Improved BP Neural Network Algorithm for Predicting Structural Parameters of Mirrors

Abstract

Addressing the nonlinear correlations between input variables and output responses, in addition to the time-consuming nature of finite element analysis in mirror design, this study introduces an enhanced back-propagation (BP) neural network (BR-TLDBO-BPNN) employing Bayesian regularization and an optimized dung beetle algorithm. This novel approach facilitates rapid and efficient parameter estimations, significantly reducing the computational overhead. Utilizing an integrated analysis platform, the study obtained training and test samples, and the BR-TLDBO-BPNN model is used to predict the reflector’s mass and root mean square (RMS). The optimization mathematical model is built, and the nonlinear planning function ($f\min con$) is utilized to solve the problem and find an ideal set of structural parameters. The outcomes demonstrate that the prediction model is accurate enough to predict the mirror characteristics to optimize the mirror structural parameters. Empirical validation demonstrates that the proposed model achieves an over 99% accuracy in predicting mirror characteristics against finite element simulations. As a result, the BR-TLDBO-BPNN algorithm studied in this article not only broadens the application scope of neural networks, but also provides a new practical technique for engineering design.

1. Introduction

With the continuous advancement in technology, optical mirrors are increasingly being used in fields such as astronomy, laser technology, and precision measurement [1]. However, due to the complex structure and meticulous manufacturing processes of mirrors, accurately predicting their structural parameters has become a significant research challenge [2,3,4]. Traditional methods for structural optimization include topology, shape, and size optimization [5,6,7,8,9]. Shape and size optimization, commonly known as parametric optimization, can be limited by the designer’s experience and lack of insight. Topology optimization seeks to determine the optimal layout of a structure under given constraints and load conditions. However, it can require extensive iterative processes and computational resources, making it challenging to implement. Therefore, new optimization methods must be introduced to address these limitations to improve efficiency and accuracy.

The BP neural network, as a typical artificial neural network model, is widely used in data prediction and pattern recognition due to its powerful self-learning and self-adaptation capabilities [10,11,12,13]. However, traditional BP neural networks are prone to getting trapped in local minima during training and have a slow convergence rate, which affects their prediction performance [14,15,16,17]. Additionally, the highly nonlinear relationships between input and output variables in structural optimization require robust numerical models to solve these problems accurately [18,19]. Despite the progress in neural-network-based prediction methods, there is still a notable research gap in addressing the limitations of traditional BP neural networks, particularly their susceptibility to local minima and slow convergence rates. This study aims to fill this gap by proposing a novel back-propagation neural network model that integrates advanced optimization algorithms, thereby guiding the optimization design of mirrors.

In this study, an improved dung beetle algorithm (DBO) and Bayesian regularization (BR) were combined into the BP neural network, to create the BR-TLDBO-BPNN model. The DBO algorithm is utilized to optimize the initial weights and thresholds of the neural network, enhancing the BP network ability to escape local minima and accelerate convergence. BR is incorporated to prevent overfitting, ensuring the model generalizes well to unseen data [20]. The inputs to the neural network include the thickness of the center, the substrate, the fascia, and the outermost wall thickness of the mirror, while the outputs are the weight of the mirror and the RMS accuracy of the mirror’s faceted shape under gravity.

An integrated analysis platform is used to obtain training and test samples, facilitating the construction of an optimized mathematical model of the mirror. The BR-TLDBO-BPNN model quickly solves for the optimal structure of the mirror, offering a novel approach and direction for the structural design of optical mirrors. This improved algorithm significantly enhances the prediction accuracy and efficiency, providing crucial technical support for mirror manufacturing and quality control. By addressing the gap in intelligent structural design, this study provides a new solution for future optimization designs of similar structures.

2. Basic Theory

2.1. BP Neural Network

The BP neural network is a multi-layer feed-forward model trained according to the error back-propagation algorithm with a strong nonlinear mapping capability. The structure of the BP neural network consists of three parts, namely, the input layer, the hidden layer, and the output layer, as shown in Figure 1.

The neural network training process consists of forward propagation and BP. In the forward propagation phase, the input data reaches the output layer from the input layer through the hidden layer and produces the output value [21]. The calculation produces an error if the output value does not match the actual value. In the BP stage, the error is propagated backward from the output layer to the input layer, and the weights and biases of the neural network are adjusted layer by layer during the process [22]. After repeated propagation, the network training stops when the minimum error requirement or the maximum number of training times is met. The goal of the BP neural network is to continuously reduce the error between the output value and the actual value through the learning process, thus improving prediction accuracy [23].

