Optimizing Aircraft Route Planning Based on Data-Driven and Physics-Informed Wind Field Predictions

Abstract

Upper-air wind fields play a crucial role in aircraft navigation, directly impacting flight safety and operational efficiency. In this study, we propose an advanced route planning framework that integrates wind field predictions derived from a neural network-based approach. Specifically, we leverage the PredRNN Sequence-to-Sequence algorithm to predict wind fields up to 10 h in advance. The model is trained on grid-based wind speed data at an altitude of approximately 5500 m, focusing on major airline routes over China. Our approach demonstrates superior accuracy in wind field forecasting when compared to other neural network architectures. To achieve route planning in dynamic wind environments, we employ the A* algorithm. The results demonstrate that the proposed method effectively identifies routes that approximate the ideal trajectory while successfully avoiding areas with drastic wind speed changes, thereby enhancing both the efficiency and safety of flight operations.

1. Introduction

Wind speed is a crucial meteorological factor that influences aircraft operations [1,2,3]. Near the ground, decisions regarding landing and takeoff are strongly influenced by prevailing wind conditions at airports [4,5,6]. At higher altitudes, wind plays a significant role in fuel consumption and flight durations, particularly for long-haul passenger routes [7]. Therefore, accurate wind speed prediction is crucial for ensuring efficiency and safety in aviation operations.

The traditional method for wind forecasting is numerical weather prediction (NWP) [8,9], which predicts future meteorological conditions based on historical data by solving equations derived from fluid mechanics and thermodynamics. However, it faces challenges in short-term forecasting, particularly for periods under six hours. Jacobs’s research [6] incorporates statistical and physical post-processing of the NWP model, integrating data from both local and upstream station observations. Furthermore, prediction methods that incorporate satellite observations [10,11,12] have been developed to enhance forecasting accuracy. For instance, the Doppler wind lidar on the Aeolus satellite allows for direct measurements of vertical profiles of horizontal winds, as well as other thermodynamic and dynamic parameters [13]. On the other hand, the equation-solving process relies on supercomputers and is highly computationally intensive, often resulting in delays. Consequently, NWP is less suitable for real-time applications.

Unlike NWP, which faces challenges in short-term accuracy due to limited real-time data integration, neural networks leverage large datasets and real-time inputs to improve short-term wind field predictions [14]. Additionally, they offer greater flexibility in handling diverse input data and excel at capturing multi-scale features within wind field data [15,16]. A significant body of contemporary research has acknowledged the advantages of neural networks in predicting wind speed and other meteorological parameters. In the study by Y. D. Mammedov [17], wavelet transforms and recurrent neural networks were utilized as innovative methods to improve the accuracy of wind speed predictions for energy analysis. However, most of these studies relied on discrete weather station measurements as their primary datasets [17,18], whereas route planning requires predictions that produce a more uniform grid of wind fields.

Meanwhile, greater emphasis has been placed on the accuracy in time-serial prediction of neural networks [19]. In the study by W. F. Xu [20], the Long Short-Term Memory (LSTM) neural network [21] was used to correct errors that arise after applying the Weather Research and Forecasting (WRF) model [22] with Runge–Kutta integration schemes. While this approach ensures that the prediction aligns with numerical simulations and real-world observations, the computational cost of numerical calculations remains high. It has been demonstrated that using neural networks requires less computational power and manual supervision compared to using a RANS solver with a GPU for flow-field calculations [23]. To further enhance physical interpretability, physics-informed neural networks (PINNs) [24,25] are constructed, where the loss function is based on the partial differential equation (PDE) to be approximated [26]. Choi [27] developed a surrogate model of a continuous stirred tank reactor (CSTR) with van de Vusse reaction using PINN, training the governing equations of the system to improve both accuracy and computational efficiency. Similarly, Hamidreza [28] applied PINN with Navier–Stokes equations to super-resolve flow-field data in both time and space, effectively enhancing resolution and reducing noise in sparse experimental measurements. Darioush [29] utilized PINN by integrating relevant physical laws into the network’s loss function, achieving improved tracking accuracy of phase interfaces and velocity fields, with a maximum error of just 5.2% at the interface. These studies collectively demonstrate that PINNs significantly enhance prediction accuracy and physical interpretability.

