Optimizing process parameters for the production of intercalated melt-blown nonwoven materials for face masks based on machine learning algorithms

Abstract

Currently a new type of coronavirus is raging around the world, and many countries have relaxed the control of the epidemic. Wearing a mask has become the best self-protection measure for people to travel. Intercalated melt-blown nonwoven materials are in short supply as filter layers for daily-worn masks. This paper studies the relationship between the process parameters and structural variables of intercalated melt-blown nonwoven materials, and creatively uses machine learning-related algorithms to solve its nonlinear relationship. The optimized back propagation neural network model is the most suitable in this field, and the goodness of fit can reach more than 99.99%. Based on various limitations of actual industrial production, this model is used to traverse the process parameters, and the intercalated melt-blown nonwoven material is obtained. The best process parameters, in which the receiving distance is 27 cm, and the hot air velocity is 890 r/min, in this case, the thickness and porosity of the material produced are very low, while the compression resilience is very high, considering the filtration efficiency of the mask and comfort.

Keywords

Melt-blowing manufacture mask machine-learning factor analysis

Currently, novel coronavirus pneumonia (COVID-19) is ravaging the world, with more than 600 million cumulative confirmed cases of new coronavirus pneumonia and more than 6.67 million confirmed deaths worldwide as of 20 December 2022. Data show that the United States has the highest cumulative number of confirmed and fatal cases to date, with more than 100 million cumulative confirmed cases and more than one million cumulative deaths. Countries with more cumulative confirmed cases include India, Brazil, France, Germany, the United Kingdom, and Russia, and countries with more cumulative deaths include Brazil, India, Russia, Mexico, and Peru.¹ The December 2019 outbreak of novel coronavirus pneumonia began to concern people about the impact of the outbreak on society, the economy, and their daily lives while taking safety precautions. 2020 Novel coronavirus pneumonia The epidemic has caused widespread catastrophe in countries around the world, with the most severe overall outbreaks in countries with large populations, relatively widespread outbreaks in developed countries in Europe and the United States, and higher rates of illness and death in developing countries with larger populations.² In 2022 novel coronaviruses have undergone many mutations, from the early alpha and beta to the more recent familiar delta and omicron.³ Many countries have experienced outbreaks of neo-coronavirus to varying degrees due to the continuous mutation of the virus, but with the current level of human medicine, it is temporarily impossible to eradicate completely the virus that continues to mutate itself.

For novel coronaviruses, transmission by respiratory droplets and close contact are the main routes of transmission, as well as transaerosol transmission that exists in relatively closed environments with prolonged exposure to high aerosol concentrations.⁴ Therefore, wearing a mask has become a necessary option for self-protection in daily travel. Especially when living in countries where epidemic control is liberalized, wearing masks is the most direct protection measure.

Along with the sharp increase in the market demand for masks, the mask industry has become a hot topic in industrial production on how to optimize mask performance, process parameters, and production costs while ensuring the quality of masks. The masks that people carry every day are divided into three layers as shown in Figure 1, the outermost two layers are spun-bond cloth, the inner side is in contact with human skin, the outer side is in contact with air, the filtering role in the whole mask structure is small, only blocking the larger particles, and so on; the key filtering role is the middle insert layer part. A three-ply surgical mask may help block the transmission of large droplets, splashes or sprays, or other hazardous fluids.⁵

Figure 1.

Illustration of mouthpiece insertion layer.

The protective effect of masks is mainly characterized by using filtration efficiency,⁶ and comfort is mainly characterized by the indexes of air permeability and moisture permeability.⁷ For the filtration performance of masks, domestic and foreign scholars have studied the relationship between structural parameters of filter materials and filtration efficiency, such as fiber diameter,⁸ pore structure,⁹ pore shape,¹⁰ pore size,¹¹ and pore size distribution,¹² respectively. For the comfort performance of masks, Chen et al.¹³ conducted a study on the comfort of 10 representative masks on the market for breathability and moisture permeability, and found that the filtration performance of functional masks is generally better and the comfort is relatively poor, and the filtration effect of gauze masks is poor and the breathability and moisture permeability are better.

As early as the severe acute respiratory syndrome (SARS) outbreak in 2003, masks with melt-blown technology as the core filter material had taken the mainstream position among all masks because of their remarkable protective effect.¹⁴ Insert melt-blown nonwoven material is a special material obtained by melt-blown nonwoven technology. It is characterized by a simple process, high yield, low fiber content, large surface area, and dense pores,¹⁵ and can be used in modern electronic equipment, biological products, and other places with high air requirements, which has a wide market and is also gaining attention from manufacturers and customers worldwide.¹⁶ Melt-blown nonwoven technology is a method of direct polymer web formation, which was first developed by the United States in the 1950 s and was first applied in the military field. After more than two decades of development, this technology was transferred to civilian use, and melt-blown nonwoven technology gradually became one of the main production methods for direct polymer web formation.

