1. Introduction

2. Methodology

2.4. Wake Parameter Prediction Model Based on PA-TLA

TCN (Temporal Convolutional Network) and LSTM (Long Short-Term Memory) networks each have their advantages and are suitable for different scenarios. TCN employs a multi-stacked, dilated causal convolution process to focus on the local features of time series data. On the other hand, LSTM uses gated units to maintain a global perception of the entire sequence, storing previous information in hidden layers to transmit contextual information. However, classical serial hybrid framework models increase the demand for computational memory and time, neglecting the increase in the number of model parameters and complexity, leading to higher computational costs during training and reduced timeliness. Therefore, this study proposes a hybrid deep learning model based on a parallel architecture that combines Temporal Convolutional Networks (TCN) and Long Short-Term Memory (LSTM) neural networks through a tensor fusion module based on attention mechanisms for predicting aircraft wake vortex parameters and safety intervals. TCN and LSTM operate independently to capture key features, such as long-term dependencies and contextual information. Finally, through tensor fusion based on the attention mechanism, the output matrices of TCN and LSTM are assigned different attention weights. These are dynamically weighted according to task requirements and merged in the feature dimension to achieve feature information fusion. The depth of the model in this parallel mode is only affected by the depth of each individual network layer. The structure of the model is illustrated in the figure, and this configuration is referred to as the Parallel Architecture LSTM-TCN-Attention (PA-TLA), as shown in Figure 6.

2.4.1. Sequence Space Feature Representation Based on TCN

The Temporal Convolutional Network (TCN) is a model based on the CNN architecture designed for processing time-series data. In this study, we employ the TCN network as a branch of the PA-TLA prediction network to extract global spatial features of sequences. The TCN consists of a stack of multiple residual units. Each residual module includes two convolutional units and a nonlinear mapping unit, comprising one-dimensional dilated causal convolution, weight normalization, ReLU activation function, and Dropout operation.

TCN mainly includes three basic features: causal convolution, dilated convolution, and residual connections. The dilated causal convolution illustrated in Figure 6 is obtained by convolving the input with a causal relationship to generate network output. In sequence modeling, causal convolution means that the current moment $\mathrm{T}$ is only related to the inputs at moment $\mathrm{T}$ and before. Its mathematical form is described as follows:

TCN primarily features three fundamental characteristics: causal convolution, dilated convolution, and residual connections. In sequence modeling, causal convolution ensures that the output at the current time T is only related to inputs at time T and prior times. The mathematical form is described as follows:

$$h_r=f(|x_1|,|x_2|,|\cdots |,|x_T|)$$

(16)

where $x_T$ is a one-dimensional vector containing $n$ features, and $h_r$ is the variable to be predicted. $f$ is the function that establishes the relationship between $x_T$ and $h_r$. Causal convolution is a unidirectional structure, ensuring the sequentiality of data processing by handling the values at moment $t$ and only using data from before moment $t$ For a one-dimensional series input, the dilated convolution operation, $F_{\left(S\right)}$ on its elements and the filter $f:\{0,\dots ,\mathrm{k}-1\}$ is defined as follows:

$$F_{(S)}=(x\ast df)(s)=\sum _{i=0}^{k-1}f\left(i\right)·x_{s-d\ast i}|$$

(17)

Given $s$ as the input sequence information, $d$ as the dilation factor, and $k$ as the size of the filter, $s-d·i$ represents the location of certain information in the history. As the dilation factor $d=(\mathrm{1,2},4)$ increases, the receptive field of the TCN increases, leading to an increase in the amount of information received. The expanded convolution structure is shown in Figure 7.

Residual connections have proven their effectiveness in training deep networks by allowing information to be transmitted across layers. Each residual block includes a branch that leads out a series of transformations F, whose outputs are added to the input X of the block:

$$\beta =Activation(X+F(X))$$

(18)

As shown in Figure 8, within the residual block, the TCN comprises two layers of dilated causal convolutions and nonlinear processing, where Rectified Linear Units (ReLU) are used. For normalization, weight normalization is applied to the convolutional filters. Additionally, after normalization through dilated convolution, a spatial dropout is incorporated in each layer to deactivate some neurons, thus avoiding overfitting.

2.4.2. LSTM

Long Short-Term Memory neural networks (LSTM) are an optimized variant of Recurrent Neural Networks (RNN), designed to address the issues of vanishing and exploding gradients that are present in traditional neural networks. LSTMs aim to mitigate long-term dependency problems, making them particularly suitable for tasks that require learning from sequences of data over long periods. They have been refined and expanded by numerous scholars in various works and have shown impressive performance across a wide range of time series problems.

An LSTM neural network unit includes three gates: the input gate, output gate, and forget gate. These gates work together to regulate the flow of information into and out of the cell, deciding what to keep in or omit from the cell's state. The architecture of a basic unit and the LSTM network architecture are depicted in the figures (not provided here). As represented in Figure 7 (hypothetically, since the figure is not displayed), the equations for an LSTM unit are as follows:

Figure 9. This is a figure. Schemes follow the same formatting.

