Prediction of Temperature Distribution on an Aircraft Hot-Air Anti-Icing Surface by ROM and Neural Networks

Abstract

To address the inefficiencies and time-consuming nature of traditional hot-air anti-icing system designs, reduced-order models (ROMs) and machine learning techniques are introduced to predict anti-icing surface temperature distributions. Two models, AlexNet combined with Proper Orthogonal Decomposition (POD-AlexNet) and multi-CNNs with GRU (MCG), are proposed by comparing several classic neural networks. Design variables of the hot-air anti-icing cavity are used as inputs of the two models, and the corresponding surface temperature distribution data serve as outputs, and then the performance of these models is evaluated on the test set. The POD-AlexNet model achieves a mean prediction accuracy of over 95%, while the MCG model reaches 96.97%. Furthermore, the proposed model demonstrates a prediction time of no more than 5.5 ms for individual temperature samples. The proposed models not only provide faster predictions of anti-icing surface temperature distributions than traditional numerical simulation methods but also ensure acceptable accuracy, which supports the design of aircraft hot-air anti-icing systems based on optimization methods such as genetic algorithms.

1. Introduction

When an aircraft flies under icing conditions, ice may accumulate on its surface and pose a severe threat to flight safety and even lead to accidents [1]. Effective anti-icing measures enhance the ability of aircraft to handle icing conditions. Hot-air anti-icing systems, widely used in large civil airliners [2], are particularly suitable for areas requiring extensive anti-icing, such as wings and engine inlets. The hot-air anti-icing system uses compressor bleed air to prevent icing by conducting heat through the skin to the outer surface [3]. The piccolo tube structure within the hot-air anti-icing cavity mainly determines the temperature distribution on the surface and the performance of the hot-air anti-icing system [4]. Therefore, a rapid and accurate prediction method of anti-icing surface temperature related to the piccolo tube structure is crucial for designing and optimizing the hot-air anti-icing system.

With advancements in icing wind tunnel experiments, researchers have investigated the effects of the internal and external aspects of the hot-air anti-icing system on surface temperature distribution. The anti-icing characteristics of engine inlet guide vanes and other components were studied in the YBF-02 icing wind tunnel, and the effects of liquid water content and hot-air mass flow rate on the performance of the anti-icing system were analyzed [5,6]. The wing hot-air anti-icing system was studied in the China Aerodynamics Research and Development Center Icing Wind Tunnel (CARDC IWT), and the effects of icing meteorological conditions, hot-air mass flow rate, and hot-air temperature on the double-skin heat transfer enhancement hot-air anti-icing system were investigated [7]. Experiments in icing wind tunnels provide controllable environments and accurate data, but the preparation cycle is long and costly.

Advanced numerical methods have become crucial for the design and verification of anti-icing systems due to the development of thermodynamic models and numerical simulations. A series of numerical calculations using the SST $k-\omega$ turbulence model were performed [8], comparing three injection methods—impingement jets, offset jets, and swirl jets—and demonstrating that the swirling effect is able to enhance internal heat transfer. The influence of the velocity, height, temperature, liquid water content (LWC), and median volume diameter (MVD) on the anti-icing thermal load was investigated [9]. Numerical simulation for predicting the surface temperature of the anti-icing cavity skin involves calculating flow fields and anti-icing performance, which is computationally intensive and time-consuming. These limitations indicate that existing numerical simulation methods cannot meet the requirements for the design of aircraft hot-air anti-icing systems based on optimization methods such as genetic algorithms.