While simpler neural network-based PINNs often require the use of Runge–Kutta methods to iteratively process time dimensions, more complex neural networks can perform function approximation and time iteration simultaneously, avoiding the need to separate these processes. For instance, Mammedov et al. [17] introduced a new loss function based on the governing equation of the Lorentz chaotic system for their echo state network, enabling efficient predictions in dynamic systems. Similarly, Z. H. Chen [30] developed a generative neural network with a mask loss function based on the convection–diffusion equation, incorporating a jump pattern strategy to improve tropospheric temperature prediction.

Building upon these advancements, the Predictive Recurrent Neural Network (PredRNN) [31,32] offers notable advantages over traditional approaches, such as stacked LSTMs and other recurrent architectures, particularly in spatiotemporal prediction tasks. Its unique memory flow mechanism enables efficient interaction between low-level and high-level features, allowing the model to capture both short-term dynamics and long-term dependencies—both critical for accurate wind field predictions. By decoupling inter-layer spatiotemporal memory from intra-layer temporal memory, PredRNN provides a more effective representation of complex temporal and spatial variations, surpassing stacked LSTMs in capturing the intricate dynamics of wind fields.

In this paper, we propose a physics-informed prediction model based on PredRNN and further enhance a flight route optimization approach by integrating the improved A* algorithm. Wind field grid data from ERA5 are used as input for model training to predict short-term upper-air wind fields, while physics-based methods derived from the Navier–Stokes equations are employed to construct the loss function. Subsequently, we employ the A* algorithm, incorporating wind speed effects, to optimize the flight path. The results demonstrate that our proposed approach effectively reduces flight costs and enhances flight safety.

The paper is organized as follows: Section 2 introduces the integration of physical information from the Navier–Stokes equations into the training of PredRNN, along with an improved A* algorithm that incorporates wind speed. Section 3 presents the wind field prediction results obtained using the proposed model. Section 4 demonstrates the results of the route optimization. Section 5 provides a conclusion of the paper.

2. Methodology

The primary objective of this research is to develop a heuristic airplane route planning algorithm based on physics-informed neural network (PINN) wind field prediction. The overall architecture of the proposed method is illustrated in Figure 1.

In this study, we developed a neural network model based on the PredRNN architecture, which is widely used for spatiotemporal analysis, to predict the wind field. PredRNN aims to capture the modular structure of visual dynamics through compositional subsystems. For the input data, wind field data from time steps $t_1$ to $t_{10}$ in a specific area are used to provide sufficient information about the evolution of the wind field.

To improve the accuracy and reliability of the predictions, a physics-informed neural network (PINN) is utilized to derive the loss function. The PINN enables more physically accurate wind field predictions, providing a better approximation of the true conditions. The output data, including predictions from $t_{11}$ to $t_{20}$, are obtained based on the designed loss function.

Based on the wind field predictions, the A* algorithm is employed to perform route planning. A cost function is designed to quantify the impact of the wind field, consisting of two components: the distance cost and the wind force cost. The optimization process seeks to strike a balance between these two costs while incorporating the wind field predictions. As a result, the optimized route deviates from the direct line connecting the starting and ending points in order to minimize overall operational costs and risks.

2.1. PredRNN-Based Wind Field Prediction

In this study, we developed a neural network model for predicting upper-air wind fields. The Predictive Recurrent Neural Network (PredRNN) [31] is an advanced framework specifically designed to learn spatial and temporal dependencies in sequential data. PredRNN extends the Convolutional Long Short-Term Memory (ConvLSTM) network by retaining its convolutional operations for spatial pattern recognition while introducing a novel parameter, $M_t^l$. This parameter enhances the model’s ability to capture spatiotemporal memory, as illustrated in Figure 2a.

The computations within the Spatiotemporal LSTM unit are expressed as follows:

