Data Augmentation and Deep Learning Methods for Pressure Prediction in Ignition Process of Solid Rocket Motors

Abstract

During the ignition process of a solid rocket motor, the pressure changes dramatically and the ignition process is very complex as it includes multiple reactions. Successful completion of the ignition process is essential for the proper operation of solid rocket motors. However, the measurement of pressure becomes extremely challenging due to several issues such as the enormity and high cost of conducting tests on solid rocket motors. Therefore, it needs to be investigated using numerical calculations and other methods. Currently, the fundamental theories concerning the ignition process have not been fully developed. In addition, numerical simulations require significant simplifications. To address these issues, this study proposes a solid rocket motor pressure prediction method based on bidirectional long short-term memory (CBiLSTM) combined with adaptive Gaussian noise (AGN). The method utilizes experimental pressure data and simulated pressure data as inputs for co-training to predict pressure data under new operating conditions. By comparison, the AGN-CBiLSTM method has a higher prediction accuracy with a percentage error of 3.27% between the predicted and actual data. This method provides an effective way to evaluate the performance of solid rocket motors and has a wide range of applications in the aerospace field.

1. Introduction

Solid rocket motors are frequently employed in domains such as missiles. The propellant within solid rocket motors is directly stored in the combustion chamber and can be preserved for an extended period, typically ranging from 5 to 20 years. Solid rocket motors possess a simplistic structure, are facile to utilize, and entail reduced maintenance requirements [1]. In the future, solid rocket motors will continue to play an important role in missile, aerospace, and other fields, making greater contributions to military and scientific and technological development.

The ignition process is the initial step of generating power for a solid rocket motor, and it is the key for the motor to work normally. Although the ignition process generally has an extremely short duration, the pressure undergoes significant alterations during this course, completing the entire pressure-building process of the motor. Consequently, it exerts a substantial influence on the operational process of the solid rocket motor. Figure 1 illustrates the pressure-time variation curve during the whole working process of the theoretical combustion chamber. In Figure 1, t₀ is the ignition process time of the motor, t₁ is the normal working time of the motor, t₂ is the shutdown process time, P₀ is the initial pressure of the combustion chamber, and P_m is the maximum ignition pressure.

The pressure variation in the ignition of the same type of solid rocket motor under diverse conditions is also dissimilar. As illustrated in Figure 2, the solid rocket motor pressure varies under the storage temperature conditions of high temperature (+50 °C), low temperature (−40 °C), and room temperature (+20 °C).

The pressure variation in the ignition process can be acquired through ground tests, and pressure sensors are utilized to collect data during this procedure [2,3]. Since the internal working environment of solid rocket motors is extremely harsh, it is difficult to collect abundant data in practice. And relying only on empirical or semi-empirical design methods cannot satisfy the requirements of modern advanced solid rocket motor design. Therefore, numerical simulation has become one of the effective tools for solid rocket motor design.

At present, research on numerical simulation in the ignition process has gradually matured. Researchers have proposed several numerical models, such as the fluid–structure interaction model [4,5] and the two-dimensional/three-dimensional turbulence model [6,7]. With the development of computer technology, fluid simulation software is gradually being applied to the simulation of the solid rocket motor ignition process. In recent years, numerous new methods have been proposed. For instance, Francois et al. [8] proposed coupling the computational fluid dynamics (CFD) solver of the combustion chamber flow field with the one-dimensional solver of unsteady propellant combustion and proposed two solution methods. Li et al. [9] numerically studied the second pulse ignition transients in a double-pulse solid rocket motor with an elastic barrier pulse separation device, and proposed a governing equation for unsteady compressible flow, heat conduction, and structural dynamics.

