Spatial–Temporal Physics-Constrained Multilayer Perceptron for Aircraft Trajectory Prediction

Abstract

Aircraft trajectory prediction (ATP) is a critical technology for air traffic control (ATC), safeguarding aviation safety and airspace resource management. To address the limitations of existing methods—kinetic models’ susceptibility to environmental disturbances and machine learning’s lack of physical interpretability—this paper proposes a Spatial–Temporal Physics-Constrained Multilayer Perceptron (STPC-MLP) model. The model employs a spatiotemporal attention encoder to decouple timestamps and spatial coordinates (longitude, latitude, altitude), eliminating feature ambiguity caused by mixed representations. By fusing temporal and spatial attention features, it effectively extracts trajectory degradation patterns. Furthermore, a Hidden Physics-Constrained Multilayer Perceptron (HPC-MLP) integrates kinematic equations (e.g., maximum acceleration and minimum turning radius constraints) as physical regularization terms in the loss function, ensuring predictions strictly adhere to aircraft maneuvering principles. Experiments demonstrate that STPC-MLP reduces the trajectory point prediction error (RMSE) by 7.13% compared to a conventional optimal Informer model. In ablation studies, the absence of the HPC-MLP module, attention mechanism, and physical constraint loss terms significantly increased prediction errors, unequivocally validating the efficacy of the STPC-MLP architecture for trajectory prediction.

1. Introduction

In Air Traffic Control (ATC), Aircraft Trajectory Prediction (ATP) has long been recognized as a critical technology for safeguarding aviation safety and managing airspace resources [1,2], which plays a pivotal role in conflict detection [3], traffic flow optimization [4,5], and emergency response operations [6]. Over the past several decades, economic prosperity and technological innovation have propelled explosive growth in civil aviation, driving sustained surges in air traffic—particularly amid global liberalization of low-altitude airspace regulations and widespread adoption of UAV (Unmanned Aerial Vehicle) technologies [7]. Such developments have exponentially amplified the complexity of airspace ecosystems, thereby creating urgent demand for high-precision trajectory prediction systems [8]. Notably, the International Civil Aviation Organization (ICAO) forecasts a 70% surge in global air traffic volume over the next decade [1,2], which mandates significant advancements in real-time dynamic trajectory prediction algorithms to address emerging operational challenges.

Essentially, aircraft trajectory prediction is a multivariate time series problem [9] that considers the aircraft’s current state and operational environmental factors. Its primary objective is to forecast the motion attributes of discrete trajectory points, including longitude, latitude, altitude, velocity, and others. Current trajectory prediction methods are broadly categorized into kinetics-and-aerodynamics models, state estimation models, and machine learning models [10]. Kinetics-and-aerodynamics models treat the aircraft as a point mass, focusing on the relationship between forces and motion [11,12]. By incorporating the aircraft’s current state, performance parameters, and meteorological conditions, these models achieve trajectory prediction. Rooted in Newtonian mechanics and aerodynamics, their explicit modeling approach enables long-term trajectory prediction with clear physical interpretability [13,14]. For instance, the Point-Mass Model (PMM) [15] exhibits stability advantages with linearly increasing errors during civil aviation cruise phases. However, this method’s heavy reliance on aircraft performance data—which is challenging to measure in real-world operational environments—yet collected from idealized test settings, significantly constrains its actual performance [16]. State estimation models employ state-space theory to mathematically represent physical systems, converting dynamical processes into probabilistic frameworks that handle sensor noise and motion uncertainty [17]. These models demonstrate strong noise robustness in real-time maneuvering target tracking. Classic examples include the Kalman Filter (KF) and its variants [18], Particle Filter (PF) [19], and the widely adopted Interacting Multiple Model (IMM) [20]. However, state estimation models are highly susceptible to environmental uncertainties [16], leading to error accumulation during multi-step prediction that ultimately degrades prediction accuracy [10].

Thanks to the rapid development of artificial intelligence, more and more researchers are utilizing machine learning models for aircraft trajectory prediction [21,22,23]. Wu et al. [24] proposed a regression model that mines historical flight data to identify influencing factors, predicts next-flight duration, and reconstructs aircraft positions per sampling period for complete 4D trajectory prediction. Wang et al. [25] employed fuzzy clustering analysis on approaching aircraft 4D trajectory data, establishing a theoretical foundation for optimizing approach/departure procedure design. Reference [26] leveraged Backpropagation (BP) neural networks to achieve high-precision four-dimensional trajectory prediction. To characterize uncertainty in ATC environments, the Bayesian neural networks [27,28] were implemented to explicitly incorporate environmental perturbations. Fundamentally, aircraft trajectory prediction constitutes a temporal sequencing problem. Capitalizing on Long-Short Term Memory (LSTM) networks’ proficiency in processing long sequences, Shi et al. [29] established a LSTM neural network for 4D trajectory forecasting. Meanwhile, Zhang et al. [30] proposed an LSTM network optimized by the Ant Lion Optimization algorithm for trajectory prediction. In addition, Ma et al. [31] proposed a hybrid architecture to extract the spatial–temporal features of the trajectory data by combining Convolutional Neural Network (CNN) and LSTM network. Zhang et al. [27] also innovatively introduced a hybrid Deep Neural Network–Long Short-Term Memory (DNN-LSTM) framework for trajectory prediction, integrating Bayesian methods to quantify predictive uncertainty and thereby enhance en-route flight safety.

