3.1 System Reconstruction Preprocessing

The BP neural network algorithm has super extensive applications, and the fusion effect of the established model is good and received a lot of attention. The neural network fully mines the nonlinear relationship between the enterprise and its own information data and the target object according to the sample data, continuously adjusts the process network parameters through the back propagation algorithm, and integrates the knowledge in the trained model. Stored in the network parameters, the new input data are fused according to fusion knowledge to obtain the final output fusion result ^[16,17]. Because enterprise information data are voluminous, and management levels are complex with multiple dimensions, this study employed the principal component analysis technique for sample dimension reduction processing of enterprise information data to minimize information correlation and to subsequently reduce the number of neural network input nodes, thus simplifying the network structure ^[18]. Then, the firework algorithm was used to adjust and optimize the parameters involved in the neural network. This step can improve the network's calculation speed, boost the fidelity of the mold fusion network’s calculation speed, and boost mound fusion's fidelity. The fusion flow chart from constructing the model is shown in Fig. 1.

Fig. 1. Dimensionality reduction steps in the PCA method.

Fig. 2. Evaluation model based on the BP neural network.

The flowchart in Fig. 2 shows the specific dimension reduction steps of PCA. For the data collected by a total of $g$ information management systems within the same time period in $n$ units of time, a sample matrix, $X$, can be composed of $n$ rows and $g$ columns, and formula (1) transforms this sample.

In formula (1), $\mu _{ij}$ represents the parameter matrix obtained after sample matrix transformation; $x_{ij}$ represents the sample; $\overline{x_{j}}$ is the average sample. $\overline{x_{j}}=\frac{1}{n}\sum _{i=1}^{n}x_{ij}$ and $s_{j}^{2}=\frac{1}{n-1}\sum _{i=1}^{n}\left(x_{ij}-\overline{x_{j}}\right)^{2}$ can get the standardized matrix, $U$. The solution of the correlation coefficient matrix of the standardized matrix is shown in formula (2):

In formula (2), $R$ and its characteristic equation is solved. See formula (3) for details:

In formula (3), $m$ represents the eigenvalues obtained when the matrix equation is solved. Then, the selected eigenvalues are sorted in descending order, and the feature vectors corresponding to the former proper values, $p$, are picked and combined into a transformation array. The expression for $A^{T}$ is shown in formula (4):

In formula (4), $u$ represents different eigenvectors. The value for $p$ is determined from $\frac{\sum _{j=1}^{P}\lambda _{j}}{\sum _{j=1}^{m}\lambda _{j}}\geq 0.85$. That is to say, when the contribution rate of the information of the previous principal component $p$ is significantly greater than 85%, the former principal component $p$ can be used as the eigenvalue of the sample. In the process, to eliminate the adverse effects of different data on the prediction fidelity of the mold, it is necessary to normalize the sample number. So min-max normalization is used to linearly change the original data and control the data between [0,-1]. The specific expression is formula (5):

In formula (5), $x$ deputizes the basic number; $\min (x)$ deputizes the least number, $\max \left(x\right)$ deputizes the max in the basic number; $x'$ represents the valid data after normalization. During the running of the model, 90% of the normalized sample data is stochastic chosen as the workout sample, and the leftover 10% is used as a sample to be tested.

3.2 Construction of the FWA-BP Enterprise Game Machinery Bottom

After improving the BP algorithm by using the principal component analysis method, the BP neural network model still has problems such as local extreme values and a slow convergence rate. In order to better optimize and adjust the training parameters, this study introduces FWA to optimize it and improve the computing speed and fusion accuracy of the network. FWA will constantly produce fireworks during operation of the system. Each firework is regarded as a solution to the system problem in the computing space, while the process of firework explosion is regarded as the process of constantly seeking the optimal solution.

Combining FWA and BP algorithms can effectively enhance the system's global search capabilities and makes more efficient use of parallel computing resources. Compared with other algorithms, implementation of the FWA-BP neural network algorithm is simpler.