Compared to numerical simulations, there are few studies using non-numerical methods to predict the surface temperature of hot-air anti-icing systems, primarily involving ROM-based and machine learning-based approaches. ROMs integrate data reduction techniques such as POD with surrogate models like neural networks to predict low-dimensional approximations of the original data. The back propagation neural network improved with the particle swarm optimization algorithm (PSO-BP) [10] effectively maps the flight condition parameters to characteristic coefficients of surface temperature, achieving an average absolute error of 3.87 K. Additionally, a meta-model was employed based on POD and a general regression neural network to evaluate the performance of electrothermal anti-icing systems [11]. These approaches demonstrate that ROMs effectively capture the main characteristics of surface temperature, thereby improving prediction efficiency while maintaining approximation accuracy. An artificial neural network (ANN) has been employed to predict wing temperatures based on experimental and computational fluid dynamics (CFD) data [12], but this method predicts temperatures at only three lines within the anti-icing region. K-nearest neighbor (KNN) and a local linear weighted regression algorithm [13] were also employed to predict the temperature trend of the electrothermal anti-icing surface, but this method converts predicted temperature change rates into actual temperatures rather than predicting surface temperature directly. Overall, existing non-numerical simulation methods primarily focus on temperature predictions for electrothermal anti-icing systems or target only specific cross-sections within the anti-icing region, with insufficient research on predicting the complex three-dimensional surface temperatures for hot-air anti-icing systems. However, the ROMs introduce additional errors during data reduction and reconstruction, and neural network methods are currently seldom applied directly to regression prediction of high-dimensional temperature data.

This study focuses on enhancing the approximation accuracy of a ROM and developing a high-dimensional neural network model to predict the surface temperature distribution across the entire hot-air anti-icing region. Comparative experiments are conducted to validate the efficacy of the proposed methods, demonstrating their superiority over traditional approaches. The Section 2 of this paper describes the methods for obtaining and processing samples, while the Section 3 explains the basic principles and network architectures of the proposed models. The Section 4 presents the experimental results and analyses.

2. Sample Data Acquisition and Processing

This paper focuses on a turbofan engine nacelle inlet and gathers surface temperature data for neural network experiments through three-dimensional anti-icing numerical simulations. Relevant data processing operations are then performed to facilitate network training.

2.1. Data Collection

Figure 1a shows the three-dimensional model of the turbofan engine nacelle inlet. Numerical simulations are conducted on the localized hot-air anti-icing system at the 12 o’clock position of the inlet (Figure 1b) to obtain the corresponding surface temperature distribution data.

The jet holes on the piccolo tube are arranged in a diamond shape, i.e., one of the two neighboring columns has only one jet hole, and the other column has two. The design parameters are selected based on the internal structural characteristics of the inlet hot-air anti-icing system (Figure 2): (1) z-direction jet hole spacing between two neighboring columns L; (2) x-coordinate of the piccolo tube center $x_{\mathrm{pic}}$; (3) y-coordinate of the piccolo tube center $y_{\mathrm{pic}}$; (4) the outflow direction angle of jet holes in the middle row ${\theta }_0$; (5) the relative angle between the clockwise (positive y-axis direction) jet holes and the middle row of jet holes ${\theta }_1$; (6) the relative angle between the counterclockwise (negative y-axis direction) jet holes and the middle row of jet holes ${\theta }_2$.

The design variable space is six-dimensional, with parameter ranges detailed, and the baseline design presented in

Table 1. The range of design variables for inlet hot-air anti-icing system.

Design Variables	L/D	${\mathit{x}}_{\mathbf{pic}}$/D	${\mathit{y}}_{\mathbf{pic}}$/D	${\mathit{\theta }}_0$/$\Theta$	${\mathit{\theta }}_1$/$\Theta$	${\mathit{\theta }}_2$/$\Theta$
Baseline	14.6	0	0	1	0.86	0.73
Range	[10, 20]	[−10, 7.5]	[−10, 7.5]	[0.54, 1.08]	[0.54, 1.08]	[0.54, 1.08]

. A total of 5000 samples of piccolo tube design variables are generated using Latin Hypercube Sampling (LHS) [14] to ensure a near-random distribution across the unit hypercube.

On this basis, the three-dimensional anti-icing numerical simulation method [15] is used for calculating the surface temperature of the hot-air anti-icing system with the sampled structural design variables. This method includes airflow field calculation, water droplet collection efficiency calculation, and three-dimensional surface icing thermodynamics calculation. Comparing the numerical simulation results of this method with those from the traditional tight-coupled method and experimental results demonstrates the reliability of this approach [1,15]. The skin solid domain used for calculating the temperature distribution data employs a structured grid consisting of 198 grids in the chord direction and 121 grids in the span direction, totaling 23,958 grids (Figure 3). The design conditions are shown in

Table 2. Design conditions for inlet hot-air anti-icing system.