Artificial intelligence methods have received more attention because of their powerful inference and generalization abilities for complex engineering systems [10,11,12]. Data-driven machine learning methods such as the Support Vector Machine (SVM) [13] and Bayesian Network (BN) [14] are widely used. However, machine learning methods face challenges such as poor learning ability and slow learning speed when dealing with large amounts of data. Deep learning can better address these problems. Methods such as Convolutional Neural Network (CNN) [15,16], Recurrent Neural Network (RNN) [17], Deep Belief Network (DBN) [18], and Long Short-Term Memory (LSTM) [19,20] have achieved good results in speech recognition [21], image recognition [22], and natural language processing [23]. Artificial intelligence methods have been widely applied in industrial fields [24,25], such as mechanical fault diagnosis [26,27], prediction of instrument residual life [28], and evaluation of instrument health status [29,30]. With the continuous development and improvement of artificial intelligence technology, its application prospects in various fields will become even broader.

The application of deep learning methods to solve solid rocket motor problems has attracted significant attention. Surina et al. [31] proposed an image-based deep learning tool to verify the boundary layer combustion mechanism of a rocket motor with mixed fuels by predicting the fuel regression rate. Lee et al. [32,33] proposed an optimization method for solid propellant grain design based on neural networks. Six neural network-based methods were tested and compared to optimize solid propellant grain design. Liu et al. [34] proposed a Deep CNN model for defect scale prediction of solid rocket motors in the case of chamber splitting and delamination. Zhang et al. [35] proposed a data-driven method using deep learning to predict the thrust of a solid rocket motor.

Due to the difficulty of obtaining data samples for solid rocket motors, the adoption of data augmentation methods to enhance the richness of data samples is essential. Data augmentation is a technical measure that generates new data samples through various transformations and expansions of the original data, thereby augmenting the size and diversity of the dataset [36]. Data augmentation is a regularization strategy that implements functional regularization by adding noise to the intermediate activations of the network or imposing constraints on the functional form [37].

In this study, a pressure prediction method for the solid rocket motor based on data enhancement and a deep learning model is proposed by combining experimental data and simulation data. Firstly, the one-dimensional data are enhanced by the AGN method, and the sample size of the data is expanded. Then, the CBiLSTM neural network model is established to learn the pressure data under different working conditions and predict the pressure under new working conditions. This process is concluded in Figure 3.

2. Methodology

This section primarily focuses on the solid rocket motor pressure prediction method of AGN-CBiLSTM. AGN is incorporated into the data augmentation method. The network models comprise a one-dimensional convolutional layer, a BiLSTM layer, and Dropout. The pressure prediction process primarily consists of three stages: data preparation, network model training, and network model testing. This section will introduce the AGN method, the CBiLSTM model structure, and the network training process respectively.

2.1. Data Augmentation Method

Through data augmentation technology [38], various operating environments and working conditions can be simulated to generate data samples that are closer to reality. This can not only improve the training effect of the deep learning model but also enhance the model generalization ability so that it can better adapt to various complex operating environments. Therefore, in the field of solid rocket motors, it is very necessary and meaningful to use data augmentation methods to increase the richness of data samples.

AGN is a type of Gaussian noise that adaptively adjusts as the maximum difference in sequence data changes [39]. This noise is applied directly to the original sequence data, assuming that the input sequence is $x_{input}=[x_1,x_2,\cdots ,x_n]$, The augmentation sequence is $x_{AGN}$, ${\alpha }_{gauss}$ is the Adaptive Gaussian Noise coefficient.

$$x_{AGN}=x_{input}+{\alpha }_{gauss}G(x_{max}-x_{min})$$

(1)

$$G={\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{x^2}{2}}}$$

(2)

Different pressure data can be obtained given different sizes of ${\alpha }_{gauss}$. Applying the AGN method with the same ${\alpha }_{gauss}$ multiple times to the same sequence of data can also generate different pressure data, thus improving the diversity of data to a certain extent. Figure 4 illustrates the change of a set of pressure data after adding Gaussian noise. The number of data augmentation is 1, the ${\alpha }_{gauss}$ size is 0.1.

Data augmentation relies on specific assumptions for Gaussian noise coefficients. However, this approach based on specific assumptions may lead to poor performance when generalized to other application scenarios or different datasets. The characteristics, structure, and distribution of the dataset need to be considered in practical applications. Blindly applying the data augmentation strategy based on Gaussian noise coefficient assumptions will destroy the original characteristics of the data, making the output violate the expected results.

