CN111324989A - GA-BP neural network-based gear contact fatigue life prediction method - Google Patents
GA-BP neural network-based gear contact fatigue life prediction method Download PDFInfo
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Abstract
The invention discloses a gear contact fatigue life prediction method based on a GA-BP neural network, which comprises the following steps: 1. collecting gear contact fatigue test data and normalizing the data to be used as sample data of a BP neural network model; 2. constructing a BP neural network structure; 3. optimizing the weight and the threshold of the BP neural network by using a genetic algorithm, and training the BP neural network; 4. calculating prediction accuracy evaluation parameter determination coefficient
Acquiring an optimized weight and a threshold when the set value is met; 5. and testing the established GA-BP neural network by using a test sample set. Compared with the existing gear contact fatigue life prediction method based on the physical model, the method has the advantages of low cost and accurate predictionThe gear fatigue life prediction method has the advantages of being high in degree, free of derivation according to a failure mechanism, capable of achieving gear fatigue life prediction, improving prediction accuracy, simple to use and capable of providing a new technical means for gear design and manufacture.
Description
Technical Field
The invention belongs to the technical field of intelligent manufacturing, and particularly relates to a method for predicting contact fatigue life of a gear.
Background
The gear is an important mechanical basic part, is widely applied to various industrial fields for motion and power transmission, and the fatigue life of the gear directly determines the service performance of the whole machine. With the development of heavy equipment such as wind power, high-speed rail and aviation towards high reliability, long service life and intellectualization, higher requirements are put forward on the service life and reliability of the gear. If the gear is subjected to fatigue failure damage in the service stage, the equipment is shut down to influence the production, and the personal and property safety is endangered. The fatigue life of the gear is accurately predicted, sudden vicious accidents of equipment can be avoided, and long-term safe and reliable operation of mechanical equipment is ensured.
The conventional gear fatigue life prediction method mainly comprises a fatigue accumulated damage theory, a mathematical model and the like, wherein the fatigue accumulated damage theory considers that the maximum life of a part is N times when the part is under the action of a constant stress amplitude, namely the part is in operation for N times, and a dangerous point of the part is in fatigue failure; the mathematical model predicts the fatigue life of the gear by establishing the relationship between the gear parameters and the fatigue life; although the method can predict the fatigue life and achieve certain accuracy, various defects exist, such as imperfect accumulation theory of fatigue damage, no consideration of the influence of loading load sequence on the fatigue damage and the like, the methods are time-consuming and high in cost, and the result often has larger deviation from the test value.
The data driving method has the advantage that the internal mechanism of the material is not required to be comprehensively clarified, and a new technical means is provided for predicting the contact fatigue life of the gear. The BP neural network has strong nonlinear fitting capability and self-adaptive capability, so that the BP neural network is widely applied to the prediction of the service life of parts, but the initial weight and the threshold are randomly selected, so that the problem of falling into a local minimum value is easily caused, and therefore, the initial weight and the threshold of the BP neural network are optimized by adopting a genetic algorithm, and the accurate prediction of the contact fatigue life of the gear is realized.
The terms: the genetic algorithm optimized BP neural network is called GA-BP neural network for short.
Disclosure of Invention
Aiming at the problems in the prior art, the technical problem to be solved by the invention is to provide a gear contact fatigue life prediction method based on a GA-BP neural network, which can predict the gear fatigue life, can improve the prediction accuracy, and is simple to use and low in cost.