H	AoA	Ma	${\mathit{T}}_{\mathit{\infty }}$	MVD	LWC	${\mathit{P}}_{\mathbf{tot}}$	${\mathit{T}}_{\mathbf{tot}}$	${\dot{\mathit{m}}}_{\mathbf{single}}$
6 km	4°	0.427	263.55 K	20 μm	0.43 g/${\mathrm{m}}^3$	0.25 MPa	555 K	1.33 g/s

.

2.2. Training Data Processing

Each sample consists of two parts: anti-icing design variables and surface temperature distribution data. All 5000 samples are divided into three datasets in a ratio of 8:1:1. A total of 4000 samples are used as the training set for training network models, with 500 samples allocated as the validation set for adjusting the model hyperparameters, and 500 samples set aside as the test set for evaluating model performance. Assessing the performance of models using the test set that was not involved in training ensures their reliability in real-world environments.

Temperature distribution data are normalized using the Z-score method to facilitate network learning, as shown in the following equation:

$$B_i={\displaystyle \frac{A_i-{\mu }_i}{{\delta }_i}}i=1,2,\dots ,23958$$

(1)

Here, $B_i$ represents the standardized temperature at the i-th point, $A_i$ denotes the original temperature at the i-th point, and ${\mu }_i$ and ${\delta }_i$ are the mean and standard deviation, respectively, of the temperature at the i-th point. The mean of the temperature dataset is 0, and the standard deviation is 1 after standardization. Normalization enables data with different features to be compared on the same scale, thereby enhancing the performance and stability of networks.

3. Rapid Prediction Method of Temperature Distribution

3.1. Basic Principles

3.1.1. Proper Orthogonal Decomposition

Proper Orthogonal Decomposition (POD), also known as Principal Component Analysis (PCA), is a mathematical method used for data dimensionality reduction and pattern recognition, allowing the extraction of main feature patterns from multi-dimensional datasets. The fundamental idea of POD is to decompose complex datasets into orthogonal basis functions that capture the main features of the data. These basis functions project the original features into a new orthogonal feature space, approximating the original data with fewer parameters. This process achieves data dimensionality reduction, eliminates redundant information between features, and retains the maximum variance, thus maintaining the key features of the data.

The steps of the POD are as follows:

• Data collection: Organize the temperature distribution matrix X so that each column represents a temperature distribution sample and each row corresponds to a grid point.
• Calculate the covariance matrix: The covariance matrix C of the matrix X can be expressed using the following equation:

  $$C={\displaystyle \frac{1}{m}}X{X}^T$$

  (2)
  Here, X is an $n\times m$ matrix, where n represents the number of temperature points on the surface and m represents the number of samples; $X^T$ denotes the transpose of matrix X, resulting in the covariance matrix C being of dimension $n\times n$.
• Eigenvalue decomposition: The eigenvalue decomposition of the covariance matrix C yields the eigenvalues ${\lambda }_i$ and the corresponding eigenvectors ${\varphi }_i$:

  $$C{\varphi }_i={\lambda }_i{\varphi }_i$$

  (3)
• Select main eigenvalues and eigenvectors: The top k eigenvalues ${\lambda }_i$ and their corresponding eigenvectors ${\varphi }_i$ are selected based on the magnitude of the eigenvalues. These eigenvectors will serve as POD basis functions.
• Construct the basis mode matrix: Construct the POD basis mode matrix $\mathsf{\Phi }$ using the selected k eigenvectors ${\varphi }_i$:

  $$\mathsf{\Phi }=[{\varphi }_1,{\varphi }_2,\dots ,{\varphi }_k]$$

  (4)
• Data dimensionality reduction: Project the original data X onto the POD basis mode matrix $\mathsf{\Phi }$ to obtain a low-dimensional representation Y:

  $$Y={\mathsf{\Phi }}^{T}X$$

  (5)
  Here, Y is referred to as the matrix of fitting coefficients.

POD is commonly used in fluid mechanics to analyze complex fluid dynamics problems, such as eddy currents and turbulence [16,17]. By reducing the dimensionality of high-dimensional temperature distribution data, POD overcomes the limitations of traditional numerical calculations and improves prediction efficiency.

3.1.2. Convolutional Neural Networks

Convolutional neural networks (CNNs) [18] are deep learning models primarily designed to process data with a grid structure, such as images and videos. CNNs can automatically learn the spatial features of images through their layered architectures and local connections.