The selection of the number of data augmentations is equally important. Although too much expansion contains more feature information, it will increase the computational complexity, increase the computational cost, and aggravate the training cycle of the neural network. However, if the number of expansions is too small, it is difficult for the model to fully learn the laws and patterns embedded in the data, which is a dilemma.

In this study, a reasonable noise factor and expansion number are selected based on the characteristics of solid rocket motor pressure data.

2.2. Deep Learning Model Structure

2.2.1. One-Dimensional Convolution Structure

One-dimensional convolution is a convolution calculation method used in CNNs, which is primarily used to process one-dimensional sequence data, such as text and time series data [40]. Convolution works by sliding along the direction of the input sequence data. During each sliding process, the convolution kernel performs convolution operations on only a small portion of the sequence data, called the local connection. The design of this local connection allows the convolutional neural network to effectively extract local features from the input data. The pressure time series data are taken as the input data to obtain features by convolution. By shifting the convolution kernel, a local convolution operation can be performed on the input data to obtain local features. These local features are combined to form a feature map that contains the characteristics of the pressure changes of the solid rocket motors during the ignition process. The operation process of one-dimensional convolution is illustrated in Figure 5.

2.2.2. Long Short-Term Memory Structure

LSTM is an improvement over RNN, designed to solve the problem of gradient disappearance or gradient explosion in RNN [41,42]. The pressure data of the ignition process are long series data, which are suitable for LSTM prediction. Unlike the structure of RNN, LSTM has a memory unit to store past information. In addition, LSTM introduces three gate control units: the input gate, the forgetting gate, and the output gate. The input gate determines how much of the current input information can enter the memory cell. The forget gate controls how much of the previous moment’s information is retained in the memory cell and can selectively discard past information that is no longer important. The output gate is responsible for determining which information in the memory cell will be output for prediction at the current moment and its impact on the subsequent network state. Through the coordinated work of these three gates, LSTM can handle the long-term dependencies in long sequence data more intelligently, effectively alleviating the dilemma that traditional RNNs cannot effectively learn long-term information due to the vanishing or exploding gradients when dealing with long sequences.

The sigmoid function is used to control the importance of different elements to the current state, the degree of forgetting historical information, and the hidden state output corresponding to the current state. The basic unit of the LSTM network is concluded in Figure 6.

BiLSTM is a further improvement based on LSTM and is composed of a forward LSTM and a backward LSTM [43]. As concluded in Figure 7, pressure data are first input into the forward LSTM input layer to obtain a set of forward output pressure data change information. This change information is input into the backward LSTM layer to obtain the reverse LSTM output, and the forward and reverse outputs are spliced to obtain the final pressure change information. BiLSTM combines forward LSTM and reverse LSTM to obtain a more complete picture of pressure data changes, enabling more accurate prediction of pressure data.

2.2.3. Convolutional Bidirectional Long Short-Term Structure

The output dimension of the first layer of the convolutional layer used in this article is 8, and the output dimension of the second layer is 16. The size of the convolution kernel of both convolutional layers is 3 × 1, the stride of the convolution is 1, and the convolution formula is as follows:

$$y_1=\sum K_i·x_i(i=\mathrm{1,2},3)$$

(3)

After convolution calculation, the pressure characteristic sequence $Y_1=[y_1,y_2,\cdots ,y_n]$ is obtained. The pressure feature sequence ${\mathrm{Y}}_1$ is placed in the second layer of convolution for calculation to obtain the pressure feature sequence ${\mathrm{Y}}_2$. ${\mathrm{Y}}_2$ contains all the change characteristics of the pressure data. To effectively obtain more dimensional information, nonlinear mapping must be introduced in the modeling process. This nonlinear mapping is called an activation function. This article uses LeakyRelu as the activation function. The calculation formula of the LeakyRelu function is as follows:

$$\mathrm{L}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{y}\mathrm{R}\mathrm{e}\mathrm{L}\mathrm{u}(x)=\left\{\begin{array}{c}x,x>0\\ ax,x\le 0\end{array}\right.$$