Figure 9. This is a figure. Schemes follow the same formatting.

$$f_t=\sigma (W_f\cdot [h_{t-1},x_t]+b_f)$$

(19)

$$i_t=\sigma (W_i\cdot [h_{t-1},x_t]+b_i)$$

(20)

$${\tilde{C}}_t=\tanh (W_C\cdot [h_{t-1},x_t]+b_C)$$

(21)

$$o_t=\sigma (W_o[h_{t-1},x_t]+b_o)$$

(22)

$$h_t=o_t\ast \tanh (C_t)$$

(23)

where $W_f$, $W_i$, $W_C$ and $W_o$​ represent the weight matrices associated with the forget gate, input gate, candidate values (for updating the cell state), and output gate, respectively. These matrices play crucial roles in determining the flow and transformation of information within the LSTM cell. $h_{t-1}$ is the hidden state from the previous time step, carrying information from earlier in the sequence. $x_t$ is the input information at the current time step, introducing new information to be processed. $b_f$, $b_i$, $b_C$ and $b_o$ are biases that adjust the output of their respective gates and candidate cell state update mechanisms, adding another level of adaptability to the model. The cell states, $C_t$ and $C_{t-1}$​, represent the memory of the LSTM cell. $C_t$ is the current cell state, and $C_{t-1}$ is the cell state from the previous time step. These states are critical for the LSTM's ability to carry information across long sequences, allowing the network to remember and forget information selectively. ${\tilde{C}}_t$ represents the candidate cell state, which is a combination of the current input $x_t$ and the previous hidden state $h_t$. It suggests potential updates to the cell state, providing the model with the option to incorporate new information based on the input and the LSTM's memory. The final decision on which memories to keep or discard—and how much of the new candidate state to add to the existing cell state—is made through the input and forget gates, guided by the LSTM's learned parameters. This architecture enables LSTMs to decide what information is valuable and should be retained over long sequences, making them exceptionally suited for tasks that require understanding and processing sequential data with complex, long-range dependencies.

2.4.3. Attention-Based Tensor Concatenation Module

From Figure 1, it can be seen that the PA-TLA (Parallel Architecture LSTM-TCN-Attention) proposed in this paper is composed of two independent models and a specialized architecture. One is a TCN (Temporal Convolutional Network) with n layers, and the other is an LSTM (Long Short-Term Memory) network with m layers. These two multilayer networks operate independently but share the same input data. Afterward, a tensor fusion module combines the output results of the two networks, The principle is as follows:

$$y_i=RELU(w\sum _{i=1}^{o+\rho }{x}_i+b)$$

(24)

where $y_i$ represents the target output value, $x_i$ is the input, $w$ denotes the weight values, $b$ stands for the bias values, $RELU$ refers to the Rectified Linear Unit, $o$ is the number of filters in the last stack of the TCN, and $\rho$ is the number of units in the last layer of the LSTM. Due to the lack of sensitivity to the temporal features within the sequence data of LSTM and TCN, they fail to capture the intrinsic connections within the data, limiting their ability to extract and learn important feature information, which is not conducive to further improving the model's prediction accuracy. Therefore, this paper introduces a self-attention mechanism, connecting different positions within the sequence to enhance the model's ability to capture key temporal feature information. First, for each output, the corresponding Query, Key, and Value vectors need to be calculated through three different fully connected layers, as shown in the equation:

$$\begin{array}{c}{Q}_i=W_Q{y}_i+b_Q\\ K_i=W_K{y}_i+b_K\\ V_i=W_V{y}_i+b_V\end{array}$$

(25)

where $W_Q,W_K,W_V$ represent the weight matrix, and $b_Q,b_K,b_V$ are the bias term. Attention scores $e_{ij}$ are computed using the query and key as follows: Where $b_{att}$ is the bias term, $W_{att}$ is the weight matrix, and $\tanh$ represents a nonlinear activation function. Then, the softmax function is applied to normalize the attention scores across all time steps, resulting in the final attention weights $a_{ij}$ as shown in the equation:

$$a_{ij}=\frac{exp(e_{ij})}{\sum _{k=1}^{N}exp(e_{ik})}$$

(26)

Finally, the final output is obtained by calculating the weighted sum, as in the equation:

$$z_i={\displaystyle \sum _{j=1}^N{a}_{ij}}{V}_j$$

(27)

3. Experiments

3.4. Results Analysis

3.4.1. Wake Evolution Prediction Model Based on PA-TLA

To further demonstrate the effectiveness of PA-TLA, we constructed a TCN-LSTM structure composed of a two-layer convolutional stacked TCN and a three-layer LSTM in series. As illustrated in Figure 14a–f. This figure shows the comparison of predicted and actual vorticity characteristic parameters for four models: PA-TLA, TCN-LSTM, LSTM, and TCN, all of which exhibit high levels of fit.