The basic structure of CNNs includes convolutional layers, pooling layers, and fully connected layers [19]. The convolutional layer is the core of CNNs, extracting local features from the input data through convolution operations. The convolution operation in this layer uses a small convolution kernel (or filter) to slide over the input data, calculating the dot product between the kernel and the input data to generate a feature map. By sharing convolution kernel parameters, the convolutional layer significantly reduces the number of model parameters, enhancing computational efficiency. The pooling layer is used to reduce the size of feature maps, decrease computational complexity, and prevent overfitting. Pooling operations include max pooling and average pooling [20]. The fully connected layer is usually located at the end of CNNs, mapping the flattened feature map to output for classification or regression tasks.

Classic CNNs include LeNet [20], AlexNet [21], UNet [22], and ResNet [23], among others. The architecture of AlexNet used in this paper is illustrated in Figure 4, where the layers in the feature block follow the classic AlexNet layer configurations. In the experiments described in Section 4, a six-dimensional input vector is reshaped into a six-channel image of size 128 × 128 before being fed into AlexNet.

3.1.3. Recurrent Neural Networks

Recurrent neural networks (RNNs) [24] are a type of neural network designed to process sequence data, making them particularly suitable for time series prediction and natural language processing tasks. The basic unit of RNNs consists of an input layer, hidden layers, and an output layer. Unlike feedforward neural networks, the hidden layers of RNNs have recurrent connections. At each time step, the hidden layers receive both the input of the current time step and the hidden state of the previous time step. This recurrent structure enables RNNs to capture dynamic changes in sequence data. At each time step, the hidden layers of RNNs update their state as shown in the following equation:

$$h_t=f(W_{xh}{x}_t+W_{hh}{h}_{t-1})$$

(6)

Here, $h_t$ is the hidden state at time t; $h_{t-1}$ is the hidden state at time $t-1$; $W_{xh}$ and $W_{hh}$ are learnable weight matrices; $x_t$ is the input at the current time step; f is typically a nonlinear activation function.

To address the gradient vanishing and exploding problems in traditional RNNs, long short-term memory (LSTM) [25] and gated recurrent unit (GRU) [26] introduce gating mechanisms to control the flow of information. GRU is a simplified version of LSTM, using only two gates (update gate and reset gate), reducing the model complexity while still capturing long-term dependencies effectively. The classic structure of a GRU network is shown in Figure 5.

The reset gate $r_t$ controls how much past information influences the current candidate hidden state. Its calculation is given by Equation (7). The current candidate hidden state ${\tilde{h}}_t$ is computed as shown in Equation (8). The update gate $z_t$ manages how much information from the previous hidden state is retained and how much from the new candidate hidden state is incorporated into the current hidden state. Its calculation is provided in Equation (9), and the calculation for the current hidden state $h_t$ is shown in Equation (10).

$$r_t=\sigma (W_r·\left[h_{t-1},x_t\right])$$

(7)

$${\tilde{h}}_t=tanh(W_h·[r_t\circ h_{t-1},x_t])$$

(8)

$$z_t=\sigma (W_z·\left[h_{t-1},x_t\right])$$

(9)

$$h_t=(1-z_t)\circ h_{t-1}+z_t\circ {\tilde{h}}_t$$

(10)

Here, $W_r$ is the weight matrix of the reset gate, $x_t$ represents the input vector at time t, and $\sigma$ denotes the sigmoid activation function, whose output ranges between 0 and 1. $W_h$ is the weight matrix for the candidate hidden state, ∘ denotes element-wise multiplication, and tanh represents the hyperbolic tangent activation function. $W_z$ is the weight matrix for the update gate. The functional expressions for the sigmoid and tanh activation functions are provided in the Equations (11) and (12), respectively.

$$\sigma \left(x\right)={\displaystyle \frac{1}{1+e^{-x}}}$$

(11)

$$tanh\left(x\right)={\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}}}$$

(12)

3.2. Temperature Distribution Prediction Method Based on ROM

This section combines the POD and neural network models to predict the three-dimensional temperature distribution of the hot-air anti-icing system. First, POD is applied to the temperature distribution to obtain the basis mode matrix and fitting coefficients. Then, key design variables serve as inputs for training the neural network, with the predicted fitting coefficients as outputs. Finally, data reconstruction combines the basis mode matrix with these predicted fitting coefficients to yield the final temperature distribution prediction results. The technical workflow is shown in Figure 6.