However, melt-blown technology has difficulty solving the problem of poor performance of its fibers and products in terms of mechanics, compression resilience, permeability, and charge storage, which has largely affected the application of melt-blown nonwoven materials. As for the melt-blown jet airflow field as a production environment for melt-blown nonwoven materials, Shambaugh has been working on the melt-blown jet airflow field since as early as 1989.¹⁷ By observing the production process of melt-blown nonwoven materials, the stretching of melt-blown polymers was divided into three stages, and a model of the stretching of the relevant materials was established, containing a complete model of the set of hydrodynamic equations.^18,19 Meanwhile, Shambaugh and colleagues developed a two-dimensional polymer stretching model and a three-dimensional polymer stretching model to calculate the motion and variation of the polymer in the plane and space based on the actual measured gas velocity and temperature data.^20–22

The researchers studied the melt-blown gas flow field by theoretical analysis of the actual measured data, summarized the empirical expressions for the decay of the gas velocity, and analyzed the diffusion, vortex, and coiled suction phenomena generated during the experiments. The draft airflow of single-hole annular nozzles,^23,24 nozzle sets rectangularly arranged by annular nozzles,^25,26 and double-slotted nozzles was investigated,^27–30 respectively. The gas flow field formed by five different sizes of double-slotted nozzles was studied and the results showed that the smaller the angle of the nozzle airflow slot and the smaller the width of the head end, the greater the gas velocity. However, the processing of nozzles with small widths is difficult.

Chen and Huang^31,32 applied fluid dynamics software to study the gas velocity and gas temperature distribution in the gas flow field of double-slotted nozzles, numerical simulation of the gas field, by changing the parameters of the nozzle gas flow angle, slot width and head width. The authors concluded that melt-blown draft gas velocity decreases as the gas flow angle becomes larger, and the temperature decreases as the angle become larger; melt-blown draft gas velocity increases as the slot width increases, and the temperature decreases as the slot width increases. The melt-blown draft air velocity increases with increasing slot width and the temperature increases with increasing slot width; the melt-blown draft air velocity decreases with increasing head end width and the temperature decreases with increasing head end width.^33,34

Wu³⁵ and Ma³⁶ have simulated the airflow field of the intercalation melt-blown materials and found that the melt-blown airflow in the intercalation melt-blown jet field will be influenced by the intercalation airflow to shift in the direction of the advance of the intercalation airflow, and the angle of the shift is related to the airflow strength.

The study of the structural variables determined by the process parameters has become more complicated because of the large number of parameters in the preparation of melt-blown nonwoven materials and the interaction between them, as well as the complexity of the interlayer airflow. In this paper, the relationship between production process parameters and structural variables of intercalated melt-blown nonwoven materials is explored based on previous scientists’ studies of intercalated melt-blown nonwoven materials. Two types of representative process parameters (receiving distance and hot air velocity) and three types of representative structural variables (thickness, porosity, and compressional resilience) are selected, and samples produced under different conditions are prepared by changing the process parameters to analyze the changes in product structure with different process parameters. At the same time, this paper innovatively introduces three models in the field of machine learning, the back propagation (BP) neural network model, random forest model and support vector machine (SVM) model, to model the relationship between process parameters and structural variables. The values of the process parameters are explored through the traversal method for the best performance of the structural variables under the constraints of the actual conditions, which provides a certain theoretical basis for the establishment of the product performance regulation mechanism. The paper uses the method of factor analysis to extract the common factors of the structural variables, which avoids the repeated calculation of the same part in different structural parameters and improves the accuracy of the research. Meanwhile, it saves the experiment from changing the material parameters many times and obtains the process parameters corresponding to the optimal structure of the melt-blown nonwoven material.

Data introduction

The data used in this paper are all experimental, using the controlled variable method, with the receiving distance and hot air velocity representing the process parameter variables, and the thickness, porosity, and compressional resilience representing the structural variables. The receiving distance refers to the distance from the point of solution injection to the position of receiving the solution injected during the experiment, and the unit is cm; the hot air speed refers to the speed of solution injection in air, and the unit is r/min; the thickness is the thickness of the melt-blown nonwoven material, and the unit is mm; the porosity is the ratio of voids to the volume of the material in the melt-blown nonwoven material. The unit in the data used is %; compression resilience is the resilience against compression, the greater the elasticity means the stronger the ability to resist compression, the unit is %. In the experiments, the receiving distance was taken as 20 cm, 25 cm, 30 cm, 35 cm, and 40 cm, and the hot air speed was taken as 800 r/min, 900 r/min, 1000 r/min, 1100 r/min, 1200 r/min (as these parameters are always used in actual industrial production), and the same process parameters were experimented three times to avoid the occurrence of chance conditions. Based on the analysis of the experimental data and the machine learning model, the process parameters of the intercalated melt-blown nonwoven material were optimized. Table 1 shows the descriptive statistics of the variables involved in the experiment.

Table 1.

The descriptive statistics of the variables involved in the experiment

Characteristics	Description	Mean/N	SD
Process parameter variables
Receiving distance	The distance from the solution injection point to the location receiving the sprayed solution during the experiment (cm)	30.00000	7.07107
Hot air velocity	The speed at which a solution is sprayed in air (r/min)	1000.00000	141.42126
Structural variables
Thickness	Thickness of melt-blown nonwovens (mm)	2.60707	0.46908
Porosity	The ratio of the voids in the melt-blown nonwoven material to the volume of the material (%)	95.87915	0.81474
Compressional resilience	Resilience against compression. The greater the elasticity, the stronger the compressive capacity (%)	86.60788	1.20857

Methodology

Factor analysis-based method to eliminate the correlation between structural variables

In industrial production, the magnitude of the process parameters determines the magnitude of the structural variables of the melt-blown fabric of the mask, so the magnitude of the structural variables can be predicted by the process parameters. From the preliminary analysis,³⁷ there is a certain linear correlation between the three variables representing the structure. Therefore, this paper first eliminates the linear correlation between the structural variables to improve the model prediction accuracy.