The technical problem to be solved by the invention is realized by the technical scheme, which comprises the following steps:
s1, collecting test data of the gear contact fatigue test, taking the structural characteristic parameters and the material parameters of the gear as input parameters and the gear contact fatigue life as output parameters, and carrying out normalization processing on each group of test data to take the test data as a training sample and a test sample;
s2, respectively determining the number of nodes of an input layer and an output layer according to the input parameters and the output parameters in the S1, determining the number of hidden layers and the number of nodes of the hidden layers, selecting the type of an activation function between each layer of the neural network, and constructing a BP neural network structure;
s3, optimizing the initial weight and threshold of the BP neural network by using a genetic algorithm, setting initial training parameters, and training the BP neural network structure by adopting the training samples in S1;
s4, calculating prediction accuracy evaluation parameter determination coefficient R2If the weight value meets the set value, extracting the trained weight value and threshold value to obtain a GA-BP neural network model; if not, returning to S3 to train again until determining the coefficient R2The set value is met;
and S5, testing the established GA-BP neural network by using the test sample set, and evaluating the network performance and the prediction accuracy.
Therefore, a GA-BP neural network model is obtained to predict the contact fatigue life of the gear.
Compared with the prior art, the invention has the technical effects that:
compared with the existing gear contact fatigue life prediction method based on a physical model, the gear contact fatigue life prediction method is low in cost, high in prediction precision, free of derivation according to failure mechanisms, capable of achieving gear fatigue life prediction, capable of improving prediction accuracy, simple to use, capable of providing a new evaluation means for manufacturing and using of gears, saving a large amount of time and cost, and has popularization and application values.
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The drawings of the invention are illustrated as follows:
FIG. 1 is a flow chart of one embodiment of the present method;
FIG. 2 is a block diagram of one embodiment of the present method;
FIG. 3 is a diagram of prediction values and decision coefficients according to an embodiment of the present invention;
FIG. 4 is a comparison graph of predicted values and test values according to one embodiment of the present invention.
Detailed Description
The invention is further illustrated by the following examples in conjunction with the accompanying drawings:
as shown in fig. 1, the present embodiment includes the following steps:
s1, collecting test data of the gear contact fatigue test, analyzing the influence of gear contact fatigue life by using gear parameters, carrying out normalization processing on each group of test data by using the structural characteristic parameters and the material parameters of the gear as input parameters and the gear contact fatigue life as output parameters, and using the normalized test data as training samples and test samples.
143 sets of gear contact fatigue life test data were collected from a number of colleges and research institutes. The selected input parameters include but are not limited to tooth number, roughness, contact stress, tooth surface hardness and material parameters, the gear contact fatigue life is taken as an output parameter, and the unit of test data is shown in table 1:
TABLE 1 units of test data
| Parameter(s) | Unit of | Parameter(s) | Unit of |
| Lubrication parameter | Hardness of tooth surface | HV | |
| Roughness of | Ra | Parameters of the material | |
| Stress of contact | MPa | Contact fatigue life of gear | Cycle |
Mapping the collected data to a (0, 1) interval through a normalization formula, wherein the normalization formula of the data is as follows:
in the formula, xmIs normalized number, x is number to be normalized or number output by training setminOutputting the minimum value, x, for the minimum value or training set to be normalizedmaxThe maximum value of the input to be normalized or the maximum value of the output of a training set is obtained, (wherein the output predicted value of the testing set is subjected to inverse normalization to obtain the contact fatigue life of the gearAnd (5) predicting the value. )
The data after the normalization process are divided into 108 training sample sets and 35 testing sample sets.
The partial training sample set before normalization is shown in table 2.
TABLE 2 partial training sample set data before normalization
Note: the material type and lubrication mode are numerically distinguished, wherein material 1 represents 20MnCr5, material 2 represents 18Cr2Ni4WA, material 5 represents AISI 4118, material 10 represents 16MnCr5, lubrication 1 represents no lubrication, and lubrication 2 represents oil lubrication.
And S2, determining the number of nodes of an input layer and the number of nodes of an output layer according to the input parameters and the output parameters in the S1, determining the number of hidden layers and the number of nodes of the hidden layers, selecting the type of an activation function between the layers of the neural network, and constructing the BP neural network structure.