• Data acquisition and preprocessing stage: First, the LHS method is used to sample, generating 5000 design variable samples of the piccolo tube. Then, the three-dimensional anti-icing numerical simulation method calculates the surface temperature of the hot-air anti-icing cavity in the turbofan engine inlet, resulting in the formation of the temperature distribution matrix. Subsequently, the POD method, described in Section 3.1.1, reduces data dimensionality on the temperature distribution matrix, obtaining the basis mode matrix and the matrix of fitting coefficients. During this process, the top 128 eigenvectors, chosen based on the magnitude of their eigenvalues, are compiled into the basis mode matrix. The truncated number, 128, is determined based on the cumulative energy ratio of the modes, as shown in Figure 7.
• Network model training and predicting stage: As illustrated in Figure 8, during the training of the neural network model, the design variables serve as inputs, while the decomposed fitting coefficients are the target values. For predictions, the trained neural network uses the provided design variables to predict the fitting coefficients.
• Data post-processing stage: The predicted fitting coefficients from the neural network, combined with the basis mode matrix obtained from the POD decomposition, are used for data reconstruction, resulting in the final predicted temperature distribution.

3.3. Temperature Distribution Prediction Method Based on High-Dimensional Data

In this section, a neural network model predicts the anti-icing surface temperature distribution using structural design variables as inputs, with the surface temperature distributions as the outputs. The neural network architecture MCG proposed in this section primarily consists of two-dimensional convolution layers (Conv), two-dimensional transposed convolution layers (ConvTranspose), GRU layers, LeakyReLU activation layers, fully connected layers (Linear), and other structures. Figure 9 illustrates the network structure of MCG.

MCG includes two different paths to process input data:

• Path 1: The initial part of this path expands a 1 × 6 input vector into a 32 × 32 matrix through a fully connected layer and a reshaping operation, facilitating subsequent processing. Subsequently, a CNN block composed of multiple convolution layers is applied to capture features and patterns in the expanded vector. Following the CNN block, a TCNN block, made up of several transposed convolution layers, aims to enlarge the dimensions of the feature map while retaining feature correlations. Finally, a fully connected layer processes the output to promote high-level abstraction and feature aggregation.
• Path 2: This path employs three layers of GRU to extract relevant features from the input sequence, effectively capturing patterns and correlations within it. It learns the mapping relationship between the anti-icing design variables and the temperature distribution at the z = 0 m position.
• Integration and output: The output of Path 1 is reshaped into a 198 × 121 format. Then, the central column is replaced with the output of Path 2 to obtain the final result.

All activation functions in the model use LeakyReLU, which is defined by the following equation:

$$f\left(x\right)=\left\{\begin{array}{c}\hfill x,x>0\\ \hfill \alpha x,x\le 0\end{array}\right.$$

(13)

The LeakyReLU function, with its non-zero output in the negative region, helps prevent neuron death that can occur with ReLU, enabling the network to learn a broader range of features. Additionally, compared to other more complex activation functions, LeakyReLU is computationally simpler, thus providing higher efficiency during training.

4. Experiments and Results Analysis

This paper employs the PyTorch framework to build prediction network models. The computer configuration used for model development and training includes an Intel (R) Core (TM) i9-10900 CPU running at 2.80 GHz × 20, 32 GB of RAM, and an NVIDIA GeForce RTX 3080 Ti graphics card. Both network training and prediction are performed using the GPU.

4.1. Loss Function

SmoothL1Loss is chosen as the loss function during the training process, and its functional expression is as follows:

$$\mathrm{SmoothL1Loss}(x,y)=\left\{\begin{array}{c}\hfill 0.5{(x-y)}^2,|x-y|<1\\ \hfill |x-y|-0.5,|x-y|\ge 1\end{array}\right.$$

(14)

When the difference between the predicted value x and the true value y is less than 1, SmoothL1Loss behaves as L2 loss (squared error), making it more sensitive to small errors. When the difference is greater than or equal to 1, the loss behaves as L1 loss (absolute error), thereby reducing the impact of larger errors.