(4)

Input the convolved pressure feature sequence into BiLSTM. The input dimension of the BiLSTM layer is 16, the hidden layer dimension is 100, and the number of network layers (Num Layers) is 1. Output the change information h_t of pressure data:

$${\displaystyle \genfrac{}{}{0pt}{}{{\overrightarrow{h}}_t=LSTM(x_t,{\overrightarrow{h}}_{t-1})}{\begin{array}{c}{\overleftarrow{h}}_t=LSTM(x_t,{\overleftarrow{h}}_{t-1})\\ h_t=W_{fd}{\overrightarrow{h}}_t+W_{bd}{\overleftarrow{h}}_t+b_t\end{array}}}$$

(5)

Among them, ${\overrightarrow{h}}_t$ is the forward hidden layer state, ${\overleftarrow{h}}_t$ is the backward hidden layer state, $W_{fd}$ and $W_{bd}$ represent the weight parameters of the forward hidden layer and backward hidden layer, respectively, and $b_t$ is the bias parameter of the hidden layer.

Enter the calculated result $h_t$ into the fully connected layer. The linear layer plays a key role in neural networks. It fully connects all neurons in the previous layer with all neurons in the current layer and realizes the combination and transformation of features by learning weight parameters. Output predicted pressure data after full connection layer integration.

The structure of the CBiLSTM model established in this study is shown in Figure 8. The model contains two convolution layers and activation function layers, one BiLSTM layer and one fully connected layer.

CBiLSTM is designed to combine the advantages of CNN and BiLSTM to better process sequence data. Traditional LSTM networks may encounter the problem of gradient vanishing or gradient explosion when dealing with long sequences, while BiLSTM can improve the model’s ability to capture contextual information by considering both forward and backward information of sequences. However, BiLSTM may encounter computational efficiency problems when processing high-dimensional data. In contrast, CNNs perform well in processing images or high-dimensional features with feature extraction and dimensionality reduction capabilities. Therefore, by combining CNN and BiLSTM, CBiLSTM aims to improve the ability of the model to process sequential data while maintaining computational efficiency. This combination allows the CBiLSTM model to have higher accuracy and robustness in processing complex sequence data.

The model designed in this study includes two convolutional layers because convolutional layers are designed to extract useful features from the input data and learn more complex and abstract feature representations by stacking multiple convolutional layers. These high-level feature representations help subsequent BiLSTM layers to better understand and process sequence data. By stacking multiple convolutional layers, the CBiLSTM model learns richer feature representations, which helps the model improve its performance on complex tasks. In addition, additional convolutional layers increase the depth of the model, which enhances the model’s learning and generalization capabilities.

The modification of CBiLSTM is to combine the advantages of CNN and BiLSTM to improve the model performance, while the introduction of 2 convolutional layers is to improve the feature extraction ability as well as the learning ability of the model. These changes make CBiLSTM more efficient and accurate in processing sequence data.

2.3. Model Training Process

Neural network prediction is mainly divided into three stages: data preparation stage, model training stage, and model testing stage. As shown in Figure 9, the data preparation stage involves processing the ignition test data and simulation data of the solid rocket motor into one-dimensional data, dividing the data into a training set and a test set, and performing data augmentation processing on the training set data. The model training phase requires establishing a network model and determining model parameters such as convolution kernel size, stride, and number of LSTM hidden layers. Backpropagation is used to optimize and update the weight parameters, and the Adam optimizer is used. The learning rate is adjusted during training. Initially, a larger learning rate is used for fast convergence, and then a smaller learning rate is used to continue training until the training round is complete. In the model testing phase, the input portion of the test set is fed into the model, the pressure prediction is performed, and the prediction results are compared with the actual results to analyze the errors.