(1)$\begin{array}{c}\hfill \begin{array}{cc}\hfill g_t& =tanh({\mathcal{W}}_{xg}\ast {\mathcal{X}}_t+{\mathcal{W}}_{hg}\ast {\mathcal{H}}_{t-1}^l+b_g),\hfill \\ \hfill i_t& =\sigma ({\mathcal{W}}_{xi}\ast {\mathcal{X}}_t+{\mathcal{W}}_{hi}\ast {\mathcal{H}}_{t-1}^l+b_i),\hfill \\ \hfill f_t& =\sigma ({\mathcal{W}}_{xf}\ast {\mathcal{X}}_t+{\mathcal{W}}_{hf}\ast {\mathcal{H}}_{t-1}^l+b_f),\hfill \\ \hfill {\mathcal{C}}_t^l& =f_t\circ {\mathcal{C}}_{t-1}^l+i_t\circ g_t,\hfill \\ \hfill g_t^{\prime }& =tanh({\mathcal{W}}_{xg}^{\prime }\ast {\mathcal{X}}_t+{\mathcal{W}}_{mg}\ast M_t^{l-1}+b_g^{\prime }),\hfill \\ \hfill i_t^{\prime }& =\sigma (W_{xi}^{\prime }\ast {\mathcal{X}}_t+{\mathcal{W}}_{mi}\ast M_t^{l-1}+b_i^{\prime }),\hfill \\ \hfill f_t^{\prime }& =\sigma (W_{xf}^{\prime }\ast {\mathcal{X}}_t+{\mathcal{W}}_{mf}\ast M_t^{l-1}+b_f^{\prime }),\hfill \\ \hfill M_t^l& =f_t^{\prime }\circ M_t^{l-1}+i_t^{\prime }\circ g_t^{\prime },\hfill \\ \hfill o_t& =\sigma ({\mathcal{W}}_{xo}\ast {\mathcal{X}}_t+{\mathcal{W}}_{ho}\ast {\mathcal{H}}_{t-1}^l+{\mathcal{W}}_{co}\ast {\mathcal{C}}_t^l+{\mathcal{W}}_{mo}\ast M_t^l+b_o),\hfill \\ \hfill {\mathcal{H}}_t^l& =o_t\circ tanh({\mathcal{W}}_{1\times 1}\ast [{\mathcal{C}}_t^l,M_t^l]).\hfill \end{array}\end{array}$

Unlike conventional ConvLSTM architectures, where parameters are propagated either within the same layer across time steps or deeper into the network within the same time step (see Figure 2b, gray flow), PredRNN allows its spatiotemporal memory $M_t^l$ to propagate in a zigzag manner (see Figure 2b, orange flow). This zigzagging memory flow ensures that the lowest layer at the next time step can access information from the deepest layer of the previous time step, enabling the model to track spatial features as they evolve over time and preventing the loss of critical details from deeper layers.

In this study, we employ PredRNN within an encoder–decoder framework, where the input consists of the u- and v-component wind fields from time steps $t_1$ to $t_{10}$. The encoder processes these past wind field sequences, capturing their spatiotemporal patterns. The decoder then uses the encoded information to predict the wind fields in both the u- and v-directions for future time steps, from $t_{11}$ to $t_{20}$. This setup enables the model to capture the complex spatiotemporal interactions within the wind fields.

2.2. PINN

PINN methods integrate the governing partial differential equations (PDEs) directly into the model, enabling wind field predictions based on physical principles.

Given that the cruising altitude of civil aviation aircraft is approximately 5500 m, the incompressible Navier–Stokes equations can be used to describe the atmospheric motion at this altitude. Since most of the motion occurs predominantly in the horizontal directions, u and v, with minimal vertical convection, the vertical velocity component, w, can be neglected. For data on the same isobaric surface, the governing equations are expressed as follows:

(2)$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}-fu=0,\hfill \\ \hfill & {\displaystyle \frac{\partial v}{\partial t}}+u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}-fv=0.\hfill \end{array}\end{array}$

where f is the Coriolis force coefficient, defined as $f=2\mathrm{\Omega }sin\left(\varphi \right)$. Here, $\mathrm{\Omega }$ is the angular velocity of the Earth, with $\mathrm{\Omega }={15}^{\circ }/\mathrm{h}$, and $\varphi$ is the latitude [33].

The main idea of PINNs is to incorporate the integral form of the governing PDEs into the loss function. M. Raissi [26] demonstrated examples of embedding the Schrödinger equation, Navier–Stokes equation, and Burger’s equation into the loss function, where the parameters are learned by minimizing the mean squared error loss that combines both the initial data and the PDE components. The loss function of a PINN typically includes contributions from the PDE, measurements, initial conditions, and boundary conditions, each with corresponding weighting coefficients. For the PDE loss, automatic differentiation is used to compute the partial derivatives.