2.2. BR Algorithm

Since traditional BP neural networks will fit the sampling error contained in the training samples during the training process, it will cause the network overfitting phenomenon; i.e., it exhibits a minor error in the training set and a more significant error in the test set [24]. The BR algorithm is a Bayesian-theory-based regularization method mainly applied to neural networks and other machine-learning models to improve their generalization ability and prevent overfitting [25]. The core of the BR algorithm lies in transforming the traditional objective of minimizing the error function to maximizing the posterior probability while considering the probability distribution of the model weights. This study uses the BR algorithm to control the model complexity and constrain the parameters to be optimized to improve the model’s generalization ability.

In the traditional BP neural network training process, the error function is usually a mean square error function (MSE), denoted as:

$${\mathrm{E}}_{mse}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{(y_i-t_i)}^2}$$

(1)

where $y_i$ denotes the desired output value, $t_i$ denotes the actual output value, and $n$ denotes the number of samples. The BR algorithm constrains the model complexity by adding a regularization term to the error function. The regularization error function ${\mathrm{E}}_{\mathrm{BR}}$ is denoted as:

$${\mathrm{E}}_{BR}=\frac{\mathsf{\beta }}{n}{\displaystyle \sum _{i=1}^n{(y_i-t_i)}^2}+\frac{\mathsf{\alpha }}{n}{\displaystyle \sum _{i=1}^n{w}^2}$$

(2)

The first term is the prediction error, the second term is the regularization penalty of the weights, and $\mathsf{\alpha }$ and $\mathsf{\beta }$ are the regularization coefficients.

In the Bayesian framework, the BR algorithm considers the merit of data fitting and the model complexity [26]. It maximizes the posterior probability of the model by adjusting the regularization parameters, $\mathsf{\alpha }$ and $\mathsf{\beta }$. The regularization parameters, $\mathsf{\alpha }$ and $\mathsf{\beta }$, are the regularization coefficients. If $\mathsf{\alpha }>\mathsf{\beta }$, the training error is appropriately reduced to prevent overfitting; if $\mathsf{\alpha }<\mathsf{\beta }$, the scale of weights is appropriately limited to reduce the effective network parameters. A larger $\mathsf{\alpha }$ will increase the weight penalty, simplifying the model and reducing the risk of overfitting. Nevertheless, it may underfit the data and reduce the performance of the model. A smaller $\mathsf{\alpha }$ will lessen the weight penalty, making the model more complex and able to better fit the training data, but increasing the risk of overfitting. A larger β will increase the weight of the data error term, making the model fit the training data more accurately, but it may lead to overfitting, especially on noisy datasets. A smaller β will reduce the weight of the data error term, allowing the model to have a certain degree of error, thereby reducing overfitting, but may lead to the underfitting of the model. From the above, it can be seen that the BR algorithm is an iterative process, each iteration of which optimizes the regular coefficients $\mathsf{\alpha }$ and $\mathsf{\beta }$. This iterative process ensures that the network training error is as small as possible to make the network’s effective weights as small as possible, to reduce the size of the network, improve the network’s generalization ability, and make up for the shortcomings of the BP neural network that is prone to overfitting.

2.3. Improving the Dung Beetle Optimizer

2.3.1. Dung Beetle Optimizer

DBO is a population intelligence optimization strategy inspired by the biological behavior of dung beetles, which centers on simulating the natural behaviors of dung beetles, such as rolling, foraging, stealing, and reproducing. To form an effective search mechanism [27]. This algorithm shows unique advantages in optimizing the parameters of neural networks, especially in optimizing the BPNN’s initial weights and threshold settings. With the DBO algorithm, the initial parameters of the BPNN can be set more precisely, thus improving the efficiency and accuracy of network training [28,29]. When dealing with complex data patterns and nonlinear problems, the DBO-assisted BPNN shows a better performance and generalization ability, which is significant to deep learning and artificial intelligence research [30].

The specific process of the DBO algorithm is as follows: firstly, the population initialization is performed—i.e., a set of initial solutions is randomly generated, and its fitness is calculated; then, it enters the iterative updating phase—in each generation, a set of individuals is selected as the parent, a mutation operation is performed on the parent to generate a new set of solutions, and then a crossover operation is performed to generate the next-generation population; then, the fitness of the solutions of the new-generation population is evaluated and compared with that of the previous generation; and, when the preset termination conditions are satisfied, the iterative process is ended and it results in the optimal solution.

2.3.2. Improved Dung Beetle Optimizer

The DBO optimization algorithm is excellent in optimality-seeking ability and convergence speed. However, it needs to be improved regarding the imbalance between global exploration and local exploitation, the ease of falling into local optimal solutions, and the need for global exploration ability. This study proposes an improved dung beetle optimizer (TLDBO), which is mainly improved in the following three aspects:

• Logistic-tent chaotic mapping initializes the population and maintains diversity, enhancing the algorithm’s optimality-seeking ability;
• The dynamic bipartite group convergence factor with an adaptive inertia threshold balances the algorithm’s local and global optimization-seeking ability;
• The Levy flight mechanism is used to optimize the foraging behavior and improve the global search ability.