4.2. Performance Metrics

In the experiments, the root mean square error (RMSE), mean relative error (MRE), and mean absolute error (MAE) are chosen as performance metrics to compare the performances on the test set of various prediction methods. The equations for each metric are as follows:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{m}}\sum _{i=1}^m\left(\sum _{j=1}^l{(T_{i,j}-{\widehat{T}}_{i,j})}^2\right)}$$

(15)

$$\mathrm{MRE}={\displaystyle \frac{1}{m}}\sum _{i=1}^m\sum _{j=1}^l\left|{\displaystyle \frac{T_{i,j}-{\widehat{T}}_{i,j}}{T_{i,j}}}\right|$$

(16)

$$\mathrm{MAE}={\displaystyle \frac{1}{m}}\sum _{i=1}^m\sum _{j=1}^l\left|T_{i,j}-{\widehat{T}}_{i,j}\right|$$

(17)

Here, m is the total number of samples, l is the total number of grids, and $T_{i,j}$ and ${\widehat{T}}_{i,j}$ represent the target and predicted values of i-th sample at the j-th grid, respectively. In order to better evaluate the established method, the mean prediction accuracy (MPA) [27] is proposed:

$$\mathrm{MPA}={\displaystyle \frac{1}{m}}\sum _{i=1}^m\left({\displaystyle \frac{N_{s,i}}{l}}\times 100\%\right)$$

(18)

Here, an absolute error less than s serves as the standard for accuracy. If this standard is met, the temperature prediction for the grid point is deemed accurate. Let $N_{s,i}$ denote the number of grids with accurate predictions for the i-th sample. With $s=5$, $N_{s,i}/l$ represents the prediction accuracy (PA) of the i-th sample. The PA for all test samples is averaged to calculate the MPA on the test set.

4.3. Temperature Distribution Prediction Based on ROM

4.3.1. Comparative Test Results Analysis of ROMs

This section compares multiple classic neural networks, including AlexNet, UNet, ResNet, LSTM, GRU, multi-layer perceptron (MLP) [28], and variational autoencoder (VAE) [29], in predicting POD fitting coefficients and evaluates their performance on the test set. Figure 10 illustrates the variation in loss values during the ROM training process.

All models have reached convergence, indicating their training losses have stabilized after a certain number of epochs. However, the extent to which the loss has decreased varies across different models. The ROMs based on GRU, LSTM, and VAE, respectively, show little improvement in their training losses, suggesting that these models may not have effectively learned the underlying data patterns as effectively as others. The POD-MLP reaches a moderate level after convergence. In contrast, the ROMs based on ResNet, UNet, and AlexNet, respectively, show the most significant reduction in training losses, with rapid and effective convergence. Notably, POD-AlexNet stands out by reducing the loss more rapidly than other networks and consistently maintains the lowest loss value throughout the training process, which further highlights its learning efficiency compared to other models.

Table 3. Performance evaluation results of methods based on ROM.

Networks	RMSE	MRE	MAE	MPA	${\mathit{t}}_{\mathit{test}}$
POD-MLP	3.27	0.80%	2.44	88.25%	4.0 ms
POD-LSTEM	11.19	2.46%	7.74	57.28%	0.7 ms
POD-GRU	8.98	2.07%	6.45	58.81%	0.4 ms
POD-VAE	4.16	0.93%	2.87	83.57%	0.4 ms
POD-UNet	4.41	1.15%	3.43	78.48%	1.2 ms
POD-ResNet	2.81	0.69%	2.11	90.86%	0.5 ms
POD-AlexNet	1.99	0.47%	1.45	95.83%	1.0 ms

Note: Bold shows the performance evaluation results of the model proposed in this study.

presents the test performance evaluation results of different ROMs, where $t_{test}$ denotes the average prediction time for a single sample. The comparative results indicate that convolution-based models outperform sequence-based models in predicting the fitting coefficients, which are essential for reconstructing the temperature distribution. This disparity arises from the inherent characteristics of these models and the nature of the data involved.