The algorithm first determines the number of starting fireworks, $N$, and the dimensions of different fireworks, $n$. The upper and lower limits of the fireworks dimensions are then determined, and fireworks are randomly generated within that range. Then, the explosion operation is carried out based on the strategy ^[19-21]. Sparks generated in the explosion process are randomly distributed, and the number and range of sparks are mainly determined according to the detonation semidiameter, $A_{i}$, and the date of detonation spark-over,$S_{i}$. The specific expression is formula (6).

In formula (6), $f\left(x_{i}\right)$ represents the fitness value for each firework; $y_{\min }$ and $y_{\max }$ represent the worst-error and optimal-error values for solving the objective function; $E_{r}$ represents the parameter that controls the range of sparks; $E_{n}$ represents the parameter that controls the number of sparks; and $\varepsilon $ is the coefficient to avoid 0 in the denominator. The explosion operation, $D_{select}=rand*D$, is carried out for each firework, randomly determining the number of dimensions, $x_{i}$, that needs to be offset. formula (7) shows the explosion sparks generated:

In formula (7), $x_{ik}$ means firework $i$ is selected and needs to be offset; $k$ denotes the explosion spark generated after firework $i$ explodes; $h$ is the random value generated within the explosion radius. The specific expression is formula (8):

The introduction of Gaussian sparks in the FWA increases the difference in the study plant or animal community. The figure of Gaussian sparks is defined as $g$. Fireworks are randomly selected from the initial fireworks population, and a dimension multiplied by random number $e$, which conforms to the Gaussian distribution, is defined as shown in formula (9):

In formula (9), the range of $e$ belongs to the Gaussian scatter, $N\left(1,1\right)$. For sparks beyond the boundary generated during the explosion, mapping rules are used to constrain them, as shown in formula (10):

In formula (10), $x_{ik}$ represents the boundary line that firework $i$ exceeds in dimension $k$. Then, the initial fireworks group, the explosion fireworks group, and the Gaussian sparks group together form the fireworks group, which is defined as $K$. The one with the best fitness performance is picked from the plant or animal community, and then, one is picked from the remaining fireworks groups according to the selection strategy, $N-1$. The $N$ selected fireworks are used as the fireworks population for the next iteration ^[22,23]. The strategy formula is (11):

In formula (11), $R\left(x_{i}\right)$ represents the sum of the European distances between other fireworks and current fireworks $x_{i}$. The range for Euclidean distance is ^[0,1]. In the set of all candidates, $p\left(x_{i}\right)$ represents the probability that fireworks $x_{i}$ are selected. When calculating the fitness value of the fireworks population generated in each iteration, if the optimal fitness satisfies the maximum number of iterations, the search for the optimal solution ends; otherwise, it jumps to the first step of the process. Combining the fireworks algorithm with the neural network, a BP neural network is created, the fireworks population is initialized, and the goal of optimization is actually all the network parameters in the neural network. The fitness worth of every firework is computed according to the above steps of the FWA. The sum of squared error (SSE) of the neural network is taken as the fitness function, and the expression for $f\left(x_{i}\right)$ is shown in (12):

In formula (12), $t$ represents the anticipated export value under the BP approach, the quantity of nerve cells in the neural net's export layer, and the actual export value under the BP calculation. According to the steps, optimization of the fireworks population and judgment on the termination conditions are carried out, and the optimal individual satisfying the conditions is decoded and assigned to the neural network. The LM algorithm is then used to train and optimize the neural network weights and thresholds with higher precision. The process involves setting the target error function to the SSE, and specifying the number of training iterations and target error beforehand, followed by initiating the training process. When the target error or the max quantity of iterations is achieved during training, it means the training is over, and establishment of the optimal model is completed. If the conditions are not met, the previous step is conducted to continue the operation.