To understand these aspects better, a brief introduction is discussed regarding them.

• Logistic-Tent Chaotic Mapping

Since the size of each dimension of dung beetle individuals is randomly generated in the initialization stage, this leads to the initial solution being prone to aggregation, and having poor coverage in the solution space and low variability among individuals. At the same time, chaotic mapping initializes the population and can effectively improve the problem. Tent chaotic mapping, logistic chaotic mapping, and circle chaotic mapping are more common. The logistic-tent chaotic mapping adopted in this study integrates the complex, chaotic dynamics of logistics and the faster iteration speed, more autocorrelation, and applicability to many sequences of the tent chaotic system. The mathematical expression of logistic–tent chaotic mapping is:

$$x_{n+1}=\left\{\begin{array}{ll}\left[r{x}_n(1-x_n)+\frac{(4-r)}{2}{x}_n\right]\mathrm{mod}1,& \hfill if{x}_n<0.5\\ \left[r{x}_n(1-x_n)+\frac{(4-r)(1-x_n)}{2}\right]\mathrm{mod}1,& \hfill if{x}_n\ge 0.5\end{array}\right.$$

(3)

where $r$ is the control parameter, $r\in (0,1)$, and $x_n$ is the position information of the dung beetle.

Figure 2 shows the initialization of the logistic–tent chaotic mapping population. The improved logistic–tent chaotic mapping chaotic values are more evenly distributed. Therefore, the improved logistic–tent chaotic mapping is used to initialize the population to enhance the diversity of the population, which, in turn, enhances the algorithm’s ability to find the best.

• Dynamic Bipartite Group Convergence Factors and Adaptive Inertia Thresholds

In the DBO, as the convergence factor R decreases linearly, the area of the dung beetle breeding area and dung beetle foraging area decreases gradually, which makes it challenging to co-ordinate the global search ability with the local search ability. Therefore, this study proposes a dynamic community convergence factor and a dynamic community strategy suitable for the dung beetle optimizer.

During the iterative process of dung beetle reproduction and dung beetle foraging populations, the Euclidean distances between the current dung beetle position $x_i^t$ and the optimal dung beetle position $x_{gbest}^{\prime }$ and the worst dung beetle position $x_{gworst}^{\prime }$ are calculated as $D_{gbest}^{\prime }$ and $D_{gworst}^{\prime }$, respectively. A threshold ($D_{gbest}^{\prime }\le D_{gworst}^{\prime }/3$) is set to classify dung beetles in this range as better subpopulations, while the others are categorized as worse. According to the characteristics of the two subgroups, this study adopts different nonlinear convergence factors, $R_1$ and $R_2$, as, respectively:

$$R_1=2-\left\{4.95\cdot \frac{\ln \left[1+{(t/T)}^{0.5}\right]}{e-1}\right\}$$

(4)

$$R_2=2-\left\{4.95\cdot \frac{\ln \left[1+{(t/T)}^2\right]}{e-1}\right\}$$

(5)

This study uses different convergence factors for the better subpopulation with the higher adaptation and the worse subpopulation with the lower adaptation. The nonlinear convergence factor $R_1$ of the better subpopulation decreases rapidly at the beginning of the iteration to promote the dung beetle in approaching the optimal solution quickly. It decreases slowly at the end of the iteration to prevent local optimization. The convergence factor $R_2$ of the worst subgroup decreases gradually at the beginning of the iteration to expand the dung beetle’s search range. It decreases rapidly at the end of the iteration to achieve convergence. Figure 3 shows the change in the convergence factor after the improvement.

Inspired by the idea of inertia weights in the particle swarm optimization (PSO), an adaptive inertia weighting strategy is added to the dung beetle daisy ball position update and the foraging small dung beetle position update to reduce the probability of the algorithm falling into a local optimum and the adaptive inertia weights are calculated as follows:

$$w(t)=w_{\min }+(w_{\max }-w_{\min })\cdot {(1-\frac{t}{t_{\max }})}^{\delta }$$

(6)

where $w_{\max }$ is the maximum value of the inertia weights, $w_{\min }$ is the minimum value of the inertia weights, $t_{\max }$ is the maximum number of iterations, and $\delta$ is a positive accurate number parameter that controls the decay rate. When $\delta >1$, the inertia weights decline slower at the beginning of the iteration, which helps the algorithm to perform a broader global search, while the rate of decline is accelerated at the later stages of the iteration, which helps the algorithm to perform a more focused local search. When $\delta <1$, the inertia weights fall faster at the beginning of the iteration, allowing the algorithm to focus more on local search early on, and fall slower later in the iteration, helping the algorithm to maintain the diversity of the search, thus avoiding falling into a local optimum solution too early.

