Fair-faced concrete apparent porosity prediction method based on GA-BP neural network

Abstract

The invention discloses a method for predicting the apparent porosity of bare concrete based on a GA-BP neural network. The method comprises the following steps: firstly, collecting key parameters such as initial slump of the bare concrete, the mixing amount of a water reducing agent and the use amount of slurry; then establishing an initial apparent porosity prediction equation based on a concrete rheological relationship, and constructing a BP neural network model; and finally, optimizing the initial weight and structural parameters of the BP neural network by adopting a GA (Genetic Algorithm), and predicting the final apparent porosity of the bare concrete. According to the method, through deep fusion of the rheological equation and the GA-BP neural network, a reliable basis is provided for bare concrete mix proportion optimization, process parameter adjustment and repair-free facing effect pre-control, and the intelligent level of engineering quality control is remarkably improved; the problems that in previous bare concrete porosity prediction, a traditional empirical formula is large in prediction error and limited in statistical model fitting capacity, and a single machine learning model is prone to local optimum and depends on a large amount of data or expensive equipment are solved.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of artificial intelligence and material engineering, and particularly relates to a method for predicting the apparent air void ratio of fair-faced concrete. BACKGROUND

[0002] As a core performance of architectural modernism, fair-faced concrete directly exhibits the natural texture and mechanical beauty of concrete through casting, and abandons the traditional finishing process. The decorative effect of the fair-faced concrete highly depends on the accurate control of the apparent quality, and the concrete is required to achieve a high degree of compaction and uniformity in one casting process. The process is extremely sensitive to the raw material ratio, engineering environment and process parameters, such as water-binder ratio, aggregate gradation, and vibration mode, and any slight deviation may cause surface defects. Especially in high-rise buildings, art galleries and other projects with strict aesthetic requirements, the fair-faced concrete needs to meet the dual standards of structural strength and visual performance, and the construction difficulty is much higher than that of ordinary concrete.

[0003] The apparent air void ratio is one of the most core quality indicators of the fair-faced concrete, and directly affects the decorative effect and long-term durability. The air voids form irregularly distributed point defects on the surface of the hardened concrete, which destroys the uniformity of light reflection of the material and causes visual mottling; at the same time, the open air voids become the channels for the penetration of water and erosion medium, accelerating the process of freeze-thaw damage, steel corrosion and carbonation. Therefore, accurate prediction of the air void ratio has a decisive significance for optimizing the mix design of the fair-faced concrete (such as the complex proportioning of air entraining agent and defoamer), adjusting the vibration process parameters, and realizing the repair-free finishing effect.

[0004] At present, the prediction of apparent air void content of fair-faced concrete mainly relies on empirical formula, traditional statistical model or machine learning method. In engineering, the apparent air void content is usually estimated by empirical formula based on single parameter such as slump or air content. However, the formation of air void is also affected by many complex factors such as water reducing agent content, paste amount, defoamer and air entraining agent content, etc. There is often a nonlinear coupling relationship between these factors, and it is difficult for empirical formula to quantitatively analyze these complex interactions comprehensively and accurately. The fitting ability of traditional statistical model for nonlinear relationship is limited. For example, the prediction error of the multiple linear regression model established in some studies exceeds 25% due to the nonlinear characteristics of concrete materials. In recent years, machine learning methods (convolutional neural network, BP neural network) of artificial intelligence have also been gradually applied to the prediction and control of air void content. However, they also have their own limitations. Convolutional neural network (CNN) requires a large number of labeled images, but the surface texture of fair-faced concrete with different strengths differs greatly, which leads to insufficient generalization ability across projects. The model is easily disturbed by light and surface water stains, and the recognition error under backlight condition may reach 20%. High-precision CNN needs to be equipped with unmanned aerial vehicles or scanners, which is costly. BP neural network has the defects of being easy to fall into local optimal solution and slow convergence speed, and is sensitive to initial weights, which requires a large amount of trial data support. The existing methods have limitations in quantifying complex nonlinear coupling factors, and machine learning methods face problems such as insufficient generalization ability, poor model stability and large data requirements. SUMMARY

[0005] The present application provides a fair-faced concrete apparent air void content prediction method based on GA-BP neural network, which aims to accurately predict the apparent air void content of fair-faced concrete and provide reliable basis for optimizing the mix design of fair-faced concrete, adjusting engineering process parameters and ensuring the non-repairing finishing effect.