In this embodiment, the number of nodes of the input layer of the BP neural network is 5, the number of nodes of the hidden layer is 2, the number of nodes of the output layer is 1, and the range of the number of nodes of the hidden layer is determined according to an empirical formula, where the empirical formula is:
in the formula, n is the number of hidden layer nodes, m is the number of input layer nodes, p is the number of output layer nodes, and a ranges from 1 to 10. And determining the number of the optimal hidden layer nodes to be 8 by a trial and error method.
The BP neural network structure is '5-8-8-1', an activation function between an input layer and a first hidden layer selects a hyperbolic tangent function Tanh, an activation function between the first hidden layer and a second hidden layer selects an S-shaped function Sigmoid, an activation function between the second hidden layer and an output layer selects a linear function Purelin, and the activation functions between the layers are as follows:
tanh activation function:
sigmoid activation function:
purelin activation function: f. of3(x)＝x
x is the total input of any node of the hidden layer or the output layer, and f (x) is the total output of any node of the hidden layer or the output layer.
Fig. 2 shows a structure of a BP neural network, where in this embodiment, a transfer formula from an input layer to a first hidden layer is:
in the formula, H1jJ is the output of the jth node of the first hidden layer, 1,2, …,8, f1As a function of Tanh activation, ωijIs the weight value, x, between the ith node of the input layer and the jth node of the first hidden layeriIs an input value of the ith node of the input layer, ajIs the threshold of the jth node of the first hidden layer.
The transfer formula from the first hidden layer to the second hidden layer is:
in the formula, H2kFor the output of the kth node of the second hidden layer, k is 1,2, …,8, f2For Sigmoid activation functions, ωjkIs the weight between the jth node of the first hidden layer and the kth node of the second hidden layer, bkIs the threshold of the kth node of the second hidden layer.
The second hidden layer to output layer transfer formula is:
wherein y is the output value of the output layer, f3Activating functions for Purelin，ωkIs the weight between the kth node of the second hidden layer and the output layer, and c is the output layer threshold.
S3, optimizing the initial weight and threshold of the BP neural network by using a genetic algorithm, setting initial training parameters including iteration times, minimum error of a training target, learning rate and maximum failure times, and training the BP neural network by using training samples.
The initial parameters of the genetic algorithm are: 70 population scale, 100 evolution times, 0.8 cross probability and 0.2 mutation probability.
The specific process of genetic algorithm optimization is as follows:
step 1) population initialization, wherein according to a network structure of a BP neural network 5-8-8-1 shown in fig. 2, the total number of 5 × 8+8 × 8+8 × 1 is 112 weight values, and the total number of 8+8+1 is 17 threshold values, so that the individual coding length of a genetic algorithm is 112+ 17-129, the individual number population scale is 70, an individual coding method is real number coding, each individual is a real number string and consists of the BP neural network weight value and the threshold value.
Step 2) selecting a fitness function
The fitness function is the inverse of the sum of the square of the output layer prediction output and the expected output error:
in the formula, FlAnd the fitness function value of the ith individual, y is the predicted output of the BP neural network, and d is the expected output.
Step 3), selecting operation: genetic algorithm selection roulette method, selection strategy based on fitness proportion, probability p of each individual l being selectedlComprises the following steps:
wherein k is a coefficient, FlIs the first oneFitness function value of the volume.
Step 4), cross operation: individuals employ real number encoding and cross-manipulation methods employ real number cross-manipulation, see the literature "a genetic algorithm project," Whitley, darrell. 65-85 ("genetic Algorithm guide", Whitley, Darrell. statistics and computing4.2 (1994): 65-85.): the r individual hrAnd the s-th individual hsThe method of interleaving at t bits is as follows:
hrt′＝hrt(1-e)+hste
hst′＝hst(1-e)+hrte
in which e is [0,1]A random number in between; h isrt' is the r-th individual after the crossover operation, hst' is the s-th individual after the crossover operation.