The dung beetle daisy ball position update formula after introducing adaptive inertia weights is:

$$B_i(t+1)=\mathrm{w}(t)\cdot X^{*}+b_1\times (B_i(t)-L{b}^{*})+b_2\times (B_i(t)-U{b}^{*})$$

(7)

where $B_i(t)$ is the position of the $i$th dung beetle at the $t$th iteration, $b_1$ and $b_2$ are the random variables of $1\times D$, and $D$ is the optimization dimension.

The position update formula of the foraging dung beetle after introducing adaptive inertia weights is:

$$x_i(t+1)=\mathrm{w}(t)\cdot x_i(t)+C_1\times (x_i(t)-L{b}^{*})+C_2\times (x_i(t)-U{b}^{*})$$

(8)

where $x_i(t)$ is the position of the $i$th daisy at the $t$th iteration, $C_1$ is a random number that follows a normal distribution, and $C_2$ is a random number belonging to (0, 1).

• Levy flight mechanism

The Levy flight mechanism is a stochastic search strategy commonly used in optimization algorithms that mimics the natural behavior of certain animals, such as the migratory paths of birds, which exhibit a specific type of random walk with step sizes that obey a Levy distribution. As shown in Figure 4, compared to Gaussian and Cauchy distributions, the Levy distribution is heavy-tailed, converges faster, and provides extensive global search capabilities in optimization. The formula for generating the Levy flight random walk is:

$$Levy(d)=\frac{u}{{\left|r_2\right|}^{1/\beta }}$$

(9)

where $\mu \in (0,{\sigma }_{\mu }^2),\nu \in (0,{\sigma }_{\nu }^2)$. ${\sigma }_{\mu }$ and ${\sigma }_{\nu }$ are:

$${\sigma }_{\mu }={\left\{\frac{\Gamma (1+\beta )\sin (\pi \beta /2)}{\Gamma \left[(1+\lambda )/2\right]\beta 2^{(\beta -1)/2}}\right\}}^{1/\beta },{\sigma }_{\nu }=1$$

(10)

where $\Gamma$ is the gamma function, $\beta$ is a constant, and $\beta \in \left(0,2\right)$.

2.3.3. Performance Evaluation of Improved Dung Beetle Optimizer

To verify the performance of the improved algorithm, the improved TLDBO algorithm in this study is compared with gray wolf optimization (GWO), northern goshawk optimization (NGO), the whale optimization algorithm (WOA), and DBO in terms of the test functions. The main purpose of optimizing algorithm testing functions is to evaluate and validate the performance and applicability of the optimization algorithm. These test functions are usually mathematical functions designed to simulate practical optimization problems and have known global optimal solutions or known optimal solution properties. By conducting experiments on these test functions, we can understand the performance of optimization algorithms in different situations, and discover their advantages, disadvantages, and applicability. Six benchmark test functions were selected for the experiment, and the dimensions and search ranges of the single-peak and multi-peak benchmark test functions are shown in

Table 1. Single-peak and multi-peak benchmark test functions.

Function Categories	Formula Expression	Dimension	Search Scope	Optimum Value
Single-peak function	$f_1(x)={\displaystyle \sum _{i=1}^n{x}_i^2}$	30	[−100,100]	0
	$f_2(x)={\displaystyle \sum _{i=1}^n\left|x_i\right|+}{\displaystyle \prod _{i=1}^n\left|x_i\right|}$	30	[−10,10]	0
	$f_3(x)={\displaystyle \sum _{i=1}^n({\displaystyle \sum _{j=1}^i{x}_j}}{)}^2$	30	[−100,100]	0
Multi-peak function	$f_4(x)={\displaystyle \sum _{i=1}^n-x_i\sin \left(\sqrt{\left|x_i\right|}\right)}$	30	[−500,500]	0
	$f_5(x)={\displaystyle \sum _{i=1}^n}\left[x_i^2-10\cos (2\pi x_i+10)\right]$	30	[−5.12,5.12]	0
	$f_6(x)=\frac{1}{4000}{\displaystyle \sum _{i=1}^n{x}_i^2}-{\displaystyle \prod _{i=1}^n\cos (\frac{x_i}{\sqrt{i}})+1}$	30	[−600,600]	0

, where $f_1$, $f_2$, and $f_3$ are single-peak benchmark test functions and $f_4$, $f_5$, and $f_6$ are multi-peak benchmark test functions. The number of populations of each algorithm is set to 30, and the number of iterations is 500. In order to reduce the chance of the experiment and increase the persuasive power of the experimental results, each algorithm is run independently 50 times on each of the six benchmark test functions, and the optimal value, average value, and standard deviation of the function are calculated. Figure 5 shows the three-dimensional view of the single-peak and multi-peak functions.

