. The model can overcome a series of problems of basic genetic algorithm, such as the slow convergence speed, premature convergence, and bad computational stability to further improve the performance of the BP neural network controller.
The main idea of BP algorithm is to divide learning process into two stages, that is, positive communication of signal and back propagation of error. In the stage of positive communication, input information is from the input layer to output layer through the hidden layer [
The simplest BP neural network is with three layers as is shown in Figure
Topology structure of BP neutral network.
The connection weight matrix between input layer nodes and hidden layer nodes is
The learning algorithm of BP network can be divided into two stages that are forward and back propagation. It is a gradient descent algorithm which can make error of per connection weights of the neural network reduce. At the beginning of learning, assign the random numbers in the range of
First, enter the learning samples
The data forward propagates to the output layer through the input layer and hidden layer. The produced weight value of output pattern classification is the learning result. The following steps are mainly included.
The input value of nodes in hidden layer is
The output value of node
The input values of nodes of output layer
The output value of node
Calculate the error of output value and expected value of output layer. The error back propagates from the output layer through the hidden layer to the input layer. The connection weight values are modified. Steps are as follows.
The error between the learning value
If
The learning process is ended when all learning samples meet the aforementioned conditions.
The learning error of node of output layer
Learning error of node of hidden layer
Set the weight value at time
Consider
Threshold of nodes of output layer is
Threshold of nodes of hidden layer
Run the BP neural network after learning and carry out pattern classification for the input vector
Assign the value revised by BP learning algorithm to the connection weight matrices
Input the input vector
Apply formulas (
Use formulas (
In quantum computation, the smallest unit of information is represented by quantum bit whose state can be expressed as
The complex numbers
Probability amplitude of quantum bit.
Obviously, in Figure
The coding scheme is
Obviously, in Figure
Decomposition of probability amplitude of quantum bit.
Formula (
Formula (
Similarly,
Increase of the supporting angle constantly like this can conclude the encoding scheme based on
It can be concluded from Figure
The process of the quantum genetic optimization based on multichain code is limited within the unit space
The update of the quantum rotation gate
Set
Set the maximal evolutionary generation to be
The mutation of quantum chromosome is realized by quantum Not gate
The specific form of
The main realization steps of multichain quantum genetic algorithm are as follows.
Generate population with
Map the multiple approximate solutions of each chromosome from unit space
Calculate the fitness of the approximate solution to obtain the contemporary optimal solution Best
Make Best
Do the iteration cycle,
Transform solution space of
Evaluate
If
Go back to Step
Connect each weight and threshold of BP network in order to form a long string of an array of real number. The string is as a chromosome. The decoding value of individual is the corresponding weights and thresholds.
The purpose of using multichain quantum algorithm to optimize BP network is to simplify the structure of the network to minimize the error of network. Therefore, the fitness function is
The combination of multichain quantum optimization algorithm and neural network is realized by using multichain quantum optimization algorithm to optimize the parameters of the neural network. The specific ways are as follows.
Use the algorithm to optimize the initial weights and thresholds of neural network.
Use the algorithm to optimize structure parameters of BP network, the main of which is to optimize the unit number of hidden layers of network.
Use the algorithm to optimize the selection the learning rate and momentum factor of BP network.
The first optimization way is adopted to realize the optimization of neural network in this paper. The specific implementation process is as below.
The basic idea of using multichain quantum optimization algorithm to optimize the initial weight and threshold value of BP network is as follows. The multichain quantum optimization algorithm is applied to ensure the appropriate initial weight and threshold value to find the optimal solution of BP algorithm.
The specific implementation steps of MCQABP (BP neural network based on multichain quantum optimization) algorithm are as follows.
Initial population with
Enter the neural network module. Apply BP algorithm to do the learning process and apply multichain quantum optimization algorithm to do the fitness inspection. If the fitness is unqualified, update the chromosome until the fitness meets the requirements. The individuals which meet requirements are updated by multichain quantum optimization algorithm to calculate fitness. The unqualified individuals are updated until they meet the accuracy requirement. Then end the algorithm.
Apply MCQABP algorithm to diagnose fault state of machine. The data sample after normalization of machine status is shown in Table
The training sample data.
Sample order  Characteristic value of sample  State 