Step 5), mutation operation: selecting the p gene h of the o individualopCarrying out mutation by the following operation method:
in the formula, hmaxIs gene hopUpper bound of, hminIs gene hopLower boundary of (f), (g) r2(1-g/Gmax)2，r2Is a random number, G is the current iteration number, GmaxFor maximum evolutionary number, r is [0,1 ]]Random number of cells, hop' is the p-th gene of the o-th individual after mutation operation.
And 6), repeating the step 3), the step 4) and the step 5) until the evolution times reach 100, and decoding the optimal individual to be used as the weight and the threshold value after the BP neural network is optimized.
The weight and the threshold after the BP neural network optimization are as follows:
TABLE 3 weight ω between input layer and first hidden layerij
| 0.4218 | 0.2447 | 0.3578 | 0.4907 | 0.3844 | -0.2171 | 0.2578 | 0.4580 |
| -0.2863 | 0.9478 | 0.1458 | 0.7965 | 0.1166 | -0.8059 | -0.2495 | -0.8747 |
| -0.7241 | -0.1998 | 0.7721 | 0.6914 | -0.5112 | -0.0388 | 0.4085 | 0.9574 |
| 0.7924 | 0.2281 | 0.8561 | 0.3451 | 0.3491 | 0.3151 | 0.6033 | 0.5980 |
| -0.8740 | 0.6742 | 0.8846 | 0.2431 | -0.5790 | -0.0948 | 0.4490 | 0.9830 |
Note: the ith row and the jth column of values represent the weight omega between the ith node of the input layer and the jth node of the first hidden layerij
TABLE 4 weight ω between first and second hidden layersjk
| 0.7752 | 0.9144 | -0.9630 | 0.5424 | -0.1378 | 0.4338 | -0.0500 | 0.6697 |
| 0.2636 | 0.0761 | -0.9036 | 0.2976 | -0.9921 | 0.3810 | -0.5291 | 0.9403 |
| 0.2336 | 0.3169 | -0.3379 | -0.3860 | 0.6275 | 0.0070 | 0.4118 | -0.6723 |
| 0.8509 | 0.5511 | -0.8442 | -0.7990 | -0.2139 | 0.4650 | 0.3608 | 0.4789 |
| -0.3446 | -0.0218 | 0.6606 | -0.2344 | -0.0162 | 0.6991 | 0.0150 | 0.5359 |
| -0.5811 | 0.3932 | 0.5897 | 0.9315 | -0.4478 | 0.5806 | -0.4985 | 0.9826 |
| -0.2206 | 0.7731 | -0.7684 | -0.0700 | -0.3670 | -0.5834 | 0.7251 | -7277 |
| -0.5653 | -0.4367 | -0.1178 | -0.6378 | -0.2122 | -0.1526 | -0.7138 | 0.9720 |
| -0.8740 | 0.6742 | 0.8846 | 0.2431 | -0.5790 | -0.0948 | 0.4490 | 0.9830 |
Note: the jth row and kth column values represent the weight ω between the jth node of the first hidden layer and the kth node of the second hidden layerjk
TABLE 5 weight ω between the second hidden layer and the output layerk
| 0.9144 | 0.0761 | 0.3169 | 0.5511 | -0.0218 | 0.3932 | -0.7731 | -0.4367 |
Note: the kth column value represents the weight ω between the kth node of the second hidden layer and the output layerk
TABLE 6 threshold a of the first hidden layerj
| -0.0736 | -0.3354 | 0.1699 | 0.5334 | -0.4973 | -0.6228 | -0.1609 | 0.5524 |
Note: the jth column value represents the threshold value a of the jth node of the first hidden layerj
TABLE 7 threshold b for the second hidden layerk
| 0.6615 | 0.1516 | -0.8594 | -0.5865 | -0.3272 | 0.1648 | 0.5183 | -0.3050 |
Note: the kth column value represents the threshold b of the kth node of the second hidden layerk
Output node threshold: and c is 0.2342.