The convergence curves of different optimization algorithms on the six benchmark test functions are shown in Figure 6. From the convergence curves and the experimental results, it can be visualized that the improved TLDBO algorithm performs better in finding optimization in different test functions. Compared with the other four algorithms, TLDBO has a better solution accuracy and faster convergence speed in single-peak test functions; TLDBO has a better ability to jump out of the local optimum in multi-peak test functions. The improvement strategy proposed above is effective, and the solution effect and robustness of TLDBO are better and more stable than the other algorithms. It also has a solid ability to find the optimum.

3. BPNN Prediction Model Construction

3.1. Integration Analysis

The structural form of the mirror directly affects its structural rigidity, thermal performance, and optical surface face type accuracy. In this study, the mirror’s diameter is 600 mm, the back of the mirror adopts the back-opening structure, and the shape of the aperture is a triangular aperture, which is a better structural form of craftsmanship and has good stability. Figure 7 is the finite element model of the mirror; according to the finite element analysis results, the structural parameters affecting the precision and stability of the optical surface pattern include the center thickness of the mirror body, the thickness of the substrate, the thickness of the ribbed plate, and the thickness of the outermost wall, which determine the quality of the mirror and the mirror surface shape error.

3.2. Training Sample Confirmation

Using the Latin hypercube sampling method to sample the input random variables, 400 sets of input variable data were obtained. Based on 400 sets of input and output data, we generate 320 sets of training samples and 80 sets of testing samples. Then, we determine the structure of the neural network, set prediction errors, and establish a prediction model using an improved DBO algorithm, BR algorithm, and BPNN method. We normalize and preprocess the training and testing sample data. The number of hidden neurons in the M model is 9, and the number of hidden neurons in the improved dung beetle algorithm is 40.

The inputs to the neural network are the thickness of the overall center, the substrate, the fascia, and the outermost wall. The outputs of the neural network are the weight of the reflector and the RMS accuracy of the mirror’s face shape under gravity. A neural network consisting of one hidden layer and one output layer is used to build a prediction model for the structural parameters of the reflector. As shown in Figure 8, x₁, x₂, x₃, and x₄ represent the thickness of the center, the substrate, the fascia, and the outermost wall of the reflector, respectively, and M and RMS represent the weight of the reflector and the RMS of the mirror face shape accuracy under gravity, respectively.

Because of the high cost of manually modifying finite element model parameters and finite element analysis, as well as the need for different tools and software, the Isight integration platform is one of the most mature platforms in the field of optomechanical integration optimization, which can easily and effectively integrate multidisciplinary software (such as optical error calculation and finite element analysis) with automated processes. Firstly, parameterizing the 3D model through UG and using coarse mesh partitioning can effectively save computational time, while completing attribute settings and constraint definitions to generate a finite element model. Subsequently, utilizing the recording of macro programs and generating loop scripts to achieve automated processes, the application of integrated analysis platforms can greatly improve efficiency. We substitute input variable data into the finite element model for modal analysis, and obtain 400 sets of output responses.

3.3. Establishment of BR-TLDBO-BP Prediction Model

Although BP neural networks have a strong nonlinear mapping ability, they often fall into local optimal solutions, making the prediction less accurate. The BR-TLDBO-BPNN model utilizes the global optimality-searching ability of the DBO algorithm to search for the optimal initial weights and thresholds of the network in its solution space. The position of the dung beetle on each dimension is the solution seeking to establish the mapping relationship between the dimensions of the dung beetle and the weights and thresholds. The BP neural network is trained and examined using the initial weights and thresholds optimized by the TLDBO method and Figure 9 shows the specific flowchart of the BR-TLDBO-BPNN model.

The BR-TLDBO-BPNN model proposed in this study consists of four main parts, as follows:

• The acquisition of samples: The application of the integrated analysis platform can significantly improve efficiency. The input variable data are substituted into the finite element model for simulation and analysis, and 400 output variable data are obtained.
• The TLDBO model: Three improvement strategies are proposed in this study to aim at the problems of the original DBO. First, the logistic–tent chaotic mapping initializes the population to maintain diversity, enhancing the algorithm’s ability to find the optimal. Second, the dynamic bipartite group convergence factor with an adaptive inertia threshold balances the algorithm’s local and global optimization-seeking ability. Finally, the Levy flight mechanism is used to optimize the foraging behavior and improve the global search ability.
• The BR-TLDBO-BPNN model: TLDBO-BPNN utilizes TLDBO to obtain the initial weights $w$ and initial thresholds $b$ of the BPNN, and TLDBO keeps updating the initial weights $w$ and initial thresholds $b$ until it finds the global best position, i.e., the optimal solution. The BR algorithm is used to control the model complexity and constrain the parameters to be optimized to improve the model’s generalization ability.
• Objective optimization: First, we determine the objective function, then set the constraints and parameter weights, and, finally, use TLDBO to optimize the objective to obtain the best optimization result.