In this paper, factor analysis is used to eliminate the linear correlation of the structural variables. The factor analysis method can simplify the huge amount of high-dimensional data into multiple factors independent of each other using correlation matrix eigenvalue calculation, while the loss of original information is within the controllable range.³⁸ The indexes after factor analysis not only reflect the original information better but also simplify the data structure and eliminate the multicollinearity among variables.

$Step 1$ : Preliminary statistical analysis of sample data

In the preliminary statistical analysis of sample data, it is first necessary to select $n$ samples to be tested, and at the same time select $p$ indicator variables for these samples, thus forming an $n \times p$ -dimensional sample space. The group number and structural variables thus constitute the sample data, thereby constructing the sample space X required for the model. In the space $X$ , $x_{i j}$ represents the $j$ index value in the $i$ sample.

X = (\begin{matrix} x_{11} x_{12} \dots x_{1 j} \\ x_{21} x_{22} \dots x_{2 j} \\ ⋮ ⋮ \dots \dots \dots ⋮ ⋮ \\ x_{i 1} x_{i 2} \dots x_{i j} \end{matrix})

(1)

\frac{x_{i j} - E (x_{j})}{\sqrt{D (x_{j})}}

(2)

$Step 2$ : Determine whether the data set meets the prerequisites for factor analysis

The $Bartlett$ sphericity test starts from the correlation coefficient matrix corresponding to the original data set, and its null hypothesis $H_{0}$ is: the correlation coefficient matrix is a unit matrix. Second, the $KMO$ test statistic is an index used to compare the simple correlation coefficient and partial correlation coefficient between variables, which can be obtained by calculation:

KMO = \frac{\sum_{i} \sum_{i \neq j} r_{i j}^{2}}{\sum_{i} \sum_{i \neq j} r_{i j}^{2} + \sum_{i} \sum_{i \neq j} p_{i j}^{2}} = 0.886

(3)

Among them, $r_{i j}$ is the correlation coefficient between variable $x_{i}$ and variable $x_{j}$ , and $p_{i j}$ is the partial correlation coefficient between variable $x_{i}$ and variable $x_{j}$ after controlling the remaining variables.

According to Table 1 in the case of the significance level $α = 0.05$ , the probability P value corresponding to the $Bartlett$ sphericity test is less than 0.05 at this time, so the null hypothesis is rejected, and there is a significant difference between the correlation coefficient matrix and the unit matrix. Second, at this time, the value of $KMO$ is 0.886. According to the $KMO$ metric standard given by $Kaiser$ , it can be known that the original variables are suitable for factor analysis.

$Step 3$ : Factor extraction

To analyze the number of factors that need to be extracted, the cumulative contribution rate of the factor should be calculated. As the original three variables have been standardized to eliminate the influence of dimensions, the formula for the cumulative contribution rate of the factor can be obtained as:

Φ = \frac{\sum_{j = 1}^{2} λ_{j}}{\sum_{i = 1}^{3} λ_{i}} = 0.96 \geq 0.85

(4)

From Table 2, it can be reflected that the factors explain the total variance of the original variables.

Table 2.

$KMO$ and $Bartlett$ ’s test

$Kaiser - Meyer - Olkin$ measure of sampling adequacy		0.886
$Bartlett$ 's test for sphericity	Approximate chi-square	39.863
	$df$	3
	$sig$	0.000

According to Table 2, two factors are extracted in the initial solution, and the cumulative variance contribution rate of these two factors has reached $96.5 %$ , which means that $96.5 %$ of the information in the original variables can be represented by these two factors. In general, the information loss of the original variables is relatively small, and the effect of factor analysis is ideal.

$Step 4$ : Rotate the factor loading matrix and calculate factor scores

The two factors are rotated orthogonally by the varimax method to ensure the nomenclature of the factors. In this paper, the regression method is used to estimate the factor score coefficient and output the factor score coefficient, namely:

According to Table 3 thickness and porosity account for a large proportion of factor $F_{1}$ , it can be considered that the information on the factor $F_{1}$ is mainly provided by thickness and porosity, so factor $F_{1}$ is named as the product setting attribute. Factor $F_{2}$ accounts for a relatively large proportion, so it is considered that the information of factor $F_{2}$ is mainly provided by the compression resilience rate, so the factor $F_{2}$ is named the compression resilience rate. According to the component coefficients, the following factor score functions are obtained:

F_{1} = 0.434 y_{1} + 0.433 y_{2} + 0.294 y_{3},

F_{2} = - 0.352 y_{1} - 0.364 y_{2} + 1.054 y_{3}

(5)

Table 3.

Total variance explained

Composition	Initial characteristic value			Extract square sum load
Composition	Total	Variance %	Accumulate %	Total	Variance %	Accumulate %
Factor 1	2.164	72.120	72.120	2.164	72.120	72.120
Factor 2	0.732	24.394	96.514	0.732	24.394	96.514
Factor 3	0.105	3.486	100.000

Among them, $y_{i} (i = 1, 2, 3)$ represent the thickness, porosity, and compression resilience, respectively. Therefore, the use of factors $F_{1}$ and $F_{2}$ can represent the predicted filtration efficiency of the product.