In the step, a Levenberg-Marquardt algorithm is adopted for training to train the BP neural network, and initial training parameters are set as follows: iteration number 1000, training target minimum error 1e-4Learning rate 0.001, maximum number of failures 15, concrete training process includes:
step 1), carrying out forward propagation on the training sample, and calculating the square sum of the predicted output and the expected output error of the output layer.
The forward propagation specifically includes: training samples are input from an input layer, processed by an activation function of a first hidden layer, output by a node of the first hidden layer, processed by an activation function of a second hidden layer and output by a node of the second hidden layer; and after the activation function processing of the output layer, the actual output value is obtained through the output of the output layer node, and the error sum of the square of the predicted output value and the expected output value of the output node is calculated.
Randomly selecting the u-th input sample and the corresponding output test data, wherein the sum of the square of the error between the predicted output and the expected output is represented by the formula:
E(u)＝(d(u)-y(u))2
where e (u) is the sum of the square of the predicted output and the expected output error of the u-th input sample, d (u) is the expected output of the u-th input sample (i.e., the output test data), and y (u) is the predicted output of the u-th input sample.
Step 2), judging whether the sum of the square of the error between the predicted output and the expected output is less than the minimum error of the training target (1 e)-4) The requirements of (1); if not, entering the step 3) to carry out a back propagation process, and if so, finishing the training of the BP neural network modelAnd (4) bundling.
And 3) performing a back propagation process by adopting a gradient descent method (gradient is a vector formed by the error function on the weight and the partial derivative of the threshold), obtaining a weight updating formula from the second hidden layer to the output layer, a weight updating formula between the input layer and the first hidden layer and a weight updating formula between the first hidden layer and the second hidden layer by utilizing the square sum function E (u) of the predicted output and the expected output error to obtain a weight composite derivative, and respectively updating the weight of the output layer, the weight between the input layer and the first hidden layer and the weight between the first hidden layer and the second hidden layer according to the updating formulas.
Wherein, the weight ω between the input layer and the first hidden layerijThe update formula is:
weight ω between first hidden layer and second hidden layerjkThe update formula is:
weight ω from the second hidden layer to the output layerkThe update formula is:
in the formula, η represents the learning rate (0.001).
Step 4), inputting the updated output layer weight obtained in the step 3), the weight between the input layer and the first hidden layer, the weight between the first hidden layer and the second hidden layer into the input layer and the hidden layer in the forward propagation in the step 1) to continue the forward propagation, and returning to the step 1).
S4, calculating prediction accuracy evaluation parameter determination coefficient R2If the weight value meets a set value (generally greater than 0.99), extracting the weight value and the threshold value after training to obtain a GA-BP neural network model; if not, go back to S3 to train again until determiningCoefficient R2The set value is met;
in this step, a prediction accuracy evaluation parameter determination coefficient R2The calculation formula is as follows:
in the formula (I), the compound is shown in the specification,
in order to predict the value of the model,
is the sample average value, yoIs the sample data value, and v is the number of training samples.
S5, testing the established GA-BP neural network by using the test sample set, performing inverse normalization on the prediction result to obtain a predicted gear contact fatigue life value, and calculating a prediction precision evaluation parameter decision coefficient R2And evaluating the network performance and the prediction accuracy, and testing partial data of the sample set as shown in a table 8:
table 8 partial test sample set
Note: the material type and the lubrication mode are numerically distinguished, wherein material 1 represents 20MnCr5, material 2 represents 18Cr2Ni4WA, material 5 represents AISI 4118, lubrication 1 represents no lubrication, and lubrication 2 represents oil lubrication.