The input variables of the sample data define the nodes in the net. There is no specific method to be sure of the number of nodes in the concealing layer. Therefore, a common empirical formula is selected, and the number of nodes is determined according to the change in the selected formula to carry out the experiment. The expression is shown in (13):

In (13), $n'$ is the number of nodes in the hidden layer in the neural network, $m'$ represents the number of nodes in the input layer, and $p$ represents the quantity of nodes in the output layer. When network performance no longer changes with an increase in the quantity, it could mean the number of conceal nodes at this moment is the optimal quantity of conceal nodes. After that, all indicators would be preprocessed accordingly, as seen in Fig. 2.

Fig. 3. Flow chart of the information game based on the fusion model for the FWA-BP neural network algorithm.

In Fig. 3, all indicators are preprocessed first, and then BP neural network detection is performed through a series of operations. The neural network deals with enterprise information, and thus, conducts games among the enterprises. Before fusion, it is necessary to clarify the relationship between the data and the target object, which is not a simple linear relationship. Therefore, it is necessary to add nonlinear factors to the activation function in the hidden layer to improve the nonlinear expression power of the mold. The hidden layer usually adopts a nonlinear sigmoid function, and the output layer adopts a linear purelin effect. When comparing the constructed models, two or more models are usually compared, and the same test samples are used to test the effect fusion. Most researchers select the mean absolute error (MAE) and the root mean square error (RMSE) as test indexes. The specific expression is formula (14):

In formula (14), $N'$ is the test date of the pattern, and $y$ is the predicted output date. Using the above formula to build the enterprise game system bottom process in the FWA-BP fusion model is revealed in Fig. 3.

4.1 Cumulative Contribution of Principal Components and Sample Training

There are many variables in the enterprise information management system, and the data from these variables are collected by different game systems. There are strong correlations among the data. If they are directly input to the neural network without processing, it may increase the complexity of the network, which will eventually affect the accuracy of the model. Therefore, in the early stages of model construction, PCA should be performed on the obtained combined dataset. The princomp function in MATLAB was called to carry out PCA processing ^[24,25]. The integrated proffer speed of the most important module of the variables in the combination is shown in Fig. 4.

Fig. 4. Cumulative contributions of principal components.

In Fig. 4, the dataset was reduced to six principal components that accounted for 94% of the variance. This dimensionality reduction helped streamline the enterprise information management game system and simplify the network structure, leading to the construction of the FWA-BP fusion model. To showcase the effectiveness of the game system developed in this study, the program was executed using a MATLAB 2017 environment. First, BP was structured as 6-12-1, with the number of input variables determined to be six due to panel data; the quantity of panel points in the hidden layer was set at 12 according to model configurations. The output layer variables represented the information management of game system effectiveness, hence requiring a single output neuron. The hidden layer activation function in BP utilized logsig, while the output layer selected the purelin effect. The main parameters of FWA-BP were set as follows: population size, $N=10$; adjustment constant of the radius of the firework explosion, $E_{r}=5$; adjustment constant of the quantity of sparks in the fireworks detonation, $E_{n}=30$; the quantity of Gaussian variation sparks, $g=10$; and the maximum number of iterations, $T=100$.

After using the FWA to update the coarse precision of BP, it was necessary to use the practicing record to further train the mold. The number of training sessions was 300; the training function selected was trainlm; the learning rate of the algorithm was expressed as 0.01, and the target error of training was set at 0.01. The specific training situation is shown in Fig. 5.

As shown in Fig. 5, the number of iterations for training the constructed model was set to 300 in advance, but in practice, the model mostly reached the target error after 110 iterations. This showed the FWA was effective for parameter optimization of the BP neural network.

Fig. 5. The training situation of the sample.

4.2 Comparison of Output Effects, and Evaluation Scores of Different Models

Then, the LM algorithm was used to train the model with higher precision, which was conducive to the model jumping from within the local min, improving practice fidelity and speeding up the training. The model was then tested, and the results are shown in Fig. 6.