Optimization of production parameters of intercalated melt-blown nonwovens based on BP neural network model

As the process parameters (receiving distance and hot air velocity) directly impact the structural variables (thickness, porosity, compression resilience), the permutation and combination of different process parameters can effectively predict the corresponding melt-blown nonwoven structure variable. For example, as far as the thickness of the sample is concerned, it will increase with the increase of the receiving distance. The research results show that with the increase in the receiving distance, the melt-blown fiber can be fully stretched and cooled for a long time. In other words, when the conditions remain the same, the number of fibers increases, and at the same time, the temperature when reaching the net curtain is relatively low, so the obtained product structure is fluffier, and the thickness is relatively increased. Due to the interactive influence between different process parameters, to make the relationship between process parameters and structural variables easier to observe, Matlab is used to visualize the relationship between different process parameters and thickness, porosity, and compression resilience. The nonlinear relationships are shown in Figure 2, Figure 3, and Figure 4.

Figure 2.

The relationship between process parameters and thickness.

Figure 3.

The relationship between process parameters and porosity.

Figure 4.

The relationship between process parameters and compression resilience rate.

It can be seen from the above relationship diagrams that there is a nonlinear relationship between process parameters and structural variables. In the face of more complex nonlinear problems, the BP neural network model is an excellent solution method. Before establishing the neural network model, it is necessary to define the activation function, define the $Loss$ function, and select the optimization process. Therefore, this article will analyze and answer the above three steps according to the nature of the topic data.

$Step 1$ : Define the activation function

From Figure 1, Figure 2, and Figure 3, it can be observed that there is a special nonlinear relationship between different process parameters and thickness, porosity, and compression resilience. If the basic functions are directly used for fitting, a large model error will often be generated. Therefore, to fit these curves more accurately, this paper introduces a soft Sigmoid function that is smoother than the $ReLu$ function³⁹ to represent them, that is:

y = b + \sum_{i} c_{i} S igmoid (b_{i} + \sum_{j} w_{i j} x_{j})

(6)

Among them, $y$ represents the structural variables (thickness, porosity, compression resilience), $x_{i}$ represents the receiving distance of process parameters and hot air velocity, $b$ represents the offset of the function, $c_{i}, b_{i}, w_{i j}$ represent the height of the $Sigmoid$ function, respectively, intercept, and slope values.

$Step 2$ : Define the Loss function

The Loss function can evaluate the quality of a model, that is, it can effectively evaluate the quality of hyperparameters and parameters, so an excellent model often has a loss function that can reach the global minimum. But in fact, the definition of the Loss function often has a greater impact on the optimization process, so this paper chooses the root mean square error as the Loss function, and its expression is:

RMSE = \sqrt{\frac{\sum_{i = 1}^{n} (y_{i} - \hat{y_{i}})^{2}}{n}}

(7)

Because the root mean square error (RMSE) is particularly sensitive to items with large deviations between the true value $\hat{y_{i}}$ and the predicted value $y_{i}$ , choosing this standard can more effectively reflect the degree of the fitting.

$Step 3$ : Choose an optimization process

By choosing an appropriate optimization process, the parameters can be iterated continuously to satisfy the minimum loss function. In machine learning, the gradient descent method is often used as the optimization process, but the gradient descent method often cannot effectively and quickly update the loss function without local minimum. Therefore, this paper optimizes the update direction and learning rate of the gradient descent method, uses the $Adam$ optimization process to update the parameters, and then minimizes the loss function, namely:

w_{i j}^{*}, b_{i}^{*}, c_{i}^{*} = \arg \min_{w, b, c} L (y, \hat{y})

(8)

$Adam$ ^40–42 is a first-order optimization algorithm that can replace the traditional stochastic gradient descent process. It can iteratively update the weight of the neural network based on the training data. It is mainly composed of $RMSProp + Momentum$ , and its specific update path is:

θ_{i}^{t + 1} \overset{update}{\leftarrow} θ_{i}^{t} - \frac{η}{σ_{i}^{t}} m_{i}^{t}

(9)

The specific process of $Momentum$ is as follows:

\{\begin{array}{l} m^{t + 1} = λ m^{t} - η \frac{\partial L (y, \hat{y})}{\partial θ}, \\ m^{0} = 0 \end{array}

(10)

where

λ, η

are hyperparameters, and the direction of the vector

m

represents the direction in which the parameters are iterated.

As shown in Figure 5, $Momentum$ fuses all the past gradients of the parameters jointly to determine the update direction. This method can better avoid local optimal solutions than traditional stochastic gradient descent.

Figure 5.

$Momentum$ iterative logic diagram.

$RMSProp$ changes the learning rate by combining the size and influence of the gradient. Often the nearest gradient has a greater influence, and the greater the combined gradient, the learning rate will decrease accordingly. Ensure the accuracy of gradient descent. The specific process is as follows:

θ_{i}^{t + 1} \overset{update}{\leftarrow} θ_{i}^{t} - \frac{η}{σ_{i}^{t}} m_{i}^{t}

\{\begin{array}{l} σ_{i}^{0} = \sqrt{(g_{i}^{0})^{2}}, \\ σ_{i}^{t} = \sqrt{α (σ_{i}^{t - 1})^{2} + (1 - α) (g_{t})^{2}} . \end{array}

(11)

Therefore, based on the above three steps, a BP neural network model based on $Adam$ optimization can be established. The basic principle is to use momentum and adaptive learning rate to speed up the convergence speed, thereby optimizing the efficiency of the neural network.⁴³ Its specific structure is as follows:

As shown in Figure 6, the $BP$ neural network consists of three parts, namely the input layer, hidden layer, and output layer. The input layer inputs the receiving distance and the speed of hot air, and the $Sigmoid$ neurons in the hidden layer can fit the regression curve of thickness, porosity, and compression resilience with high quality, and output the thickness, porosity, and compression resilience through the output layer. Among them, the BP neural network continuously iterates the parameter values through the back propagation algorithm, thereby making the predicted value more accurate.