1  0.2286, 0.1292, 0.072, 0.1592, 0.1335, 0.0733, 0.1159, 0.094, 0.0522, 0.1345, 0.009, 0.126, 0.3619, 0.069, 0.1828  Normal 
2  0.209, 0.0947, 0.1393, 0.1387, 0.2558, 0.09, 0.0771, 0.0882, 0.0393, 0.143, 0.0126, 0.167, 0.245, 0.0508, 0.1328  Normal 
3  0.0442, 0.088, 0.1147, 0.0563, 0.3347, 0.115, 0.1453, 0.0429, 0.1818, 0.0378, 0.0092, 0.2251, 0.1516, 0.0858, 0.067  Normal 
4  0.2603, 0.1715, 0.0702, 0.2711, 0.1491, 0.133, 0.0968, 0.1911, 0.2545, 0.0871, 0.006, 0.1793, 0.1002, 0.0789, 0.0909  Crack 
5  0.369, 0.2222, 0.0562, 0.5157, 0.1872, 0.1614, 0.1425, 0.1506, 0.131, 0.05, 0.0078, 0.0348, 0.0451, 0.0707, 0.088  Crack 
6  0.0359, 0.1149, 0.123, 0.546, 0.1977, 0.1248, 0.0624, 0.0832, 0.164, 0.1002, 0.0059, 0.1503, 0.1837, 0.1295, 0.07  Crack 
7  0.1759, 0.2347, 0.1829, 0.1811, 0.2922, 0.0655, 0.0774, 0.02273, 0.2056, 0.0925, 0.0078, 0.1852, 0.3501, 0.168, 0.2668  Defect 
8  0.0724, 0.1909, 0.134, 0.2409, 0.2842, 0.045, 0.0824, 0.1064, 0.1909, 0.1586, 0.0116, 0.1698, 0.3644, 0.2718, 0.2494  Defect 
9  0.2634, 0.2258, 0.1165, 0.1154, 0.1074, 0.0657, 0.061, 0.2623, 0.2588, 0.1155, 0.005, 0.0978, 0.1511, 0.2273, 0.322  Defect 
The test sample data.
Sample order  Characteristic value of sample  State 

1  0.2101, 0.095, 0.1298, 0.1359, 0.2601, 0.1001, 0.0753, 0.089, 0.0389, 0.1451, 0.0128, 0.159, 0.2452, 0.0512, 0.1319  Normal 
2  0.2593, 0.18, 0.0711, 0.2801, 0.1501, 0.1298, 0.1001, 0.1891, 0.2531, 0.0875, 0.0058, 0.1803, 0.0992, 0.0802, 0.1002  Crack 
3  0.2599, 0.2235, 0.1201, 0.0071, 0.1102, 0.0683, 0.0621, 0.2597, 0.2602, 0.1167, 0.0048, 0.1002, 0.1521, 0.2281, 0.3205  Defect 
Use BP, GABP, DCQABP (BP neural network based on doublechain quantum optimization algorithm), and MCQABP (BP neural network based on multichain quantum optimization algorithm, where the number of chain is 4). In GABP and DCQABP, the population number is set to be 50; the maximal evolutionary generation is set to be 30. The 10 times average values of the performance of algorithms are shown in Table
The experiment results of the four algorithms.
Algorithm  Error  1  2  3  4  5  6  7  8  9  10  Average value 

BP  Error 1  0.9455  1.0252  1.1493  1.0368  0.8473  0.9781  1.0185  1.0967  0.9795  1.0955  1.0172 
Error 2  1.9559  1.9604  2.0556  1.9821  1.8676  1.9592  1.9683  2.0947  1.9839  1.9868  1.9815  


GABP  Error 1  0.8828  0.3893  0.5888  0.6893  0.8178  0.7885  0.3947  0.5838  0.5635  0.8384  0.6537 
Error 2  1.6535  1.421  1.3859  1.5857  1.8434  1.6883  1.4641  1.6759  1.5937  1.8758  1.6187  


DCQABP  Error 1  0.6463  0.4027  0.5394  0.4859  0.6673  0.6375  0.3375  0.5473  0.5532  0.6391  0.5456 
Error 2  1.5874  1.3867  1.3589  1.4845  1.5838  1.5423  1.5984  1.5873  1.4839  1.5868  1.52  