3. Results and Comparisons

In this section, the AGN-CBiLSTM method is used to predict the pressure data during the solid rocket motor ignition process. The sources of pressure data are solid rocket motor ground experiments and fluid simulations. These two types of data are processed into one-dimensional data and normalized, and finally the training set and test set are divided to complete the data preparation process. The superiority of the AGN-CBiLSTM method is highlighted by comparing it with mainstream deep learning methods.

3.1. Data Preparation

3.1.1. Solid Rocket Motor Turbulence Simulation Model

The main structure of a solid rocket motor consists of a propellant, a combustion chamber, an igniter, a long tail nozzle, and a nozzle plug cap. After receiving the ignition command, the igniter generates a high-temperature flame to ignite the propellant. After the propellant is ignited, it begins to burn vigorously, producing high-temperature, high-pressure gas. The long-tailed nozzle has a special design shape that can efficiently convert the thermal energy of the gas into kinetic energy, so that the gas is ejected at high speed, generating a huge reaction force. The nozzle cap plays the role of sealing and protecting the internal structure of the nozzle before the motor is started, and after the motor is ignited, it will be broken under the strong pressure of the gas, and the gas can be sprayed out of the nozzle.

Due to the problems of long computation time using 3D simulation, some necessary simplifications of the model structure of the solid rocket motor are made in this study. Considering the axisymmetric distribution of the solid rocket motor structure, only the upper part is simulated. Therefore, a 1/2 axisymmetric two-dimensional model is used to simulate the transient ignition process. The entire flow field region is discretized into a mesh, and each region is treated accordingly based on the mesh density. The meshes of the wall and inlet regions should be relatively dense. Since the simulation time is relatively short, the opening of the plug cover is not considered, and the nozzle expansion part is truncated during the simulation to simplify the calculation. The mesh structure of the solid motor is shown in Figure 10.

The two main parts in Figure 11 are the combustion chamber and the long tail nozzle. The igniter is placed inside the combustion chamber during the simulation, and three locations are selected to be zoomed in to show the mesh boundaries more clearly.

The S-A model is a one-equation turbulence model with the turbulent viscosity transport equation as the core. The model equation is:

$${\displaystyle \genfrac{}{}{0pt}{}{\frac{\partial }{\partial t}\left(\rho v_1\right)+\frac{\partial }{\partial x_i}\left(\rho v_1{u}_i\right)=C_{b1}\rho S_1{v}_1-C_{w1}\rho f_w{\left(\frac{v_1}{d}\right)}^2}{+\frac{1}{{\sigma }_{\tilde{v}}}\left\{\frac{\partial }{\partial x_j}\left[\left(\mu +\rho v_1\right)\frac{\partial v_1}{\partial x_j}\right]+C_{b2}\rho {\left(\frac{\partial v_1}{\partial x_j}\right)}^2\right\}}}$$

(6)

$$f_{v1}={\displaystyle \frac{{\chi }^3}{{\chi }^3+C_{v1}^3}},\chi \equiv {\displaystyle \frac{v_1}{v}},f_w=g{\left({\displaystyle \frac{1+C_{w3}^6}{g^6+C_{w3}^6}}\right)}^{{\displaystyle \frac{1}{6}}},g=r+C_{w2}\left(r^6-r\right),r\equiv {\displaystyle \frac{v_1}{S_1{\kappa }^2{d}^2}}$$

(7)

$$C_{b1}=0.1355,C_{b2}=0.622,C_{w2}=0.3,C_{w3}=2,C_{v1}=7.1,\kappa =0.41$$

(8)

In this study, an axial igniter is used with black powder as the propellant. During ignition, the igniter injects high-temperature gas along the axis of the combustion chamber to ignite the propellant surface.

The source of the simulation pressure data is the normal temperature ignition simulation under different igniter propellant weights. Different igniter propellant weights will affect the ignition mass flow rate of the igniter, thereby affecting the pressure change process. The ignition mass flow rate equation used in this article is:

$${\displaystyle \genfrac{}{}{0pt}{}{\dot{m}=a{t}^2+bt+c}{\begin{array}{c}{\left.\dot{m}\right|}_{t=0}=0\\ {\left.\dot{m}\right|}_{t=t_{i\mathrm{g}}}=0\end{array}}}$$

(9)

$${\int }_0^{t_{i\mathrm{g}}}\dot{m}dt=M_{i\mathrm{g}}$$

(10)

M_ig is the mass of igniter propellant weight, t_ig is the time when ignition ends, and $\dot{m}$ is the mass flow rate of ignition propellant.