[0006] Technical scheme: A fair-faced concrete apparent air void content prediction method based on GA-BP neural network, the method comprises establishing a prediction equation of the initial apparent air void content of fair-faced concrete; and constructing a GA-BP model based on GA genetic algorithm and BP neural network to predict the final apparent air void content of fair-faced concrete, comprising the following steps:

[0007] Step (1): Collecting the related parameters of fair-faced concrete;

[0008] Step (2): Establishing the prediction equation of the initial apparent air voids of the exposed-finish concrete, and calculating the initial apparent air voids of the exposed-finish concrete;

[0009] Step (3): Building a BP neural network model, and inputting the data set into the BP neural network;

[0010] Step (4): Optimizing the BP neural network by using the GA genetic algorithm, and predicting the final apparent air voids of the exposed-finish concrete.

[0011] Further, the related parameters of the exposed-finish concrete in step (1) include the density, the initial slump, the water-reducing agent dosage, the paste dosage, the 1h slump loss, the defoaming agent dosage, and the air-entraining agent dosage of the exposed-finish concrete.

[0012] Further, the prediction equation of the initial apparent air voids of the exposed-finish concrete is established based on the rheological relationship model of the concrete fluidity and the air voids in step (2) as follows:

[0013] , wherein: L q is the initial apparent air voids (%) of the exposed-finish concrete; τ y is the yield stress (Pa) of the exposed-finish concrete; e is a natural constant; s is the initial slump (mm) of the exposed-finish concrete; is the water-reducing agent influence factor; is the paste dosage influence factor; is the slump loss influence factor; is the defoaming agent and air-entraining agent complex mixing influence factor;

[0014] It should be particularly noted that the above prediction equation of the initial apparent air voids of the exposed-finish concrete is a dimensionless expression, and each term on the right side of the equation, including the constant term, the exponential term, and the influence factor term, is a dimensionless value. When calculating, all terms only participate in pure numerical calculation, and the numerical values of each parameter in the prescribed unit are directly substituted, and the obtained result L q is the percentage (%) prediction value of the initial apparent air voids of the exposed-finish concrete, without unit conversion.

[0015] Further, the specific expression of each term in the prediction equation is as follows: τ y The specific expression is , wherein ρ is the density (kg / m 3 ) of the exposed-finish concrete, H is the slump cone height, which is 300 mm, and s is the initial slump (mm) of the exposed-finish concrete; The specific expression is , wherein w is the water-reducing agent dosage (%); The specific expression is , wherein k is the paste dosage (m 3 ). The specific expression is Wherein The 1h slump loss (mm) is 1h; The specific expression is Wherein d is the dosage of defoaming agent (%), and a is the dosage of air entraining agent (%);

[0016] Further, the specific steps of constructing the BP neural network model and inputting the data set in step (3) are as follows:

[0017] Step (31): determining the network structure and node number of the BP neural network;

[0018] Step (32): selecting the activation function of each layer;

[0019] Step (33): setting the learning rate;

[0020] Step (34): setting the training parameters;

[0021] Step (35): inputting the data set;

[0022] Further, the method for determining the node number of the hidden layer of the BP neural network in step (31) is as follows: first, the node number is preliminarily estimated according to the empirical formula Preliminary estimation, wherein I is the node number of the input layer, O is the node number of the output layer, and C is an empirical constant, which is usually taken as 5-10, then the one-by-one testing method is adopted to traverse between the preliminarily estimated node numbers, the mean square error (MSE) of the training set corresponding to each node number is compared, and finally the node number making the MSE minimum is selected as the node number of the hidden layer.

[0023] Further, the specific steps of optimizing the BP neural network by adopting the GA genetic algorithm and predicting the final apparent air void ratio of the fair-faced concrete in step (4) are as follows:

[0024] Step (41): setting the initial population;

[0025] Step (42): fitness evaluation;

[0026] Step (43): performing selection, crossover and mutation operations;

[0027] Step (44): completely training the GA-BP neural network;

[0028] Step (45): predicting the final apparent air void ratio of the fair-faced concrete;

[0029] Beneficial effects: compared with the prior art, the method has the following remarkable effects:

[0030] (1) The present application firstly deeply integrates rheological equation and GA-BP neural network, constructs an initial prediction equation based on key parameters such as yield stress and paste dosage, and optimizes the initial weight and structure parameters of the neural network through the GA genetic algorithm, effectively overcoming the defects of traditional BP neural network such as easy to fall into local optimal solution and slow convergence speed.