LSTM and GRU are well-suited for sequence data, excelling at capturing temporal dependencies [30]. However, the primary challenge in this study is not modeling temporal sequences but accurately capturing spatial patterns in the fitting coefficients. LSTM and GRU struggle to model these spatial relationships effectively, which results in suboptimal performance.

POD-MLP and POD-VAE exhibit weaker performances. MLP, as a fully connected model, struggles to efficiently capture spatial dependencies inherent in the data [31], leading to slower and less accurate predictions. On the other hand, VAE, which aims to learn a probabilistic latent representation, often suffers from information loss and tends to produce overly smooth outputs [32,33]. This limitation hinders its ability to capture fine spatial details in the temperature distribution.

In contrast, convolution-based models like AlexNet are inherently better suited for spatial data. The POD-AlexNet method developed in this paper, despite its relatively simple architecture, demonstrates significant improvements across all performance metrics. Furthermore, the average prediction time is just 1 ms, considerably lower than that of traditional numerical simulation methods.

4.3.2. Prediction Results and Analysis

Figure 11 compares the predicted results of POD-AlexNet with the target results. The three contour plots on the left illustrate the predicted results, target results, and absolute errors of the surface temperature distribution. In these plots, the vertical coordinate s = 0 m represents the stagnation point at the leading edge of the inlet. Positive values represent the upper surface of the inlet, while negative values correspond to the lower surface. The line graph on the right illustrates the predicted and target temperature results at z = 0 m (the 12 o’clock position, as shown in Figure 1). It can be observed that the predicted results align closely with the target results, and the predicted values, along with their change patterns at various positions on the anti-icing surface, are notably accurate.

4.4. Temperature Distribution Prediction Based on High-Dimensional Model

4.4.1. Comparative Test Results Analysis of High-Dimensional Model

This section conducts comparative experiments on several classic neural networks, including LeNet, AlexNet, UNet, VAE, LSTM, and GRU, as well as our proposed model, MCG. Their effectiveness and superiority are evaluated by comparing their performance on a test set.

Figure 12 illustrates the variation in loss values during the high-dimensional model training process. In this figure, some of the first epochs have been ignored for the sake of a graphical representation of data. All models’ training losses have converged, but there are some variation in their final loss values. The VAE exhibited the largest loss after convergence, followed by UNet with the second-largest loss, showing slightly better performance than VAE, but still not ideal compared to the other models. AlexNet, LeNet, LSTM, and GRU all converged to similar loss values, occupying the middle ground, which suggest that the performance of these models is comparable. The MCG model demonstrated the greatest reduction in loss and achieved the best convergence, maintaining superior performance throughout the training process.

Table 4. Performance evaluation results of methods based on high-dimensional model.

Networks	RMSE	MRE	MAE	MPA	${\mathit{t}}_{\mathit{test}}$
LeNet	2.85	5.26‰	1.66	91.76%	7.5 ms
AlexNet	2.80	4.98‰	1.59	92.23%	8.5 ms
UNet	2.84	5.26‰	1.66	91.82%	1.3 ms
VAE	3.62	6.40‰	2.05	88.46%	4.6 ms
LSTM-3L	2.56	4.68‰	1.48	93.18%	5.4 ms
GRU-3L	2.47	4.56‰	1.45	93.64%	5.5 ms
MCG	1.75	3.23‰	1.02	96.97%	5.5 ms

Note: Bold shows the performance evaluation results of the model proposed in this study.

presents the performance evaluation results for various networks. Comparative tests reveal that sequence-based models, such as LSTM and GRU, outperform convolution-based models like LeNet, AlexNet, and UNet when directly predicting high-dimensional temperature distribution data. Among all models, VAE exhibits the lowest accuracy. These performance differences arise from the distinct characteristics of the models and the nature of the high-dimensional data.

Convolution-based models like LeNet, AlexNet, and UNet are highly effective in tasks dominated by local spatial features, such as image classification or segmentation, as shown in numerous studies [19,21,22]. However, predicting high-dimensional temperature distributions in this study requires capturing complex dependencies across the entire spatial field, which may limit the effectiveness of convolutional models that prioritize local features extraction.

Conversely, LSTMs and GRUs capture long-term dependencies and patterns across data effectively [34]. Their gating mechanisms dynamically adjust the importance of different data points, which make them particularly suited for handling complex, high-dimensional data [32,33]. This likely explains their superior performance over convolutional models, which focus on local feature extraction and may overlook broader, non-local dependencies.