In this study, a polynomial fitting method is used to simulate the flame propagation process of the igniter, and the propellant weight of the igniter is set from 12 g to 54 g. A third-order polynomial equation is used to fit the mass flow rate of the igniter, and the ignition duration is set to 100 ms, and the mass flow rate of the igniter reaches its maximum value at 30 ms. This is then set as the mass flow rate input condition using the User Define Function (UDF) functionality.

The initial pressure of the model, the axial and radial velocities of the gas, and the temperature of the combustion chamber should also be considered in the calculation of this model. The inlet to the igniter is a conventional mass flow inlet and the injection duration must be set. The initial pressure of the model is set to 0.101325 MPa, the axial and radial velocities are set to 0, and the temperature is set to 300 K. The mass flow inlet is used at the inlet, and the injection duration is set to 0.01 s with a time step of 0.00001 s.

3.1.2. Fluid Simulation Pressure Data

As concluded in Figure 11, this paper conducted a total of 8 sets of simulations. The selected igniter propellant weights changed from 12 g to 54 g, and the length of each set of data was 900 data points. It can be seen from Figure 11 that when the same ignition device is used, the greater the igniter propellant weight, the faster the pressure changes.

Figure 11. Simulation of pressure data using different ignition propellant qualities.

Figure 11. Simulation of pressure data using different ignition propellant qualities.

3.1.3. Solid Rocket Motor Ground Experiment Pressure Data

As concluded in Figure 12, the data used in this article come from the same model of solid rocket motor ground test experiment. There are 10 sets of test data, and the length of each set of data is 900 data points. Each different-colored curve represents ignition ground experiments conducted under different solid rocket motor ignition initial conditions. The yellow lines represent all ground experiments conducted under high temperature conditions, the green lines represent ground experiments conducted under normal temperature conditions, and the blue lines represent ground experiments conducted under low temperature conditions.

3.1.4. Data Set Partition Processing

Under the conditions of high temperature and normal temperature, 8 sets of pressure experiment data were selected. When the igniter propellant weight is 12 g to 54 g, 8 sets of simulation data are selected as the input data of the model. Two sets of pressure test curves with the igniter propellant weight of 12 g at low temperatures were used as labels. The AGN method was applied to improve the data. The noise coefficient was set to 0.002, and each data set was enhanced 20 times. After data enhancement, there are 20 data sets for each working condition, with a total of 320 data sets. During the training, 1 set of pressure data was randomly selected under each working condition, and a total of 16 sets of data were used as input to the model. Then, 1 set was randomly selected from 2 sets of pressure data under low temperature and 12 g experimental conditions as a label. The ignition pressure data after data augmentation are shown in Figure 13.

3.1.5. Data Normalization Processing

To facilitate subsequent model learning, the min-max normalization method is used to normalize the input data. Min-max normalization is a linear transformation method aimed at scaling the original data into a specified range, typically [0, 1]. Through this method, the numerical values among different features become comparable, while reducing the sensitivity of the model to feature scales. Consequently, it enhances the accuracy and stability of the algorithm. The mathematical expression for Min-max normalization is:

$$x^{*}={\displaystyle \frac{x-x_{min}}{x_{max}-x_{min}}}$$

(11)

$x_{max}$ is the maximum value in the data, and $x_{min}$ is the minimum value in the data.

3.2. Different Network Prediction Results

To verify the superiority of the CBiLSTM method, this study compares it with four different neural network models, namely, Convolutional Neural Network (CNN), Long Short-Term Memory (LSTM), Bidirectional Long Short-Term Memory (BiLSTM), and Gated Recurrent Unit (GRU). After processing the ignition pressure data using the data augmentation method under the same conditions, different neural network models are used for comparison. The ${\alpha }_{gauss}$ used for experimental and simulation data is 0.002 and the number of data augmentations is 20. This verifies that the AGN-CBiLSTM model is more advantageous under the same training conditions.