[0031] (2) The present application realizes accurate quantification of complex nonlinear relationships such as slump and paste dosage by establishing an initial prediction equation that integrates multiple factor coupling, and taking the output as the input layer node of the neural network. Compared with the large prediction error of traditional empirical formula, the present application can provide high-precision basis for the mix proportion optimization of fair-faced concrete.

[0032] (3) The present application only needs conventional engineering detection parameters (slump, water reducing agent dosage, etc.) as input, without relying on expensive image acquisition equipment. The GA-BP model directly outputs the air void rate prediction value, solving the problems of insufficient cross-project generalization ability of convolutional neural network and high backlight condition missing rate, significantly reducing the equipment cost and data acquisition threshold, and being more suitable for rapid quality control in engineering site.

[0033] (4) Based on the prediction result, the present application can optimize the air entraining agent and defoamer complex mixing ratio and vibration process in advance, and inhibit the generation of air holes from the source. Compared with the existing technology which relies on image analysis after molding, the present application can predict the surface quality before concrete pouring, guarantee the non-repair effect, and promote the intelligent pre-control upgrade of fair-faced concrete engineering. BRIEF DESCRIPTION OF DRAWINGS

[0034] Figure 1 The method described in the present application is implemented as a whole flowchart.

[0035] Figure 2 The BP neural network model structure diagram constructed in Example 2.

[0036] Figure 3 The GA-BP neural network training set prediction value and actual value comparison chart in Example 2.

[0037] Figure 4 The GA-BP neural network test set prediction value and actual value comparison chart in Example 2. DETAILED DESCRIPTION

[0038] In order to explain the technical solutions disclosed in the present application in detail, the following further elaboration is made in combination with the drawings and specific examples in the specification.

[0039] Example 1

[0040] Figure 1As shown is the overall framework diagram of the GA-BP neural network-based apparent air void ratio prediction method of the present application, comprising the following steps:

[0041] 101, Collecting the relevant parameters of the fair-faced concrete: collecting the parameters of the fair-faced concrete whose apparent air void ratio needs to be predicted, the specific parameters collected including the density, initial slump, water reducing agent dosage, paste dosage, 1h slump loss, defoaming agent dosage and air entraining agent dosage of the fair-faced concrete.

[0042] 102, Establishing the prediction equation of the initial apparent air void ratio of the fair-faced concrete and calculating the initial apparent air void ratio of the fair-faced concrete: according to the prediction equation of the initial apparent air void ratio of the fair-faced concrete , the parameter values collected in step 101 are substituted into the equation to calculate the initial apparent air void ratio L of the fair-faced concrete. q .

[0043] 103, Constructing a BP neural network model and inputting the data set into the BP neural network: constructing a BP neural network model by using the neural network toolbox in the computer Matlab software, and inputting the collected multiple groups of fair-faced concrete sample data, the parameters of the fair-faced concrete collected in step 101 and the initial apparent air void ratio L q calculated in step 102 into the BP neural network. Specifically, the following steps are included:

[0044] 1031, Determining the network structure and node number of the BP neural network: the network structure of the BP neural network includes the input layer, the output layer and the hidden layer. The node number of the input layer is n, n is the specific number of the factors affecting the final apparent air void ratio of the fair-faced concrete; the node number of the output layer is 1, corresponding to the final apparent air void ratio of the fair-faced concrete; the number of the hidden layer is 1~2, and a single hidden layer is preferred, and the node number thereof is initially estimated according to the empirical formula , wherein I is the node number of the input layer, O is the node number of the output layer, and C is an empirical constant, usually taking 5~10, and then the one-by-one test method is adopted to traverse between the initially estimated node numbers, compare the mean square error (MSE) of the training set corresponding to each node number, and finally select the node number making the MSE minimum as the node number of the hidden layer.

[0045] 1032, Selecting the activation function of each layer: the input layer does not need an activation function; the activation function available for the output layer is the Sigmoid, Softmax, Purelin and other functions; the hidden layer must select a nonlinear function, and the activation function available therefor is the ReLU and its variants, tanh, Sigmoid and other functions.