MCQABP  Error 1  0.5263  0.3017  0.4194  0.3559  0.5273  0.4275  0.5175  0.6273  0.3332  0.3291  0.4365 
Error 2  1.5174  1.3167  1.4389  1.3545  1.4438  1.3123  1.5284  1.4373  1.3239  1.3268  1.4 
Apply the test method of mean difference of two normal populations (
The data of error of the four methods can be considered to be samples from normal population
The form of rejection region is
Consider
So the rejection region is
The hypotheses
The mean and variance of Error 1 of BP, GABP, DCQABP, and MCQABP are as follows:
Consider
Consider
Consider
The mean and variance of Error 2 of BP, GABP, DCQABP, and MCQABP are as follows:
Consider
Consider
Consider
The simulation results show that MCQABP is significantly superior to BP, GABP, and DCQABP. The evolutionary process of BP, GABP, DCQABP, and MCQABP is shown in Figure
The evolutionary process of BP, GABP, DCQABP, and MCQABP.
It can be seen that the fitting results of different models are roughly the same. The trend of these algorithms is the same. It shows that the method of this paper is scientific and effective. The fitting error of the model of this paper is less than results of other models. So the model in this paper is better than other models.
For the reason that MCQABP with super parallel and ultrahigh speed which can overcome a series of problems existing in the BP and GABP such as slow convergence speed, premature convergence, and bad computational stability optimizes the structure of the neural network effectively, the computational cost of the MCQABP is lower than the method of BP, GABP, and DCQABP in test case
The training data is from the results of numerical simulation of 1365 kinds of target board damage under the kinetic energy rod collision of penetration cases. 15 kinds of destruction values predicted by neural network are randomly chosen and shown in Table
Contrast between 15 kinds of simulation of damage index value and predicted value.
Order number  Value of laboratory  Prediction value of BP  Prediction value of GABP  Prediction value of MCQABP  Error value of BP  Error value of GABP  Error value of MCQABP 

1  0.5753  0.6743  0.6642  0.5912  17.21%  15.45%  2.76% 
2  0.3735  0.3574  0.3671  0.3719  4.31%  1.71%  0.43% 
3  1.0842  1.1218  1.1189  1.0901  3.47%  3.20%  0.54% 
4  1.2436  1.2120  1.1203  1.2392  2.54%  9.91%  0.35% 
5  1.8643  1.9212  1.9200  1.8765  3.05%  2.99%  0.65% 
6  0.607  0.5716  0.5825  0.5923  5.83%  3.23%  2.42% 
7  0.5394  0.6067  0.5923  0.5532  12.48%  9.81%  2.56% 
8  1.1865  1.2077  1.2065  1.1965  1.79%  1.69%  0.84% 
9  1.3987  1.3844  1.3865  1.3899  1.02%  0.87%  0.63% 
10  1.4826  1.4863  1.4861  1.4830  0.25%  0.24%  0.03% 
11  0.5436  0.5627  0.5512  0.5311  3.51%  1.40%  2.30% 
12  0.0817  0.0780  0.079  0.0813  4.53%  3.30%  0.49% 
13  0.9045  0.8847  0.8859  0.8899  2.19%  2.06%  1.61% 
14  1.4768  1.4950  1.4854  1.4801  1.23%  0.58%  0.22% 
15  1.3164  1.3490  1.3387  1.3214  2.48%  1.69%  0.38% 
It can be seen from the table that the prediction effect of MCQABP is better than BP network and GABP on the same sample data. The error range of MCQABP network is less than that of BP neural network and GABP neural network under the same learning times. The high efficiency of MCQABP network model is fully illustrated.
A kind of super parallel ultrafast BP neural network model and algorithm based on multichain quantum optimization algorithm are proposed. The algorithm makes full use of local time domain feature of the BP network and global optimization search capability of multichain quantum optimization algorithm to enhance the intelligent search ability of network. It overcomes the disadvantages of BP network to improve the effectiveness of the optimization and accelerate search efficiency and convergence speed. The model controller based on MCQABP network can evaluate damage effect of penetration target board of kinetic energy well. The antijamming ability of the model is fine. The MCQABP network controller is effective as can be seen from the actual simulation results.
The authors declare that there is no conflict of interests regarding the publication of this paper.