Figure 6.

$BP$ neural network model.

Through the above analysis, the expression of the model can be obtained as:

w_{i j}^{*}, b_{i}^{*}, c_{i}^{*} = \arg \min_{w, b, c} L (y, \hat{y}) \Leftarrow θ_{i}^{t + 1} \overset{update}{\leftarrow} θ_{i}^{t} - \frac{η}{σ_{i}^{t}} m_{i}^{t}

\{\begin{array}{l} y = b + \sum_{i} c_{i} S igmoid (b_{i} + \sum_{j} w_{i j} x_{j}), \\ L (y, \hat{y}) = RMSE, \\ m^{t + 1} = λ m^{t} - η \frac{\partial L (y, \hat{y})}{\partial θ}, \\ σ_{i}^{t} = \sqrt{α (σ_{i}^{t - 1})^{2} + (1 - α) (g_{t})^{2}}, \\ σ_{i}^{0} = \sqrt{(g_{i}^{0})^{2}}; m^{0} = 0, \\ i = 1, 2, 3; t = 1, 2, \dots, n . \end{array}

(12)

Optimization of production parameters of intercalated melt-blown nonwovens based on random forest algorithm

The random forest algorithm is an effective regression prediction method, which has high regression accuracy, can effectively prevent the regression accuracy from being too low in the case of noise and outliers and has a strong generalization ability.

A single decision tree can perform regression based on a certain accuracy, but it is difficult to process accurately the regression with only a single decision tree. To optimize the decision tree model, the easiest way to operate is to build multiple trees to form a forest. The classification accuracy can be significantly improved, and each tree in the forest participates in decision-making to determine the broadest category. In the random forest, successive trees do not rely on earlier trees; rather, they independently use a bootstrap sample of the data set.⁴⁴

The random forest algorithm is based on the decision tree ${g (D, O_{n}), n = 1, 2, \dots, N}$ on the data set $D = \{X_{i}, Y_{i}\}, X_{i} \in R^{k}, Y_{i} \in {1, 2, \dots, M}$ , and the random forest contains N decision trees. The regression results of each decision tree are voted to determine the regression results obtained by the random forest. The specific implementation process is as follows:

$Step 1$ : Selecting samples to train a decision tree

If there are $N$ samples, $N$ samples are randomly selected with replacement (one sample is randomly selected each time, and then returned to continue selection). The selected $N$ samples are used to train a decision tree as samples at the root node of the decision tree. Its specific diagram is as follows (Figure 7):

Figure 7.

Schematic diagram of a random selection of random forest data.

$Step 2$ : Determine split properties

When each sample has $M$ attributes when each node of the decision tree needs to be split, randomly select $m$ attributes from the $M$ attributes, satisfying the condition $m < < M$ . Then select an attribute from the $m$ attributes by measuring the size of the $Gini$ index as the splitting attribute of the node. The $Gini$ index represents the uncertainty of the set $D$ , and the larger the value of the Gini index, the greater the uncertainty of the sample. Under the condition of feature $A$ , the formula of the $Gini$ index of sample $D$ is as follows:

Gini (D, A) = \frac{|D_{1}|}{|D|} Gini (D_{1}) + \frac{|D_{2}|}{|D|} Gini (D_{2})

(13)

$Step 3$ : Form a decision tree

During the formation of the decision tree, each node must be split according to step 2 (if the attribute selected by the node next time is the attribute used when the parent node splits, the node has reached the leaf node and there is no need to continue splitting). Until it can no longer be split, there is no pruning during the entire decision tree formation process.

$Step 4$ ：Build many decision trees according to steps 1–3, to form a random forest

The training samples of each tree in the random forest are random, and the regression attributes of each node in the tree are also randomly selected. These two random selection processes ensure that the random forest will not produce overfitting. The diagram of the random forest is as follows:

Optimizing parameters of intercalated melt-blown nonwoven materials based on SVM

To study the nonlinear relationship between the process parameters and the structural variables of intercalated melt-blown nonwoven materials, the classic model for solving the nonlinear relationship is the SVM model. The SVM was formally proposed by Cortes and Vapnik⁴⁵ in 1995. This technology is based on the idea of statistical learning. Compared with the previous traditional pattern recognition method using the principle of experience risk minimization, it uses the structure of the model The risk is minimized to enhance the generalization learning ability of the learning machine so that the model can achieve a balance between the complexity and the handling of nonlinear problems.⁴⁶ SVMs use kernels to operate directly on the input space. This is the characteristic ‘crossness’ of SVMs. They help developers to act as though they are using a basic linear algorithm while using complex algorithms like pattern recognition, regression, or feature extraction. Gaussian kernel, linear kernel, and polynomial kernel methods were used as learning algorithms for the SVM models. Iterative single-data algorithms, quadratic programming, and sequential minimal optimization methods were used as solvers.⁴⁷ Before establishing the SVM model, it is necessary to define the hyperplane equation and classification decision function, formalize the learning strategy, and obtain the dual problem. Therefore, this paper will solve the following three steps according to the nature of the data.