And calculating through a GA-BP neural network model, performing inverse normalization on the prediction result to obtain a predicted gear contact fatigue life value, and comparing the predicted gear contact fatigue life value with a test value, as shown in fig. 3 and 4. As can be seen from fig. 3: test set decision coefficient R20.9587; the gear contact fatigue life prediction precision of the test sample set is high and stable. In FIG. 4, the abscissa of the data point is measuredThe actual value of the contact fatigue life of the gear is tested, the ordinate is the predicted value of the contact fatigue life of the gear, the linear straight line indicates that the predicted value is equal to the actual value, the closer the data point is to the straight line, the higher the prediction precision is, and the higher the prediction precision can be seen.
And introducing a group of data (roughness Ra0.6, contact stress 1997.09MPa, surface hardness HV695, material 18CrNiMo, lubricating mode oil lubrication and real life 1.4E +07Cycle) which are not concentrated in the test sample into the model to obtain a predicted value 1.51E +07Cycle, so as to meet the precision requirement.
The method improves the prediction accuracy of the contact fatigue life of the gear, particularly overcomes the defects of complex prediction and lower prediction accuracy of the existing gear fatigue life, and is convenient, effective and low in cost.
Claims (4)
1. A gear contact fatigue life prediction method based on a GA-BP neural network is characterized by comprising the following steps:
s1, collecting test data of the gear contact fatigue test, taking the structural characteristic parameters and the material parameters of the gear as input parameters and the gear contact fatigue life as output parameters, and carrying out normalization processing on each group of test data to take the test data as a training sample and a test sample;
s2, respectively determining the number of nodes of an input layer and an output layer according to the input parameters and the output parameters in the S1, determining the number of hidden layers and the number of nodes of the hidden layers, selecting the type of an activation function between each layer of the neural network, and constructing a BP neural network structure;
establishing a prediction model by using a BP neural network containing 2 hidden layers, wherein the number of nodes of the hidden layers meets the following requirements:
in the formula, n is the number of hidden layer nodes, m is the number of input layer nodes, p is the number of output layer nodes, and a ranges from 1 to 10;
the activation function between the input layer and the first hidden layer selects a hyperbolic tangent function Tanh, the activation function between the first hidden layer and the second hidden layer selects an S-shaped function Sigmoid, and the activation function between the second hidden layer and the output layer selects a linear function Purelin;
the transfer formula from the input layer of the BP neural network to the first hidden layer is as follows:
in the formula, H1jJ is the output of the jth node of the first hidden layer, 1,2, …, n, f1As a function of Tanh activation, ωijIs the weight value, x, between the ith node of the input layer and the jth node of the first hidden layeriIs an input value of the ith node of the input layer, ajA threshold value of the jth node of the first hidden layer;
the transfer formula from the first hidden layer to the second hidden layer of the BP neural network is as follows:
in the formula, H2kFor the output of the kth node of the second hidden layer, k is 1,2, …, n, f2For Sigmoid activation functions, ωjkIs the weight between the jth node of the first hidden layer and the kth node of the second hidden layer, bkA threshold value of a kth node of the second hidden layer;
the transfer formula from the second hidden layer to the output layer of the BP neural network is as follows:
wherein y is the output value of the output layer, f3As Purelin activation function, ωkC is the weight between the kth node of the second hidden layer and the output layer, and c is the threshold value of the output layer;
s3, optimizing the initial weight and threshold of the BP neural network by using a genetic algorithm, setting initial training parameters, and training the BP neural network structure by adopting the training samples in S1;
the genetic algorithm optimization process comprises the following steps:
step 1), population initialization: setting a population number q, wherein individual codes of the population are real number codes, and each individual consists of a BP neural network weight and a threshold;
step 2) selecting a fitness function
The fitness function is the inverse of the sum of the square of the output layer prediction output and the expected output error:
in the formula, FlThe fitness function value of the ith individual is obtained, y is the prediction output of the BP neural network, and d is the expected output;