To align the wind prediction results of the neural network with the laws of physics, we incorporate Equations (2) into the loss function. This leads to the construction of new functions,

(3)$\begin{array}{c}\hfill \begin{array}{cc}\hfill & F_u={\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}-fu,\hfill \\ \hfill & F_v={\displaystyle \frac{\partial v}{\partial t}}+u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}-fv.\hfill \end{array}\end{array}$

Thus, the loss function is formulated based on the mean squared error (MSE):

(4)$MSE=MS{E}_u+MS{E}_v+MS{E}_{F_u}+MS{E}_{F_v},$

where

(5)$\begin{array}{c}\hfill \begin{array}{cc}\hfill & MS{E}_u={\displaystyle \frac{1}{N}}\sum _{i=1}^N{(u_i-{\widehat{u}}_i)}^2,\hfill \\ \hfill & MS{E}_v={\displaystyle \frac{1}{N}}\sum _{i=1}^N{(v_i-{\widehat{v}}_i)}^2,\hfill \\ \hfill & MS{E}_{F_u}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{(F_{u_i}-{\widehat{F}}_{u_i})}^2,\hfill \\ \hfill & MS{E}_{F_v}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{(F_{v_i}-{\widehat{F}}_{v_i})}^2.\hfill \end{array}\end{array}$

Here, N represents the total number of data points used in the calculation.

2.3. Self-Adapted Route Planning

Based on the wind field prediction data, self-adaptive route planning is achieved by implementing the A* algorithm [34]. The A* algorithm is an extension of Dijkstra’s algorithm. Unlike Dijkstra’s algorithm, which lacks knowledge of the overall grid structure, the A* algorithm prioritizes nodes that lead toward the destination, avoiding unnecessary exploration of nodes that move away from it, thus reducing computational overhead. The A* algorithm is constructed by incorporating a heuristic function. Its cost function is defined as follows:

(6)$\begin{array}{c}\hfill F\left(n_k\right)=H\left(n_k\right)+\sum _{i=1}^{k}G\left(n_i\right),\end{array}$

where $F\left(n_k\right)$ is the cheapest cost to go to the k-th node from the starting node $n_1$, and $G\left(n_i\right)$ is the cost of reaching the i-th node $n_i$ from the previous node $n_{i-1}$ in the path. The heuristic function $H\left(n_k\right)$ represents the estimated cost from $n_k$ to the destination node $n_m$. In this research, $H\left(n_k\right)$ is computed based on the diagonal distance. This adjustment facilitates the progressive elimination of longer paths during the searching process, effectively narrowing down the nodes that must be examined before arriving at an optimal solution.

In aircraft fuel consumption, fuel emissions are positively correlated with the duration of each phase of flight [35]. To account for this, the wind speed is combined with the cost function (6) through speed decomposition and composition; see Figure 3. When considering the wind speed, $H\left(n_k\right)$ and $G\left(n_i\right)$ are given by

(7)$\begin{array}{c}\hfill \begin{array}{cc}\hfill H\left(n_k\right)& =\sum _{i=k+1}^m{d}_i{\displaystyle \frac{v_{airspeed}}{v_{ground}}},\hfill \\ \hfill G\left(n_i\right)& =d_i{\displaystyle \frac{v_{airspeed}}{v_{ground}}},\hfill \end{array}\end{array}$

where $d_i$ denotes the distance between nodes $n_{i-1}$ and $n_i$ in spherical coordinates, and $v_{airspeed}$ represents the speed of the aircraft flying in a vacuum. $v_{ground}$ is the aircraft’s actual speed relative to the ground, and it is the resultant speed composed of $v_{airspeed}$ and $v_{wind}$.

3. Wind Field Predictions

3.1. Training Datasets

The training datasets are sourced from ERA5, an atmospheric reanalysis dataset provided by the European Centre for Medium-Range Weather Forecasts (ECMWF).

ERA5 provides global hourly isobaric climate data from 1979 to the present, covering 16 parameters, including temperature, humidity, wind speed in the u-component, wind speed in the v-direction, and geopotential height. Reanalysis data integrates model outputs with global observations to create a comprehensive dataset that is grounded in the laws of physics. At ECMWF, current forecasts and new observations are optimally combined every 12 hours, with predictions updated and refined to produce the most accurate estimates.

To gather observation data from a wider range of sources for more accurate meteorological calculations, the release of reanalysis data is delayed by five days. In these experiments, we obtained 600 grid data sequences, with $80\%$ used for training, $10\%$ for validation, and $10\%$ for testing. The dataset contains three parameters: temperature, wind speed in the u-direction, and wind speed in the v-direction, covering the Chinese region $({73}^{\circ }\mathrm{E}$–${136}^{\circ }\mathrm{E},3^{\circ }\mathrm{N}$–${54}^{\circ }\mathrm{N})$ with a resolution of 0.5 degrees, along with the geopotential height at 500 hPa as the raw data.