Fig. 6. Test results from the FWA-BP mold.

Based on the FWA-BP construction experiment performed on the enterprise game system, the results depicted in Fig. 7 demonstrate that while prediction deviation for some samples exceeded 10, the remaining model predictions proved consistent with the actual expected value, the predicted curve largely aligning with the actual curve. Generally speaking, the closer the accessories degree is to 1, the better the fitting effect, and the fit was 0.9254 in the $R^{2}$ results. This showed that the constructed fusion model can well represent information management of the enterprise game system, and can also meet the system’s precision requirements. To verify the feasibility and superiority of the model, the evaluation results from the constructed model were compared with the improved particle swarm optimization algorithm BP (IPSO-BP) model and the improved genetic algorithm BP neural network (IGA-BP) model. The comparison is shown in Fig. 7.

Fig. 7. Comparison of the effects obtained from different models.

From Fig. 7, observe that the RMSE predicted by the FWA-BP neural network model is approximately 7.775; The prediction result of the IGA-BP neural network is approximately 7.941; The IPSO-BP neural network predicted a result of approximately 7.928. The results of the constructed model are not significantly different from the IGA-BP neural network, slightly smaller than the IGA-BP neural network, but slightly larger than the predicted results obtained by IPSO-BP. The reason the model produces such results may be from outliers in the predicted output that deviate too much from the true value. And due to the squared processing of the model error, when encountering outlier data, the error will be amplified, resulting in poorer RMSE results. However, in the evaluation of MAE indicators, the indicator effect of the FWA-BP neural network is superior to the other two models, with a score of 6.608. The difference between the indicator score of the IPSO-BP neural network and the proposed model’s score is relatively small, while the score of the IGA-BP neural network indicator was relatively large.

4.3 Simulation Comparison of the Enterprise Information Management Game System Model

To further verify the accuracy and effectiveness of the proposed method, a fitness index was used for analysis and comparison. The fitness curves of the FWA and IPSO algorithms are plotted in Fig. 8.

Fig. 8. Comparison curve of fitness between FWA-BP and IPSO-BP.

Fig. 8 compares the number of iterations at which the algorithms reached convergence. Between them, the FWA-BP curve reached convergence earlier, and the final value was small, whereas the IPSO-BP curve achieved convergence after 150 iterations and fell into a local minimum. The search for solutions stopped prematurely. However, FWA-BP continued to calculate downwards to find the optimal solution at the same time the number of iterations reached equilibrium. The comparison of the two phases showed that global optimization of the FWA-BP was stronger, and the best point can be found faster. This also showed that the research and construction model could quit the local optimum more easily in the process of calculation and operation, and accuracy was higher. Then, the degree of fit for the game system was analyzed, as shown in Fig. 9.

Fig. 9. Fitness curve of the game system model.

Fig. 9 shows that the error analogy result of the IGA-BP neural network was significantly higher than the FWA-BP algorithm: 2.612 versus 0.625, respectively. The maximum error value IGA-BP could reach was 4.3, which was relatively large; while the maximum error between the FWA-BP neural network and the target result was 0.01. Comparison of the two showed that FWA-BP had a very good fit, which can be applied to system construction of enterprise information management and can effectively ensure its stability and consistency. After passing the above tests, a construction enterprise was taken as an experimental object to test the performance of the information management system during construction of the enterprise. The duration of the test was 24 hours. The specific test is shown in Table 1.

Table 1 shows the system stability and use of the information management system of a construction enterprise. As time passed, the utilization rate of the CPU and system memory changed little. When the test time reached 6h, maximum CPU utilization was 21.3%; At 12h, minimum CPU utilization was 19.9%. At 24h, the system memory utilization rate had a maximum of 60.8%, but at 6h, system memory utilization had a minimum value of 59.9%. In the overall process, the difference between CPU and memory utilization rates of the built system was not more than ${\pm}$ 0.5, and stability can be guaranteed.