To further observe the prediction errors of the four models, Figure 14 presents the relative error plots for the predicted vorticity characteristic parameters. It is evident that TCN, with its capability to perceive spatial features of the sequence, performs better in terms of relative error compared to LSTM. Both PA-TLA and TCN-LSTM, which integrate the spatial feature perception of TCN with the temporal feature perception of LSTM, show good performance in predicting aircraft wake characteristic parameters, with PA-TLA outperforming TCN-LSTM in relative error.

Figure 15. Relative Error in Q-Criterion Prediction for Three Models (A for 50 meters, B for 100 meters, C for 150 meters, D for 200 meters, E for 250 meters, F for 300 meters).

Figure 15. Relative Error in Q-Criterion Prediction for Three Models (A for 50 meters, B for 100 meters, C for 150 meters, D for 200 meters, E for 250 meters, F for 300 meters).

Table 2 reveals that across different heights, the models predict wake circulation, vorticity, and Q-criterion. PA-TLA and TCN-LSTM outperform LSTM and TCN on all error metrics, exhibiting lower mean squared error (MSE), mean absolute error (MAE), root mean square error (RMSE), and higher coefficient of determination (R²), indicating superior model fit and significantly enhanced predictive accuracy on this dataset. Moreover, compared to TCN-LSTM, PA-TLA shows an average reduction in MSE by 6.80%, in MAE by 7.70%, in RMSE by 4.47%, and an average increase in R² by 0.36%. The training time for the PA-TLA model is approximately 144 minutes, and prediction time is about 292 seconds, whereas the TCN-LSTM requires 201 minutes for training and 380 seconds for prediction, demonstrating a 28% improvement in training efficiency and a 35% increase in prediction efficiency for the parallel structure over the serial model structure.

3.4.2. Analysis of Near-Ground Phase Wake Vortex Evolution Characteristics Combining Numerical Simulation and PA-TLA Model

The experiment, focusing on the A330-200 model, predicts different stages of wake evolution in the near-ground phase through the proposed PA-TLA model, selecting circulation and vortex core position as parameters for quantitative analysis of vortex characteristics. Figure 1a shows that at different height conditions, the model-predicted average vortex circulation follows a negative exponential decay rule through the initial stage, rapid decay, and gradual transition to a stable evolution pattern. Within 0-10 s, the vortex structure remains tight and concentrated, with circulation concentrated near the vortex core. For lower altitude vortices, the distance between the vortex and the ground is smaller, which might lead to stronger constraints and deformation of the vortex structure, differing in the specific form of initial circulation distribution. During the rapid decay phase of 10-20 s, air viscosity causes circulation within the vortex to diffuse and dissipate radially and axially through friction and mixing processes within micro-scale vortices, accelerating the reduction of circulation. Moreover, for low altitude vortices, the shear layer thickness between them and the ground is thinner, making the viscous effects more significant and further intensifying the dissipation of circulation. However, after 80 s, as the vortex structure gradually diffuses, part of the circulation may reorganize into secondary vortices or vortex filaments, forming more complex vortex structures. Despite the potential continued decrease in circulation, the change in vortex morphology can slow its further decay. From 10m-50m, as the height increases, the effect of ground influence weakens, increasing the rate of wake decay. From 50m-300m, the rate of wake decay is almost consistent.

Figure 18. Evolution Curve of Aircraft Wake Circulation at Different Heights.

Figure 18. Evolution Curve of Aircraft Wake Circulation at Different Heights.

The time-dependent evolution trend of vortex core position at different initial heights is shown in figures (a) and (b). The vortex core height curves show a consistent time-decreasing characteristic at different initial heights, exhibiting a sharp decline rate in line with theory due to increased viscous resistance intensifying local dissipation of vorticity in the early phase (approximately the first 25 seconds). However, as time progresses, the influence domain of the tail vortex expands, the velocity difference between the vortex core and the surrounding flow field decreases, and the induced effect and air resistance cause the interaction between vortices to weaken, leading to a gradually stabilizing vortex core sinking speed. Near the ground, the lateral spacing evolution trend of the wake vortex, as shown in figure (c), exhibits a non-linear expansion characteristic over time. The presence of the ground not only changes the flow state of the boundary layer but also enhances vortex diffusion in the near-ground region, promoting a faster growth rate of vortex spacing. Moreover, from 10 m-50 m, affected by ground effect, the higher the altitude, the faster the decay rate, and the vortex core position initially sinks briefly before showing an upward trend, while the ground effect induces an increase in the distance between two vortices, developing towards isolated vortex stages and weakening the mutual induction forces between them.

Figure 19. Evolution of Vortex Core Position of Aircraft Wake at Different Heights((a) represents the three-dimensional variation curve of the vortex center of the aircraft wake, (b) represents the evolution of the longitudinal position of the vortex center, and (c) represents the evolution of the distance between the two vortex centers).

Figure 19. Evolution of Vortex Core Position of Aircraft Wake at Different Heights((a) represents the three-dimensional variation curve of the vortex center of the aircraft wake, (b) represents the evolution of the longitudinal position of the vortex center, and (c) represents the evolution of the distance between the two vortex centers).

Conclusion

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