$Step 1$ : Define hyperplane equation and decision function

The learning goal of the support vector regressor is to find a hyperplane in the n-dimensional data space, so this paper defines the hyperplane equation as:

ω^{T} x + b = 0

(14)

Separating hyperplane:

ω^{*} x + b^{*} = 0

(15)

Decision function:

f （x） = sign （ω^{*} \cdot x + b^{*}） \Rightarrow \{\begin{array}{l} H_{1} : ω_{0} + ω_{1} * x_{1} + ω_{2} * x_{2} \geq 1 for y_{i} = + 1 \\ H_{2} : ω_{0} + ω_{1} * x_{1} + ω_{2} * x_{2} \leq 1 for y_{i} = - 1 \end{array}

\Rightarrow y_{i} (ω_{0} + ω_{1} * x_{1} + ω_{2} * x_{2}) \geq 1

(16)

As shown in Figure 8, what the regressor is looking for is the partition hyperplane with the largest interval, so this paper defines the interval as $γ$ , so the learning objective of the support vector regressor can be expressed as:

\max ‖\frac{2}{ω}‖ s . t . y_{i} (ω^{T} \cdot x_{i} + b) \geq 1, i = 1, 2, \dots, m

(17)

Figure 8.

Random forest model illustration.

$Step 2$ : Formalize learning strategies

The learning strategy of SVM is interval maximization, so this paper formalizes this learning strategy as a convex quadratic programming. As maximizing ${‖ω‖}^{- 1}$ is equivalent to minimizing ${‖ω‖}^{2}$ ; therefore, we can re-express the learning objective of the support vector regression machine as:

\max \frac{1}{2} {‖ω‖}^{2} s . t . y_{i} (ω^{T} \cdot x_{i} + b) \geq 1, i = 1, 2, \dots, m

(18)

$Step 3$ : Get the dual problem

As the original problem and the dual problem need to be found to solve the linear programming, this paper uses the Lagrange multiplier method to obtain the dual problem, in which the Lagrange function can be written as:

L （(ω, b, α)） = \frac{1}{2} {‖ω‖}^{2} + \sum_{i = 1}^{m} α_{i} (1 - y_{i} (ω^{T} x_{i} + b))

(19)

where

α_{i}

is the Lagrangian multiplier. In this paper, the dual problem obtained by using the Lagrange multiplier method is expressed as:

\max \sum_{i = 1}^{m} α_{i} - \frac{1}{2} \sum_{i = 1}^{m} \sum_{j = 1}^{m} α_{i} α_{j} y_{i} y_{j} x_{i}^{T} x_{j}

(20)

s . t . \sum_{i = 1}^{m} α_{i} y_{i} = 0, α_{i} \geq 0, i = 1, 2, \dots, m

After obtaining the original problem and the dual problem, this paper uses the $SMO$ algorithm for the original problem continuously to decompose the original quadratic programming problem into a quadratic programming problem with only two variables, and analyze and solve the quadratic programming problem until all variables. Satisfy the conditions of $Karush - Kuhn - Tucker$ . That is:

\{\begin{array}{l} α_{i} \geq 0 \\ y_{i} * f (x_{i}) - 1 \geq 0 \\ α_{i} (y_{i} * f (x_{i}) - 1) = 0 \end{array}

(21)

Through the above analysis, the expression of the model can be obtained as:

\{\begin{array}{l} ω^{T} x + b = 0 \\ m a x \frac{1}{2} {‖ω‖}^{2} s . t . y_{i} (ω^{T} \cdot x_{i} + b) \geq 1, i = 1, 2, \dots, m \\ L (ω, b, α) = \frac{1}{2} {‖ω‖}^{2} + \sum_{i = 1}^{m} α_{i} (1 - y_{i} (ω^{T} x_{i} + b)) \\ α_{i} \geq 0 \\ y_{i} * f (x_{i}) - 1 \geq 0 \\ α_{i} (y_{i} * f (x_{i}) - 1) = 0 \end{array}

(22)

From the above, we can construct the SVM model.

Results and discussion

Model solving

BP neural network model

In this paper, $Matlab$ is used to solve the BP neural network model. First, the number of hidden neurons is defined as 10. At the same time, to avoid the problem of overfitting, we divide the proportions of the training data set, test data set and inspection data set into 70%, 15%, and 15%. After normalizing the training sample data, the network is built, and the $sigmoid$ activation function is added to the neural network to make the neural network contain nonlinear factors. The $RMSE$ index is used as the loss function to determine the quality of the neural network fitting. The $ADAM$ gradient descent method is used to optimize the neural network and continuously train it. After training 10 times, the optimal neural network fitting effect is obtained. After exporting the model function, use the $sim$ function to predict the thickness, porosity, and compressive resilience rate in combination with the known receiving distance and hot wind velocity given in the question.As the data were normalized in the early stage to avoid the influence of dimensions on the fitting effect, the prediction results were finally denormalized to obtain the thickness, porosity, and compression resilience ratio required in the question. To facilitate the measurement of the degree of fitting, the $plot$ function is also set to visualize the training process and the error histogram, etc. To update the parameters more effectively, this paper adopts the mini-batch gradient descent method. The training data are divided into multiple batches, and the data in each $Batch$ are less. First, set the number of training iterations of all $Batch$ in this training forward propagation and back propagation process to 10 times, that is, $epochs = 10$ (see Figure 10).

It can be seen from Figure 9 that the gradient is very close to 0 in the seventh training, and the gradient has hardly changed after that, so it can be considered that the seventh training has been able to minimize the loss function. At the same time, it is observed that from the fifth training, the training error has no longer decreased, so there is no need to continue training.

Figure 9.

Schematic diagram of support vector machine (SVM).