4. Results and Analysis

4.1. Validation of the BR-TLDBO-BPNN Prediction Model

4.1.1. Evaluation Indicators for the Model

In order to measure the prediction effect of the model, the mean absolute error (MAE), mean square error (MSE), root mean square error (RMSE), and mean absolute percentage error (MAPE) are used as the evaluation indices, and the formulae are calculated as follows, respectively:

$$\left\{\begin{array}{l}{E}_{MAE}=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|\stackrel{\wedge }{y_i}-y_i\right|}\hfill \\ E_{MSE}=\frac{1}{n}{{\displaystyle \sum _{i=1}^n(\stackrel{\wedge }{y_i}-y_i)}}^2\hfill \\ E_{RMSE}=\sqrt{\frac{1}{n}{{\displaystyle \sum _{i=1}^n(\stackrel{\wedge }{y_i}-y_i)}}^2}\hfill \\ E_{MAPE}=\frac{1}{n_{samples}}{\displaystyle \sum _{i=0}^n\left|\frac{\stackrel{\wedge }{y_i}-y_i}{y_i}\right|}\hfill \end{array}\right.$$

(11)

4.1.2. Analysis of Cross-Validation Results

Table 2. Comparison of cross-validation results.

Evaluation Indicators	BP Predictive Model	BR-TLDBO-BPNN Predictive Modeling
$E_{MAE}$	0.171	0.118
$E_{MSE}$	0.063	0.031
$E_{RMSE}$	0.251	0.178
$E_{MAPE}$	0.793	0.545

shows the comparison results of the cross-validation results of the two models, and each evaluation index of the model is taken as the average of the K prediction results. As shown in Table 2, compared with the BP neural network, the prediction value of BR-TLDBO-BPNN is closer to the actual value. The mean absolute error ($E_{MAE}$), mean square error ($E_{MSE}$), root mean square error ($E_{RMSE}$), and mean absolute percentage error ($E_{MAPE}$) are reduced by 39.9%, 50.8%, 29.1%, and 31.27%, respectively. The above results show that the BR-TLDBO-BPNN model effectively overcomes the limitation whereby BP neural networks are prone to fall into a local optimization and improves the prediction effect of the model in dealing with complex nonlinear problems, especially in terms of accuracy and stability. This advancement provides a more reliable and efficient computational tool for the optimal design of reflector structures.

4.1.3. Analysis of Forecast Results

The regression analysis plots of BR-TLDBO-BPNN after training are shown in Figure 10, respectively. The fitting curves were obtained by fitting the test data to the predicted values, Y representing the predicted values, T representing the test data, and the dashed line indicating the ideal conditions where the predicted values and test data are equal. The regression analysis plot reflects the fit between the predicted and actual values of the data, and the R-values of all regression lines are very close to 1. The fitting accuracies of the training set, validation set, test set, and the total dataset are all above 99%. It shows an excellent agreement between the actual and predicted values, which ensures that the improved neural network model proposed in this study has a good generalization ability.

Figure 11 shows the comparison between the predicted results and the actual values of the two network models, BPNN and BR-TLDBO-BPNN, and Figure 11a,c show the actual values of each test sample, as well as the predicted values of the two models. It can be observed that the BR-TLDBO-BPNN model predicts closer to the actual values on most of the samples, with a more compact distribution of points showing a minor variance, suggesting that its prediction stability and accuracy may be higher. Figure 11b,d, on the other hand, are scatter plots showing the prediction errors of the two models. It can be seen that the BR-TLDBO-BPNN model has a minor error on most of the test samples, and the error points are mainly concentrated around zero, which indicates its higher prediction accuracy.

The experiment was carried out on a computer with an Intel i7 14600KF processor, Nvidia RTX 3080 graphics card, 32 GB RAM, and MATLAB 2019a. The time of calculating a test sample by ANSYS is about 300 s, while the method proposed in this paper only needs 0.13 s, which shows that the BR-TLDBO-BPNN model has the significant advantages of its running time and high efficiency. In general, the combination of Bayesian regularization and improved dung beetle optimization algorithms helps to improve the generalization ability of the model and avoid overfitting; the BR-TLDBO-BPNN model is superior to the standard BPNN model in prediction performance.

Overall, the BR-TLDBO-BPNN model outperforms the standard BPNN model in terms of predictive performance due to the combined application of Bayesian regularization and improved dung beetle optimization algorithms, which helps to improve the generalization of the model and avoid overfitting.