In all comparison methods in this article, the input layer and output layer of the model are the same as in the CBiLSTM method, and the optimizer used for updating the model parameters is Adam. The maximum training epochs are kept the same for all models. In the variable learning rate method, the first 25 epochs of training are set to 0.0001 to quickly optimize the network, and then a learning rate of 0.00001 is used for accurate convergence.

Table 1. Parameter Settings for neural network.

Parameter Name	Values	Parameter Name	Methods
Learning rate1	1 × 10−4	Learning rate2	1 × 10−5
Activation function	LeakyReLU	optimizer	Adam
Batch size	10	Loss function	MSE
Maximum epochs	50	Dropout	0.1

illustrates the hyperparameters adopted by all the neural networks.

To better illustrate the change in the model loss function during the training process, we plot the loss function values of the first 20 epochs for comparison, as shown in Figure 14.

The loss values of CBiLSTM decrease faster for the same epochs, indicating that the CBiLSTM model exhibits higher learning efficiency at the early stage of training. CBiLSTM can converge to lower loss values faster than comparison models (e.g., traditional BiLSTM or other variants), which usually means that its parameter optimization process is more efficient, and the model is able to capture key features in the data faster.

The predicted pressure results of 50 training courses were averaged. The prediction results of the four models are concluded in Figure 15.

It can be seen from Figure 15 that the predicted pressure data are basically consistent with the pressure data in the test data set. There is some fluctuation in the forecast pressure data at data points 200–400. Figure 16 illustrates a zoomed-in view for sequence points 200–400, and the predicted pressure of the AGN-CBiLSTM model is much closer to the real data, indicating that its prediction is more satisfactory.

Each neural network was trained 50 times and t-distributed Stochastic Neighbor Embedding (t-SNE) images of the prediction results and the test set stress data were plotted. t-SNE is a technique commonly used for downscaling and visualization and is particularly suited for exploring complex structures and distributions in high-dimensional data. In Figure 17, the distribution of the predictions of each model mapped to a two-dimensional space after 50 training sessions is shown. Through the t-SNE image, the clustering effect and distribution characteristics of the prediction results of different models can be visualized.

For the CBiLSTM model, the t-SNE images of its prediction results show a clear clustering structure, with the predicted values closely surrounding the cluster centers of the real test set stress data, with only two points deviating from the larger ones. This indicates that the prediction results of the CBiLSTM model are not only accurate but also have a good generalization ability to capture the intrinsic features of the test set data well.

It should be noted that although t-SNE images can provide intuitive visualization, they may distort the high-dimensional structure of the original data to a certain extent due to the nonlinear transformations in their dimensionality reduction process. Therefore, when interpreting t-SNE images, we need to combine the actual prediction performance of the model with other evaluation indicators to make a comprehensive judgment.

To test the prediction effect of the model, the Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Mean Absolute Percent Error (MAPE) are used as the evaluation functions of the model.

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{m}}\sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|$$

(12)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{m}}\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(13)

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{1}{m}}\sum _{i=1}^n\left|\left(y_i-{\widehat{y}}_i\right)\right|\times 100\%$$

(14)

Figure 18 and

Table 2. Error of predicted pressure data.

Error	AGN-CBiLSTM	CNN	GRU	LSTM	BiLSTM
MAE	0.1431	0.2458	0.2772	0.2390	0.2233
RMSE	0.1779	0.2989	0.3210	0.2803	0.2672
MAPE	3.27%	5.46%	6.90%	5.88%	4.43%

show the experimental results of different network models for pressure prediction. In these results, we can clearly see the differences in the performance of each model in terms of prediction accuracy. The GRU method achieves the maximum values of MAE, RMSE, and MAPE in the experiment. This indicates that the accuracy of the GRU model in predicting pressure data is relatively low, and there is a large deviation between its prediction results and the actual values.