step 3), selecting operation: genetic algorithm selection roulette method, probability p of each individual l being selectedlComprises the following steps:
wherein k is a coefficient, FlThe fitness function value of the ith individual;
step 4), cross operation: the individuals adopt real number coding, the cross operation method adopts a real number cross method, and the r-th individual hrAnd the s-th individual hsThe method of interleaving at t bits is as follows:
hrt′＝hrt(1-e)+hste
hst′＝hst(1-e)+hrte
in which e is [0,1]A random number in between; h isrt' is the r-th individual after the crossover operation, hst' is the s-th individual after the crossover operation;
step 5), mutation operation: selecting the p gene h of the o individualopPerforming mutation by the following method：
In the formula, hmaxIs gene hopUpper bound of, hminIs gene hopLower boundary of (f), (g) r2(1-g/Gmax)2，r2Is a random number, G is the current iteration number, GmaxFor maximum evolutionary number, r is [0,1 ]]Random number of cells, hop' is the p gene of the o individual after mutation operation;
step 6), repeating the step 3), the step 4) and the step 5) until the evolution times are reached, and decoding the optimal individual to be used as the weight and the threshold value after the BP neural network is optimized;
training the BP neural network by using the training set in S1;
s4, calculating prediction accuracy evaluation parameter determination coefficient R2If the weight value meets the set value, extracting the trained weight value and threshold value to obtain a GA-BP neural network model; if not, returning to S3 to train again until determining the coefficient R2The set value is met;
prediction accuracy evaluation parameter determination coefficient R2The calculation formula is as follows:
in the formula (I), the compound is shown in the specification,
in order to predict the value of the model,
is the sample average value, yoIs the sample data value, v is the number of training samples;
and S5, testing the established GA-BP neural network by using the test sample set, and evaluating the network performance and the prediction accuracy.
2. A GA-BP neural network-based gear contact fatigue life prediction method according to claim 1, wherein in S1, the input parameters are tooth number, roughness, contact stress, tooth surface hardness and material parameters.
3. The GA-BP neural network-based gear contact fatigue life prediction method of claim 2, wherein in S2, the BP neural network is constructed such that the number of nodes of the input layer is 5, the number of hidden layers is 2, the number of nodes of the output layer is 1, and the number of nodes of hidden layers is 8.
4. A method for predicting gear contact fatigue life based on GA-BP neural network as claimed in claim 1,2 or 3, wherein in S4, the step of training BP neural network with the training set in S1 is:
step 1), carrying out forward propagation on a training sample, and calculating the square sum of the predicted output of an output layer and an expected output error;
randomly selecting the u-th input sample and the corresponding output test data, wherein the sum of the square of the error between the predicted output and the expected output is represented by the formula:
E(u)＝(d(u)-y(u))2
wherein E (u) is the sum of the predicted output and the desired output error squared for the u-th input sample, d (u) is the desired output for the u-th input sample, and y (u) is the predicted output for the u-th input sample;
step 2), judging whether the sum of the square of the error between the predicted output and the expected output meets the requirement of being smaller than the minimum error of the training target or not; if not, entering the step 3) to carry out a back propagation process, if so, finishing the training of the BP neural network model, and ending;
step 3), performing a back propagation process by adopting a gradient descent method, and obtaining a weight updating formula from the second hidden layer to the output layer, a weight updating formula between the input layer and the first hidden layer, and a weight updating formula between the first hidden layer and the second hidden layer by utilizing the composite derivation of the predicted output and an expected output error square sum function E (u) degree weight:
wherein, the weight ω between the input layer and the first hidden layerijThe update formula is:
weight ω between first hidden layer and second hidden layerjkThe update formula is:
weight ω from the second hidden layer to the output layerkThe update formula is:
wherein η is the learning rate;
step 4), inputting the updated output layer weight obtained in the step 3), the weight between the input layer and the first hidden layer, the weight between the first hidden layer and the second hidden layer into the input layer and the hidden layer in the forward propagation in the step 1) to continue the forward propagation, and returning to the step 1).
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