To analyze the relationship between the influence of each input parameter on the output parameter, the Pearson correlation coefficient method was used to quantify the linear relationship between the inputs and outputs for the established prediction model, and the degree of influence between the two was assessed by a multiple regression analysis. Figure 12 shows the calculated results of the relationship between input and output parameters.

As can be seen in Figure 12a, Y₁ (mass) has a significant positive correlation with X₁ (center thickness) and X₃ (tendon thickness). At the same time, Y₂ (RMS) shows a significant negative correlation with the two input parameters, X₁ and X₃. In particular, the correlation coefficient between X₃ and Y₁ is as high as 0.81, indicating that, when the tendon thickness increases, the mass also increases, while the correlation coefficient between X₃ and Y₂ is −0.41, suggesting that an increase in the tendon thickness may lead to a decrease in the RMS. From Figure 12b, it is further known that X₃ has a significant positive effect on Y₁ with a coefficient of 2.01; on the contrary, X₃ has a negative correlation with RMS with a coefficient of −1.1403, which is consistent with the results of the thermogram. Combining the analysis of multiple regression coefficient plots and thermograms, it can be seen that the rib thickness (X₃) has the highest sensitivity to mass (Y₁) and RMS (Y₂) relative to other variables, and the influence of rib thickness on the performance of mirrors should be prioritized in the design process, which is an essential guideline for the optimization of parameter adjustments in the design and manufacturing process.

4.2. Optimization of Reflector Parameters Based on BR-TLDBO-BPNN Model

For a long time, structural design and optimization are often carried out based on the designer’s experience. Considering the optimization scheme’s accuracy and efficiency, this study approximates the relationship between the structural parameters and performance parameters of the reflector through the BR-TLDBO-BPNN model. It uses it to construct the objective function of the mathematical model for reflector optimization and obtains a better combination of structural parameters of the reflector after solving.

4.2.1. Parameter Optimization Process

• Design variables

Since the objective function in the optimization mathematical model will be used in the neural network model, the optimization design variables must be consistent with the input samples of the neural network data set. $x_1$ (the center thickness), $x_2$ (the substrate thickness), $x_3$ (the fascia thickness), and $x_4$ (the outermost wall thickness) are selected as the design variables of the optimization mathematical model, which can be expressed as follows:

$$x=(x_1,x_2,x_3,x_4)$$

(12)

• Objective function

The mirror studied in this study’s face shape accuracy RMS value under the action of gravity load comprehensively reflects the mirror’s stiffness and accuracy retention ability, and the quality affects its processing cost. In this study, the weighted combination of mass and RMS value defines the optimization variables, and the specific formula is:

$$F(x)=w_1{y}_1(x)+w_2{y}_2(x)$$

(13)

$$f(x)=F(x)$$

(14)

where $w_1$ and $w_2$ are the weight, here taken as $w_1=w_2=0.5$; $y_1(x)$ is the mass; and $y_2(x)$ is the RMS.

• Constraints

The constraints of the reflector optimization mathematical model are as follows:

$$\left\{\begin{array}{l}60 \mathrm{mm}\le x_1 \le 90 \mathrm{mm}\\ 1 \mathrm{mm}\le x_2 \le 10 \mathrm{mm}\\ 1 \mathrm{mm}\le x_3 \le 10 \mathrm{mm}\\ 5 \mathrm{mm}\le x_4 \le 15 \mathrm{mm}\end{array}\right.$$

(15)

In summary, the optimization problem in this study is to minimize the $F(x)$ of the reflector by adjusting the design variables while satisfying the relevant constraints. Therefore, the optimization mathematical model of the mirror is as follows:

$$\left\{\begin{array}{l}find x=(x_1,x_2,x_3,x_4)\\ \min f(x)=F(x) \\ s.t. 60 \mathrm{mm}\le x_1 \le 90 \mathrm{mm}\\ 1 \mathrm{mm}\le x_2 \le 10 \mathrm{mm}\\ 1 \mathrm{mm}\le x_3 \le 10 \mathrm{mm}\\ 5 \mathrm{mm}\le x_4 \le 15 \mathrm{mm}\end{array}\right.$$

(16)

4.2.2. Reflector Structure Optimization Results

The mathematical model of the control rod component optimization is solved using the $f\min con$ function, one of the optimization toolbox functions that come with MATLAB for solving multidimensional objective optimization problems with constraints. The function is called in the form:

$$x=f\min con=(fun,x_0,A,b,Aeq,beq,lb,ub,nonlcon,options)$$

(17)

where $x$ is the optimal solution of optimization; $fun$ is the objective function of minimization; $x_0$ is the initial value of optimization variables; $A$ and $b$ are the constraints of linear inequality, which satisfies $Ax\le b$; $Aeq$ and $beq$ are the constraints of linear equation, which satisfies $Aeq=beq$; $lb$ and $ub$ are the upper and lower bounds of variables; $nonlcon$ is the nonlinear constraints; and $options$ is the optimization options, including the number of iterations, the function value, and the algorithm selection.