Figure 10.

The gradient training process of the BP neural network model.

In summary, to obtain a smaller loss function and prevent over-learning, this article will design the training times to be four times. By controlling the over-fitting phenomenon, the final training effect is guaranteed to the greatest extent.²⁷

To achieve the minimum error, and the data at this time are representative to a certain extent, which can effectively provide the structural information of the data, to use the accurate structural information. The values of thickness, porosity, and compression resilience are predicted.

Finally, to make the accuracy of the test set higher, by continuously adjusting, iterating parameters and hyperparameters, and comparing them, the parameters with the highest accuracy of the test set can be obtained, thereby ensuring the accuracy of the calculation results.

As shown in Figure 11, at this time the goodness of fit of the model is close to 1, indicating that the model fitting effect is almost perfect. Therefore, using the neural network model, changing the values of the receiving distance and hot air velocity, the thickness, porosity, and compression resilience is predicted.

Figure 11.

The best verification performance and error histogram of the BP neural network model.

Random forest model

To avoid the problem of over-fitting, we divided the proportions of the training data set, test data set, and inspection data set into 70%, 15%, and 15%, respectively. After normalizing the training sample data, the random forest model was built, and the five-fold cross-validation method was used to improve the accuracy of model training. First, the number of training iterations is set to 30, and the random forest model iteration diagram about the minimum MSE is obtained.

As shown in Figure 12, when the iteration reaches the fifth time, the MSE of the training model reaches the minimum value and does not decrease significantly in the subsequent training. Therefore, to avoid the phenomenon of overfitting caused by too many training times, we set the number of training iterations to five. After the model is solved, to avoid the inaccurate evaluation of the model caused by the single MSE index evaluation. This article also uses R², mean absolute error (MAE), RMSE, and mean absolute percentage error (MAPE) to evaluate the pros and cons of the model. The calculation formula is as follows:

R^{2} = \frac{\sum_{i = 1}^{n} ({\hat{y}}_{i} - \bar{y})^{2}}{\sum_{i = 1}^{n} (y_{i} - \bar{y})^{2}}

(23)

MAE = \frac{1}{n} \sum_{i = 1}^{n} |{\hat{y}}_{i} - y_{i}|

(24)

RMSE = \sqrt{\frac{\sum_{i = 1}^{n} (y_{i} - \hat{y_{i}})^{2}}{n}} .

(25)

MAPE = \frac{1}{n} \sum_{i = 1}^{n} |\frac{{\hat{y}}_{i} - y_{i}}{y_{i}}| \times 100 %

(26)

and get the following Table 5.

Figure 12.

The goodness of fit of the dataset by the BP neural network model.

For the error per sample, there are the following images for factor 1 and factor 2, respectively.

From the data in Table 4 and the information in Figure 13, it can be concluded that the random forest model is extremely accurate in predicting the structural variables of the material according to the production process parameters of the intercalated melt-blown nonwoven material and can reach 98.8% for the two common factors The above goodness of fit, and MAE, RMSE, and MAPE have all reached very accurate standards. In the analysis of absolute percentage error, although the error has certain fluctuations for different data, the error of most data samples is maintained. Below 0.1%, the highest error is only about 1%, which can be applied to industrial production.

Table 4.

Component score coefficient matrix

Structure variable	Component
Structure variable	Factor 1	Factor 2
Thickness	0.434	−0.352
Porosity	0.433	−0.364
Compression rebound rate	0.294	1.054

Table 5.

Random forest model evaluation

Evaluation Index	Factor 1	Factor 2
$R^{2}$	0.9948427422	0.9898978655
MAE	0.0148246886	0.0351003513
RMSE	0.1283855699	0.3039779596
MAPE	0.0001094651	0.0002390107

Figure 13.

MSE graph of the random forest model.

SVM model

In this paper, $matlab$ is used to solve the SVM model, the number of iterations is set to 30, and the calculation of MSE in the iterative training process of the SVM model is performed by $matlab$ , and the error diagram is as follows (Figure 15).

It can be seen from Figure 14 that the MSE value of the SVM model is the smallest in the sixth iteration training, and the MSE of the model has reached below 0.05. In the seventh and subsequent training, the MSE of the model has no obvious downward trend and remains at around 0.03. Therefore, to avoid the phenomenon of overfitting caused by too many training times, we set the number of training iterations to 25 times, and multiple indicators are used to evaluate the prediction accuracy of the SVM model. The model prediction accuracy is shown in Table 6 below.

Figure 14.

Random forest model error plot.

Table 6.

Support vector machine model evaluation

Evaluation Index	Factor 1	Factor 2
$R^{2}$	0.7921	0.96
MAE	0.1882	0.2370
RMSE	0.2351	0.2883
MAPE	0.0001971	0.0014658

Figure 15.

MSE graph of the support vector machine (SVM) model.

Table 7.

Evaluation of three machine learning models based on different metrics

Model	$R^{2}$	MAE	RMSE	MAPE
BP neural network	0.9999	0.1991	0.2561	0.0000049
Random forest	0.9924	0.0249	0.2162	0.0001741
Support vector machines	0.8761	0.2126	0.2617	0.0008323

Result analysis

In this paper, we investigate the nonlinear relationship between material process parameters and material structure variables by using melt-blown nonwoven materials for mask inserts as the research object. To avoid repeated calculations of the common factors of the three material structure variables, factor analysis was used to obtain two common factors that could represent 96.5% of the original information: $F_{1} = 0.434 y_{1} + 0.433 y_{2} + 0.294 y_{3}$ and $F_{2} = - 0.352 y_{1} - 0.364 y_{2} + 1.054 y_{3}$ , which were derived according to the weight of each of their indicators The two common factors represent product setting properties (thickness and porosity) and compression resilience, respectively.