3.2. Prediction Results Evaluation

We evaluated our model using the ERA5 dataset and compared the results with those obtained from the Conv-LSTM and PredRNN methods, excluding the integration of physical information. Our model takes 10 time steps as input and predicts the subsequent 10 time steps. For time-series prediction, the results are assessed using the mean squared error (MSE) and Structural Similarity Index (SSIM). The details of the comparison models are summarized in Table 1.

3.2.1. Visualization of Actual Wind Field and Prediction

Figure 4 illustrates the predicted u- and v-direction wind fields using the PINN model. A comparison between the ground truth and the predicted wind fields shows that the PINN model effectively captures both the spatial and temporal patterns of the wind fields. Minor discrepancies are observed in regions with abrupt changes in wind speed and direction.

3.2.2. Analysis of Predicted Wind Speed

Figure 5 illustrates the mean and range of MSE among the test set results. The figure demonstrates that the predictions obtained using the PredRNN framework outperform those from ConvLSTM, particularly in the later time steps of u-component wind prediction, highlighting PredRNN’s superior ability to capture long-term temporal variations and features. For the u-component wind, predictions using the PINN exhibit a lower MSE in later time steps and a reduced error range, as well as higher structural similarity (SSIM) in earlier time steps, indicating better consistency in both accuracy and spatial features. For the v-component wind, the PINN achieves a predictive performance comparable to that of PredRNN, with a smaller MSE and a higher SSIM during the first four time steps, showcasing its strength in maintaining both numerical accuracy and structural similarity for short-term predictions.

In the subsequent route planning, we selected the predicted wind field of the first time step for a specific test point in the testing set to calculate the shortest path. The mean and range of absolute error for predictions at each time step for this test point are shown in Figure 6. In the u-component wind field, the absolute wind speed values are larger, leading to more significant differences among the models’ predictions. In contrast, in the v-component wind field, the absolute wind speed values are smaller, resulting in relatively minor differences among the models. From the results of the u-component wind field, it is evident that the mean and range of absolute error for the PINN model are smaller than those of PredRNN and ConvLSTM. This indicates that the PINN demonstrates superior performance in predicting larger wind speed values, achieving higher accuracy and stability in such scenarios.

4. Route Planning Optimization

After obtaining the predicted wind field, we selected two flight routes between the points with coordinates $(37,94)$ and $(5,115)$, one for each direction, to simulate the shortest paths using the A* algorithm. The simulated routes traverse wind fields with frequent variations in wind speed and direction. By comparing the shortest paths (or costs) derived from the A* algorithm, as expressed in (6), under four conditions—real wind field at time $t-1$, real wind field at time t (where t corresponds to $t_{11}$ in the wind field prediction test points), predicted wind field at time $t-1$, and predicted wind field at time t—we aim to analyze the impact of the predicted wind field on route planning. The simulated routes passing through the real wind field at time $t-1$, the real wind field at time t, and the predicted wind field at time t are shown in Figure 7. As observed in the figure, the PINN-predicted wind field effectively captures the dynamic changes in wind speed but tends to smooth the boundaries of regions with uniform wind speeds.

The shortest path lengths in both directions under the three wind field scenarios are shown in Table 2. For the route from $(37,94)$ to $(5,115)$, the shortest path length planned using the predicted wind field at time t is shorter than the path lengths planned without considering wind fields or using the real wind field at time $t-1$. For the route from $(5,115)$ to $(37,94)$, the shortest path length planned using the predicted wind field at time t falls between the path lengths planned without considering wind fields and using the real wind field at time $t-1$.

We plotted the shortest paths under no wind field and the three wind field scenarios on the real wind field at time t for comparison, as shown in Figure 8. In the wind field map, the red area indicates the wind direction opposite the aircraft’s flight direction, representing headwind conditions, while the blue area indicates the wind direction aligned with the aircraft’s flight direction, representing tailwind conditions.

For route planning from $(37,94)$ to $(5,115)$, the u-direction wind speed is mostly aligned with the flight direction across most of the region, whereas the v-direction wind speed changes direction four times along the path. Among the shortest paths, those derived with wind field input avoid the headwind region near the endpoint in the v-direction wind field and primarily travel through areas with lower absolute wind speed compared to the path derived without wind field input. The shortest paths with wind field input are similar to each other, with minor differences in path length. This may be because the absolute value of the u-direction wind speed is larger than that of the v-direction, making the route planning more influenced by the u-direction wind speed. Since the u-direction wind speed is mostly tailwind with infrequent changes, the adjustments needed in route direction are minimal.