[0046] 1033、Set learning rate: the learning rate of BP neural network can use fixed learning rate or join adaptive attenuation strategy, and set the initial learning rate to 0.01~0.1.

[0047] 1034、Set training parameters: the core training parameters of BP neural network include maximum training times, training target error, batch size, training function. The maximum training times is 500~5000 times; the training target error is set according to different error types (mean square error MSE, cross entropy loss, accuracy); the batch size is set to 32~128 according to the power of 2 principle; the available training functions are trainbfg, trainscg, trainlm and other functions.

[0048] 1035、Input data set: the data set of BP neural network includes training set, validation set and test set, and multiple groups of fresh concrete sample data collected in engineering field or laboratory are divided into training set and validation set according to proportion, normalized and input into BP neural network; the fresh concrete parameters collected in step 101 and the initial apparent air voids L q collected in step 102 are normalized and input into BP neural network as test set.

[0049] 104、Optimize BP neural network by GA genetic algorithm to predict the final apparent air voids of fresh concrete: use the global optimization toolbox in computer Matlab software to realize GA genetic algorithm to optimize BP neural network and predict the final apparent air voids of fresh concrete. Specifically, the following steps are included:

[0050] 1041、Set initial population: encode the number of hidden layer nodes, learning rate and batch size of BP neural network into binary string or real number vector, set the population size to 30~200 individuals, and randomly generate the population.

[0051] 1042、Fitness evaluation: select appropriate fitness function for fitness evaluation

[0052] 1043、Selection, crossover and mutation operation: use roulette method or tournament method and other selection strategies to select the population; use single-point crossover, two-point crossover or uniform crossover and other methods, and set the crossover probability to perform crossover operation; use random bit flipping, adaptive bit flipping or block flipping and other methods, and set the mutation probability between 0.001~0.1 to perform mutation operation.

[0053] 1044、Complete training of GA-BP neural network: set the maximum iteration times to 100~200 times, and use the data set input in step 1035 to complete the training of GA-BP neural network, and if the training meets the error target requirements, the test set can be predicted.

[0054] 1045. Predict the final apparent porosity of fair-faced concrete: Predict the final apparent porosity of the test set (i.e., the fair-faced concrete in step 101) using the trained GA-BP neural network.

[0055] Example 2

[0056] All data in this embodiment are derived from fair-faced concrete prepared by the inventors in the laboratory. The density, water-reducing agent dosage, slurry dosage, defoamer dosage, and air-entraining agent dosage of the fair-faced concrete were directly measured using experimental instruments and mix design. The initial slump and 1-hour slump loss were measured by slump tests. The actual measured value of apparent porosity was obtained by image processing using computer software Image-Pro Plus 6.0. Six groups were randomly selected from the prepared fair-faced concrete for prediction, denoted as A1, A2, A3, B1, B2, and B3. The specific mix proportions of the six groups of fair-faced concrete are shown in Table 1 below.

[0057] Table 1 Mix Proportions for Fair-faced Concrete

[0058] Step 1: Collect relevant parameters of fair-faced concrete: Collect relevant parameters for six groups of fair-faced concrete: A1, A2, A3, B1, B2, and B3. See Table 2 below for specific parameters.

[0059] Table 2 Relevant parameters of fair-faced concrete

[0060] Step 2: Establish the prediction equation for the initial apparent porosity of the fair-faced concrete and calculate the initial apparent porosity of the fair-faced concrete: Based on the prediction equation for the initial apparent porosity of the fair-faced concrete... The initial apparent porosity of the six groups of fair-faced concrete in step 1 was calculated. The calculation process and results are shown in Table 3 below.

[0061] Table 3. Calculation process and results of initial apparent porosity of fair-faced concrete.

[0062] Step 3: Construct a BP neural network model by inputting the dataset into the BP neural network:

[0063] Step 31: Determine the network structure and number of nodes of the BP neural network: The number of nodes in the input layer is determined to be 7, corresponding to the initial slump, water-reducing agent dosage, slurry dosage, slump loss over 1 hour, defoamer dosage, air-entraining agent dosage, and initial apparent porosity of the fair-faced concrete (since the density values ​​of different fair-faced concretes do not differ significantly, and given the model error in the prediction equation for the initial apparent porosity, the initial apparent porosity rather than density is chosen to be included in the input layer of the neural network). The number of nodes in the output layer is determined to be 1, corresponding to the final apparent porosity of the fair-faced concrete. The number of hidden layers is determined to be 1, and its number of nodes is based on an empirical formula. The initial estimate was 7-12. A test-and-error method was used, iterating through the number of nodes from 7 to 12, comparing the mean squared error (MSE) of the training set for each number of nodes. Finally, the number of nodes with the smallest MSE, 9, was selected as the number of hidden layer nodes. The calculation results for different numbers of hidden layer nodes in Example 2 are shown in Table 4 below.