To verify the effectiveness of the BR-TLDBO-BPNN model in practical application, this algorithm is compared with the Adam algorithm and RMSPROP algorithm which are commonly used in opto-mechanical structure optimization. According to the optimized results of BR-TLDBO-BPNN, the reflector is fabricated and assembled into the optical system, and the reflector is located in the optical axis horizontal state (working state); a 4D dynamic interferometer is used to measure the surface profile accuracy of the mirror, and the results are shown in Figure 13. The BR-TLDBO-BPNN model, Adam algorithm, RMSProp algorithm and experimental mirror parameters ${\mathrm{x}}_1{,\mathrm{x}}_{2,}{\mathrm{x}}_{3,}{\mathrm{x}}_4$, and corresponding Mass and RMS data are shown in

Table 3. Comparison results of optimization algorithms.

Algorithm	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	Mass	RMS
BR-TLDBO-BPNN	75 mm	2 mm	2 mm	6 mm	15.55 kg	16.41 nm
Adam	81 mm	3 mm	6 mm	7 mm	18.93 kg	19.17 nm
RMSProp	78 mm	4 mm	3 mm	8 mm	17.09 kg	20.38 nm
Experimental results	75 mm	2 mm	2 mm	6 mm	15.13 kg	15.19 nm

.

As shown in Figure 14, there are some errors between the three models and the real data measured by the experiment. Among them, the BR-TLDBO-BPNN model used in this paper is the closest to the experimental data after the optimization of the mirror, showing a low level of error. In contrast, the other two models have large deviations from the measured values, which shows that their prediction accuracy is not as good as the BR-TLDBO-BPNN model. Therefore, it can be concluded that the BR-TLDBO-BPNN model used in this paper has significant advantages in the mirror optimization problem, which provides an important reference for future opto-mechanical system design.

5. Conclusions

Aiming at the nonlinear characteristics between the input variables and the output response during the optimization and design process of the mirror structure and the high time cost of the finite element analysis, the BP neural network, BR algorithm, and the improved dung beetle algorithm are applied to establish the output characteristic prediction model (BR-TLDBO-BPNN) of the mirror, and the optimization process of the mirror structure is described in detail. The main conclusions of this study are as follows:

• The improved DBO algorithm is used to introduce the optimized initial weights and thresholds in BPNN, which effectively improves the prediction performance of BPNN.
• The BR-TLDBO-BPNN model incorporates the BR algorithm into the BPNN, which enables the BPNN to avoid overfitting and improves the generalization ability of the BPNN.
• In order to overcome the limitations of the original DBO algorithm, this study proposes the TLDBO algorithm. Logistic-tent chaotic mapping initializes the population and maintains its diversity, enhancing the algorithm’s optimization-seeking ability. Secondly, the algorithm’s local and global optimization ability is balanced by using the dynamic bipartite group convergence factor and adaptive inertia threshold. Again, the Levy flight mechanism is used to optimize the foraging behavior and improve the global search capability. Finally, four other meta-heuristic algorithms are used to compare with the TLDBO algorithm on six benchmark functions. The results show that TLDBO exhibits a good performance for both single-peak and multi-peak functions, verifying the effectiveness of the improved strategy and its competitiveness with other meta-heuristic algorithms.
• The BR-TLDBO-BPNN model can fit the nonlinear function between the input variables and the output response in the optimal design of mirrors with a prediction error of 99.9%, which provides a high prediction accuracy and reduces a large amount of finite element computation time.

The BR-TLDBO-BPNN model, i.e., Bayesian regularization with the improved dung beetle optimizer for back-propagation neural networks, proposed in this study, is intended for applications in the optimal design of reflector structures. The model exhibits a high prediction accuracy, effectively avoiding overfitting and saving time. The method is able to quickly and efficiently solve for the optimal structural design through predictive modeling, thus ensuring the superiority and functionality of the design prior to manufacturing. In addition, the method is able to adapt to different design requirements and conditions by adjusting the algorithmic parameters, which suggests that it is capable of solving more complex optimization problems in practical industrial applications, such as lens assembly, grating positioning, and the manufacturing design of other precision optical components. Given its excellent performance, the BR-TLDBO-BPNN model was expected to play a vital role in performing the multiphysics field analysis of complex optomechanical structures. In the future, this technique is expected to provide a generalized solution for optical design.