For route planning from $(5,115)$ to $(37,94)$, the u-direction wind speed is predominantly opposite to the flight direction, while the v-direction wind speed still undergoes four directional changes along the path. Except for the shortest path under the predicted wind field at time t, all paths cross regions with multiple v-direction wind shifts, potentially causing flight instability. In contrast, the shortest path under the predicted wind field at time t first travels through a tailwind region and then moves through areas with lower absolute wind speeds, avoiding zones with shifting wind directions or high headwind speeds, resulting in a smoother flight. Furthermore, the shortest path under the predicted wind field shows fewer directional changes. This may be due to the smoother boundaries between regions with the same wind speed in the predicted wind field, reducing the complexity of route planning.

Based on the results, it is evident that in routes with frequent wind speed changes, the predicted wind field generated by the PINN neural network may perform better than the historical wind field for route planning. It enables the selection of shorter and smoother paths. Given the strong correlation between an aircraft’s flight path length and fuel consumption, planning the shortest route based on the predicted wind field can effectively reduce flight costs. A smooth flight can also enhance safety and improve the passengers’ overall flight experience.

5. Conclusions

This study proposed an improved route planning method that integrates the A* algorithm with the wind field predicted by the physics-informed PredRNN model.

In wind field prediction, to address the shortcomings of traditional methods in terms of computational efficiency, this paper employed the PredRNN model with the advantage of spatiotemporal memory and improved the prediction accuracy and explanation by adding wind field physical constraints to the loss function. During the model comparison, this PredRNN, embedded with physical information, demonstrated superior predictive performance.

For route planning optimization, we designed a new cost function for the A* algorithm using the predicted wind field in order to account for the impact of wind speed on the flight route. The comparison of the planned routes under different embedding wind fields showed that the wind field predictions of our physics-informed PredRNN model led to ideal route planning results. These routes successfully avoided areas with drastic wind field changes and outperformed the routes planned with past wind fields in terms of fuel consumption.

Overall, this route planning optimization method achieved more fuel-efficient flight routes by combining high-accuracy wind field predictions in the planning process. It holds significant application potential and practical value, offering crucial support to airlines in reducing operating costs while enhancing flight safety.

Author Contributions

Conceptualization, J.M., Q.Y. and Z.J.; data curation, J.H. and H.L.; formal analysis, J.H.; investigation, J.H. and H.L.; methodology, J.M., P.X., Q.Y. and Z.J.; resources, Q.Y.; software, J.M.; supervision, Q.Y. and Z.J.; validation, H.L.; visualization, J.M.; writing—original draft, J.M.; writing—review and editing, P.X., Q.Y. and Z.J. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in this study are included in the article material. The dataset used in this research was derived from the following resources available in the public domain: https://cds.climate.copernicus.eu/datasets/reanalysis-era5-pressure-levels?tab=download, accessed on 7 December 2024.

Acknowledgments

We sincerely thank Zesen Chen, who previously studied at Sun Yat-sen University and is currently employed at Avic General Huanan Aircraft Industry Co., Ltd., for providing resources and his valuable discussions that meaningfully contributed to this research.

Conflicts of Interest

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

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Flowchart of wind field prediction and aircraft route planning.

Figure 2. PredRNN model.

Figure 3. Speed decomposition when expanding nodes.

Figure 4. PINN wind field prediction.

Figure 5. Comparison of wind field predictions across different models on the whole test set.

Figure 6. Comparison of wind field predictions across different models on the test point.

Figure 7. Comparison of A* paths with u- and v- direction wind fields.

Figure 8. Shortest paths in real wind field at t.

Table 1

Comparison of wind field prediction models.

	Models	Prediction Result Resolution
Model 1	Conv-LSTM	0.5 ° × 0.5 °
Model 2	PredRNN	0.5 ° × 0.5 °
Model 3	PINN (PredRNN with 1 × N-S Part of loss function)	0.5 ° × 0.5 °
Model 4	PINN (PredRNN with 10 × N-S Part of loss function)	0.5 ° × 0.5 °

Table 2

Lengths of the shortest paths for all three wind fields.

Start Point	End Point	No Wind Field	Real Wind Field at t − 1	Predicted Wind Field at t	Real Wind Field at t
(37, 94)	(5, 115)	4193.458 km	4200.286 km	4185.063 km	4177.322 km
(5, 115)	(37, 94)	4558.380 km	4592.404 km	4587.700 km	4534.746 km

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