[0064] Table 4 Calculation results of the number of nodes in different hidden layers

[0065] Step 32: Select activation functions for each layer: No activation function is used for the input layer, Purelin activation function is used for the output layer, and Sigmoid activation function is used for the hidden layers.

[0066] Step 33: Set the learning rate: The BP neural network uses a fixed learning rate, and the learning rate is set to 0.01.

[0067] Step 34: Set training parameters: Set the maximum number of training iterations to 1000, the target error type to mean squared error (MSE) and set it to 0.000001, the batch size to 32, and select the trainlm function as the training function.

[0068] Step 35: Input Dataset: Collect parameters from the remaining fair-faced concrete prepared in the laboratory, summarize and organize them into the dataset in Table 5 below, and input them into the BP neural network after normalization.

[0069] Table 5 Dataset

[0070] Continued from Table 5. Dataset Note: In the table above, numbers 1-26 represent the training set, numbers 27-33 represent the validation set, and numbers A1, A2, A3, B1, B2, and B3 represent the test set.

[0071] Figure 2 The BP neural network model with a 7-9-1 structure established in step 3 is shown.

[0072] Step 4: Optimize the BP neural network using the GA genetic algorithm to predict the final apparent porosity of fair-faced concrete.

[0073] Step 41: Set the initial population: Encode the number of hidden layer nodes (9), learning rate (0.01), and batch size (32) of the BP neural network into a real number vector [9, 0.01, 32]. Set the initial population size to 40 individuals and generate the population randomly.

[0074] Step 42, Fitness Evaluation: Select the fitness function: fitness = -MSE to evaluate fitness.

[0075] Step 43: Perform selection, crossover, and mutation operations: Use a roulette wheel selection strategy for selection, use a uniform crossover method with a crossover probability of 0.8 for crossover, and use a random bit flipping method with a mutation probability of 0.02 for mutation.

[0076] Step 44: Perform full training on the GA-BP neural network: Set the maximum number of iterations to 100 and use the dataset in Table 5 to perform full training on the GA-BP network.

[0077] Figure 3 The image shows the comparison between the predicted and actual values ​​of the GA-BP neural network after complete training. As can be seen from the graph, the difference between the predicted values ​​(red squares) and the actual values ​​(blue dots) is small, especially on most data points, where the predicted values ​​closely follow the fluctuations of the actual values. This indicates that the GA-BP neural network model has a good fit on the training set, high prediction accuracy, and small error. Although there are deviations on a few individual data points, the overall performance is still quite ideal. This demonstrates that the neural network can effectively learn the patterns in the training set and make reasonable predictions.

[0078] Step 45: Predict the final apparent porosity of fair-faced concrete: Use the trained GA-BP neural network to predict the final apparent porosity of the test set (i.e., the six groups of fair-faced concrete in Step 1). The prediction results are shown in Table 6 below.

[0079] Table 6. Final Apparent Porosity Prediction Results on the GA-BP Neural Network Test Set

[0080] Figure 4 This presentation shows a comparison between the predicted and actual values ​​of the final apparent porosity of the test set (i.e., the six groups of fair-faced concrete in step 1) based on the GA-BP neural network, combined with... Figure 4As shown in Table 6 above, the red predicted value curve and the blue actual value curve are highly consistent in overall trend. For example, the trends at sample points A2, A3, B1, and B3 are basically identical, indicating that the model can effectively capture the characteristics of data changes. Specifically, the absolute error of sample A2 is 0.0086%, and the absolute error of sample B1 is 0.0047%, with the absolute errors of most samples remaining within a small range. Overall, the model exhibits good predictive performance, strong fitting ability, high predictive accuracy, and stable and controllable error, making it a reliable and practical model in real-world applications.