Prediction of product structure variables

In this paper, three classical algorithms of machine learning are used: the BP neural network algorithm based on Adam optimization, the random forest algorithm, and the SVM algorithm for solving the problem, and all three machine learning algorithms have obtained good prediction results. Their goodness-of-fit ( $R^{2}$ ) is above 0.79, MAE is below 0.3, RMSE is below 0.4, and MAPE is below 0.0003. The evaluation of the three machine learning algorithms based on different metrics is shown in the figures below.

Analyzing Figures 16, 17, 18, and 19, and Table 5, it can be seen that the BP neural network model optimized based on Adam's algorithm performs the best in the context of melt-blown nonwoven material production with the most accurate prediction results, and its goodness-of-fit is above 0.9999, which meets the requirements of industrial production for model accuracy, after integrating the evaluation criteria of different models.

Figure 16.

Support vector machine (SVM) model error plot.

Figure 17.

Response diagram of the Adam optimized BP neural network model.

Figure 18.

Response diagram of the random forest model.

Figure 19.

Response diagram of the support vector machine (SVM) model.

Figure 20.

Absolute percentage error of three models.

Practical significance

In actual industrial production, due to the dual consideration of mask comfort and filtration efficiency, the thickness and porosity of intercalated melt-blown nonwoven materials are reduced and the compression resilience is increased. However, in actual production, due to the calculation of the production cost and the consideration of the actual performance of the machine, there are certain requirements for the receiving distance and the speed of hot air. According to the data,^16,35,36 the receiving distance should be controlled below 100 cm, the speed of hot air should be controlled below 2000 r/min, according to the requirements of the application, the thickness should not exceed 3 mm, and the compression resilience rate should not be lower than 85%. Therefore, this paper uses the model most suitable for the production scene of intercalated melt-blown nonwoven materials after analysis: the BP neural network model based on Adam optimization, for the receiving distance from 0 cm to 100 cm, the hot air speed from 0 r/min to 2000 r/min, traverse in 0.1 units to find out the situation where $F_{1}$ is the smallest and $F_{2}$ is the largest.

As we need to find the condition that the minimum value of $F_{1}$ and the maximum value of $F_{2}$ are satisfied at the same time in this paper, the objective function is re-established as:

F = - F_{1} + F_{2}

(27)

Therefore, the condition of the maximum value of equation (27) is the condition of the process parameters searched for in this paper. For the metric factors $F_{1}$ and $F_{2}$ the trend is shown below in Figures 21 and 22.

Figure 21.

Change of factor 1.

Figure 22.

Change of factor 2.

It was calculated that the maximum $F$ case was obtained when the receiving distance was 27 cm and the hot air speed was 890 r/min, and the maximum $F$ value was –10.589.

Conclusions

In this paper, three models in the field of machine learning are used to explore the nonlinear relationship between material process parameters (hot air speed and receiving distance) and material structural variables (thickness, porosity, and compressive resilience), taking melt-blown nonwoven materials with mask inserts as the research object. Meanwhile, to avoid the double calculation of the common factors of the three material structure variables, factor analysis was used to extract two common factors of the structure variables, whose contribution to the original information could reach 96.5%.

After the prediction of the Adam optimized BP neural network model, the random forest model and the SVM model, and the evaluation of the model with multiple indicators, we found that the Adam optimized BP neural network model is the most applicable in the production scenario of intercalated melt-blown nonwoven materials, and its goodness of fit can reach more than 99.99%, and the calculation of MAE, RMSE, and MAPE. Therefore, this model can be used in actual production. Meanwhile, this paper uses this model to traverse the process parameters based on the limitations of the process parameters in the actual industrial production and obtains the receiving distance and hot air speed corresponding to the best material produced in the actual situation, when the receiving distance is 27 cm and the hot air speed is 890 r/min, and the thickness and porosity of the material produced in this situation are very low, and the compression resilience is very high. In this case, the material thickness and porosity are very low, and the compression resilience is very high, considering the filtration efficiency and comfort of the mask.

Research contribution

In this paper, the machine learning algorithm is applied to the production of melt-blown nonwoven materials, and the problem of optimizing industrial production with statistical methods is innovative, which provides a new idea for the optimization of industrial production in the future.

The method of factor analysis is used to extract the common factors, which avoids the repeated calculation of the same part in different structural parameters and improves the accuracy of the research.

In this paper, after obtaining a model with a goodness of fit above 0.9999, the process parameters corresponding to the optimal structure of the melt-blown nonwoven material are obtained by using the traversal method with a unit of 0.1, which saves the experiment from changing the material parameters many times and obtains the best excellent solution.

Research limitations

Using machine learning algorithms to optimize the process parameters of intercalated nonwoven materials has poor interpretability, which is caused by the principle of machine learning algorithms.

Factors such as production cost should also be considered in actual production. Applying this theory to industrial production can have a better performance in product quality, but further exploration is needed in terms of production cost.

Suggestions for further research

Future research can bring into production cost factors and add more process parameters to fit further the research with actual production. At the same time, more attention should be paid to the interpretability of theory and in-depth analysis of the factors affecting various structural variables to obtain more optimized process production.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Zeyan Li

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