In contrast, the CBiLSTM method exhibits minimum values in all four error metrics. This fully proves that the CBiLSTM model has excellent performance in predicting pressure data, and its prediction results are not only accurate but also stable. The CBiLSTM model can fully exploit the temporal and spatial features of the data to make more accurate predictions.

The errors of BiLSTM, LSTM, and CNN, on the other hand, are intermediate between GRU and CBiLSTM. This indicates that these three models have some accuracy in predicting the stress data, but their performance could be improved relative to the CBiLSTM model.

It is worth noting that among the two models based on long and short-term memory networks, BiLSTM and LSTM, the error of BiLSTM is slightly smaller than that of LSTM. This suggests that the bi-directional structure can capture the temporal characteristics of the data more efficiently, thus improving the prediction accuracy of the model. The bi-directional structure allows the model to consider both past and future information when processing data at the current time step and therefore provides a more comprehensive understanding of the dynamics of the data.

In summary, the four errors of the CBiLSTM method are the smallest, while the error of the GRU method is the largest. The CBiLSTM method has the advantage of high precision in predicting solid rocket motor pressure data.

By comparing the performance of these four networks, the CBiLSTM method has significant advantages in accuracy and other aspects. In contrast, the four networks of CNN, LSTM, BiLSTM, and GRU have certain limitations when processing sequence data and cannot capture complex patterns and dependencies in sequence data as effectively as the CBiLSTM method. Therefore, the CBiLSTM method is a more excellent neural network model with broad application prospects and potential value.

In this way, after inputting the pressure data of similar solid rocket motors under different operating conditions, it is possible to predict the pressure data under new operating conditions. This helps to better understand the pressure dynamics inside the engine and thus optimize the structural design of the engine. If higher pressure peaks are predicted under certain operating conditions, then the combustion chamber walls need to be designed with higher-strength materials or increased wall thickness to ensure engine safety and reliability.

In the future, the failure of the model can be analyzed by introducing data from extreme working conditions (e.g., ultra-high temperatures, strong corrosive environments) and using simulation software to simulate the extreme and abnormal conditions, generating a large amount of synthetic data to supplement the lack of actual test data. Inputting the extreme working condition data to verify whether the neural network can maintain stability for extreme conditions. Develop methods to quantify the uncertainty of model predictions for extreme conditions.

4. Conclusions

An AGN-CBiLSTM solid rocket motor pressure prediction method has been proposed to solve the problem of different pressure changes during the ignition process of solid rocket motors under different operating conditions. Through the processing of experimental and simulation data, experimental pressure data and fluid simulation pressure data are simultaneously used as input data for collaborative training. A model suitable for solid rocket motor data was established with the purpose of predicting the pressure data under new working conditions by inputting the pressure data of the same type of solid rocket motor under different working conditions. By fully training the model, the method achieves good results on the test set. The AGN-CBiLSTM method has MAE = 0.1431, RMSE = 0.1779, MAPE = 3.27%. The prediction results concluded a small error and are close to the actual data.

Compared with other commonly used deep learning methods, this method has the accuracy and effectiveness of prediction. The two error indicators of the AGN-CBiLSTM method are smaller than the errors of the other three deep learning methods. The results indicate that this method can effectively reflect the changes in actual pressure.

Although the currently proposed AGN-CBiLSTM method has achieved favorable results in experiments, further optimization of the model structure is still required to reduce the prediction error and reflect the pressure changes during the ignition process of solid rocket motors more accurately. Although it has been pointed out in the research that the AGN method is helpful for expanding data samples, deep learning models still rely heavily on a large amount of training data. In the future, more efficient data augmentation strategies or methods to reduce data dependence need to be explored to ensure that the model can maintain a high level of prediction accuracy with limited data. Currently, this method has shown good performance on the test set, and it can be attempted to be widely applied in the design practices of solid rocket motors under different types and working conditions, to better provide theoretical support and practical guidance for the design of new solid rocket motors.