[0081] The above embodiments are merely illustrative of the design concept and features of the present invention, and are intended to enable those skilled in the art to understand the content of the present invention and implement it accordingly. The scope of protection of the present invention is not limited to the above embodiments. Therefore, all equivalent changes or modifications made based on the principles and design concepts disclosed in the present invention are within the scope of protection of the claims of the present invention.

Claims

1. A method for predicting the apparent air voids of fair-faced concrete based on GA-BP neural network, the method comprising: establishing a prediction equation for the initial apparent air voids of the fair-faced concrete; and constructing a GA-BP model based on the genetic algorithm (GA) and the BP neural network to predict the final apparent air voids of the fair-faced concrete, comprising the following steps: Step (1): collecting the relevant parameters of the fair-faced concrete; Step (2): establishing the prediction equation for the initial apparent air voids of the fair-faced concrete to calculate the initial apparent air voids of the fair-faced concrete; the prediction equation for the initial apparent air voids of the fair-faced concrete is as follows: Step (3): constructing a BP neural network model and inputting the data set into the BP neural network: Step (31): determining the network structure and the number of nodes of the BP neural network; Step (32): selecting the activation function of each layer; Step (33): setting the learning rate; Step (34): setting the training parameters; Step (35): inputting the data set; Step (4): optimizing the BP neural network by using the genetic algorithm (GA) to predict the final apparent air voids of the fair-faced concrete: Step (41): setting the initial population; Step (42): evaluating the fitness; Step (43): performing selection, crossover and mutation operations; Step (44): performing complete training of the GA-BP neural network; and Step (45): predicting the final apparent air voids of the fair-faced concrete. The relevant parameters of the fair-faced concrete in Step (1) include the density, the initial slump, the water-reducing agent content, the paste content, the 1h slump loss, the defoaming agent content and the air-entraining agent content of the fair-faced concrete. In Step (3), the network structure of the BP neural network is 7-9-1, the initial slump, the water-reducing agent content, the paste content, the 1h slump loss, the defoaming agent content, the air-entraining agent content and the initial apparent air voids of the fair-faced concrete are used as the input parameters, the final apparent air voids of the fair-faced concrete are used as the output parameter, the number of hidden layers is 1 and the number of nodes is 9; the input layer of the BP neural network does not use an activation function, the activation function of the output layer is the Purelin function and the activation function of the hidden layer is the Sigmoid function; the BP neural network uses a fixed learning rate of 0.01; the maximum number of training times is 1000, the training target error type is the mean square error (MSE) of 0.000001, the batch size is 32 and the training function is the trainlm function. In Step (4), the parameters of the genetic algorithm (GA) are set as follows: the size of the initial population is 40 individuals, the fitness function is the fitness = -MSE, the selection strategy is the roulette selection, the crossover probability is 0.8 and the crossover method is the uniform crossover, the mutation probability is 0.02 and the mutation method is the random bit flipping, and the maximum number of iterations is 100. ​ ​ ​ In the formula: L q is the initial apparent air void content (%) of the exposed concrete; τ y is the yield stress (Pa) of the exposed concrete; e is the natural constant; s is the initial slump (mm) of the exposed concrete; is the water-reducing agent influence factor; is the paste amount influence factor; is the slump loss over time influence factor; is the defoamer and air entraining agent complex mixing influence factor; wherein τ y The specific expression is , wherein p is the density of the clear concrete (kg / m 3 ), H is the slump cone height, the standard is 300 mm, and s is the initial slump of the clear concrete (mm); The specific expression is , wherein w is the water reducing agent dosage (%); The specific expression is , wherein k is the paste dosage (m 3 ); The specific expression is , wherein is the 1h slump loss (mm); The specific expression is , wherein d is the defoaming agent dosage (%) and a is the air entraining agent dosage (%). ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ 2. The method of claim 1, wherein: ​ 3. The method of claim 1, wherein: ​ ​ 4. The method of claim 1, wherein: The method for determining the number of nodes in the hidden layer of the BP neural network in the step (3) is: first, according to an empirical formula Preliminary estimation, wherein I is the number of input layer nodes, O is the number of output layer nodes, and C is an empirical constant, usually 5-10, then a one-by-one test method is adopted, and the mean square error (MSE) of the training set corresponding to each number of nodes is compared, and finally the number of nodes making the MSE minimum is selected as the number of nodes in the hidden layer.

5. The method of claim 1, wherein: ​

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