VAEs encode data into a latent space and then decode it, making them more prone to information loss, particularly in high-dimensional tasks [35,36]. In addition, VAEs aim to learn the data’s probability distribution rather than directly optimize prediction accuracy, often resulting in overly smooth outputs. These characteristics make VAEs less effective at capturing fine details in temperature distributions, potentially explaining their weaker performance in experimental results.

Based on the above experimental and analytical results, this study proposes the MCG model, which combines convolutional networks and GRUs in parallel, leveraging the advantages of both. The convolutional networks excel at extracting spatial features, while the GRU captures long-term dependencies in high-dimensional temperature distribution data. The results indicate that this combination significantly enhances prediction performance. The MCG model achieves notable gains in RMSE, MRE, MAE, and MPA on the test set. Moreover, its average prediction time per sample is about 5.5 ms—much faster than traditional numerical simulations that typically take hours or even days, marking a substantial improvement in computational efficiency.

4.4.2. Prediction Results and Analysis

Figure 13 compares the predicted results of MCG with the target results. The three contour figures on the left display the predicted results, target results, and absolute errors of the surface temperature distribution. The vertical coordinate s = 0 m in the figure represents the stagnation point at the leading edge of the inlet. Positive values represent the upper surface of the inlet, while negative values represent the lower surface. The line graph on the right shows the predicted and target temperature results at z = 0 m (the 12 o’clock position as shown in Figure 1). The predicted results align well with the target results. The predicted values and change patterns at various positions on the anti-icing cavity surface are accurately predicted.

4.5. Comparative Analysis of POD-Alexnet and MCG

This section compares two proposed methods for predicting the surface temperature distribution through experiments. The results indicate that both methods achieve high prediction accuracy but emphasize different aspects.

Figure 14 compares the predicted absolute errors of the two methods, POD-AlexNet and MCG, using the same samples. As indicated by the comparative figures, the error distribution of the POD-AlexNet is relatively even, with no particular local absolute errors standing out significantly. In contrast, the prediction of the MCG exhibits regions with high-error points. However, aside from these regions, MCG’s overall error distribution is lower than that of POD-AlexNet. Further analysis suggests that the errors in the POD-AlexNet primarily result from truncating the basis modes and inaccuracies in the neural network’s prediction of fitting coefficients. Since each basis mode captures specific temperature distribution features across the entire three-dimensional surface, especially the first few modes, errors in the fitting coefficients propagate across the surface, leading to a relatively uniform distribution of prediction errors at each grid point.

5. Conclusions

Two models for predicting the surface temperature distribution of aircraft hot-air anti-icing systems are developed in this paper. Specifically, a prediction model is based on POD and AlexNet, termed POD-AlexNet, and another model is based on high-dimensional data, termed MCG. Additionally, extensive comparative experiments are conducted to validate the effectiveness of these models, leading to the following conclusions:

(1)

The POD-AlexNet model enables rapid predictions of the POD fitting coefficients and obtains anti-icing temperature distributions by reconstructing the POD basis modes based on these fitting coefficients. By selecting the appropriate neural network model, AlexNet, the RMSE of test samples is less than 2, the MRE is less than 0.5%, the MAE is less than 1.5, and the MPA is higher than 95%. In addition, the time cost for predicting each sample is about 1 ms, achieving fast and efficient prediction.

(2)

The MCG model enables the rapid and direct predictions of anti-icing surface temperature distributions. It achieves an RMSE of 1.75 on the test set, an MRE of 3.23‰, an MAE of 1.02, and an MPA of 96.97%. Additionally, the average single-sample prediction time is about 5.5 ms, which is a significant improvement compared to the traditional numerical simulation method, which takes hours or even days.

(3)

The error distribution of POD-AlexNet is relatively uniform; in contrast, the MCG exhibits localized high-error points. However, the overall error distribution of the MCG is significantly lower than that of the POD-AlexNet. In general, the POD-AlexNet and MCG models can provide faster predictions of anti-icing surface temperature distributions than traditional numerical simulation methods with acceptable error, which supports the design of aircraft hot-air anti-icing systems based on optimization methods such as genetic algorithms.