As mentioned above, in the machine learning model, the trajectory prediction problem is transformed into the ordered time prediction problem. The machine learning model learns the hidden motion features from the massive trajectory data through a lot of training, and then realizes the trajectory prediction of aircraft. Although data-driven machine learning approaches have achieved certain success in the ATP task, the lack of interpretability remains unsolved, which prevents the implementation of machine learning in ATP [10,32]. For example, it is still a challenging task to examine the underlying flight patterns in sufficient details [33]. In general, the primary dynamics results from the aircraft maneuvering during the climb and descent phase, as well as the intention-driven operations.

To address the aforementioned challenges, this paper proposes a Spatial–temporal Physics-Constrained Multilayer Perceptron (STPC-MLP) model. In this method, the self-attention mechanism [34] is employed to allocate weights to different spatial features (longitude, latitude and altitude) as well as time steps to capture the spatial–temporal information. Adapted from DeepHPM [35], the hidden physics constrained neural network (HPC-MLP) is employed to distill the hidden physics relationship between the input data and the corresponding output data. Experimental results demonstrate that the STPC-MLP model reduces prediction error by 7.13% in RMSE, 40.38% in MSE, and 34.31% in MAPE compared to the state-of-the-art LSTM baseline. In ablation studies, the absence of the HPC-MLP module, attention mechanism, and physical constraint loss terms significantly increased prediction errors, unequivocally validating the efficacy of the STPC-MLP architecture for trajectory prediction.

The primary contributions of this paper are outlined as follows.

(1)

A novel spatial–temporal attention-based encoder is applied to find informative features in aircraft trajectories from both time and space dimension. By separately encoding timestamps and spatial coordinates (longitude, latitude and altitude), the model mitigates the feature ambiguity caused by mixed spatiotemporal representations in conventional methods. Simultaneously, by fusing the features of temporal attention and spatial attention, our model can extract degradation information more effectively and improve the prediction accuracy of trajectory points.

(2)

HPC-MLP is used to make predictions and capture the hidden physics between the input data and output data. Dynamic equations are integrated as physical constraints to ensure predicted trajectories strictly adhere to aircraft kinematic principles. Furthermore, the loss function is designed to penalize violations of the aircraft’s maximum acceleration and minimum turning radius, thereby preventing physically implausible predictions beyond the aircraft’s maneuverability limits. Benefiting from the physical constraint, the model can make reasonable predictions with improved interpretability and accuracy than pure data-driven methods.

(3)

By integrating a spatial–temporal separated encoder based on the attention mechanism and physical constraints derived from aircraft dynamics, the proposed STPC-MLP model exhibits significant advantages in aircraft trajectory prediction, with average RMSE, MAE, and MAPE values of only 0.5158, 0.1891, and 0.0067%, respectively. Compared to the optimal Informer model, it achieves substantial reductions in prediction errors—specifically, 7.13% in RMSE, 40.38% in MAE, and 34.31% in MAPE—highlighting its superior accuracy in capturing complex spatial–temporal dynamic data information. Furthermore, ablation experiments further validate the model’s robustness: removing the HPC-MLP module, attention mechanism, or physical constraint loss term leads to significant increases in prediction errors, confirming the critical role of these components in performance enhancement. Overall, these improvements enable the model to generate more reliable predictions that conform to physical laws, strictly adhering to aircraft kinematic principles (e.g., maximum acceleration and minimum turning radius constraints), thereby enhancing its value in real-world applications where safety and precision are paramount.

2. Related Work

Aircraft trajectory prediction methods can be categorized into kinetics-and-aerodynamics models, state estimation models, and machine learning models [10] based on their modeling principles and technical approaches.

2.1. Kinetics-and-Aerodynamics Models

The kinematics and dynamics-based approach treats the aircraft as a point mass, emphasizing the relationship between forces and motion [11,12]. This methodology necessitates the prior establishment of an aircraft performance parameter database. Utilizing the aircraft’s flight state, aerodynamic and kinematic assumptions are developed, which are subsequently employed to formulate the aircraft’s state equations, facilitating the prediction of its trajectory. The Base of Aircraft Data (BADA) database [36], which is the most widely used, provides a range of aircraft performance parameters, including Aircraft Performance Models (APM) and Operating Procedure Coefficients (OPCs), for various aircraft types. In 2008, Porretta et al. [37] represented OPCs as a set of discrete variables and developed the renowned three-dimensional Point Mass Model (PMM) for predicting four-dimensional aircraft trajectories. However, the PMM heavily relies on the aircraft’s key performance parameters, which are often considered commercial secrets and thus not publicly available; furthermore, due to the model’s sensitivity to these key performance parameters, the parameters obtained from theoretical experimental environments often deviate from those in real-world conditions. These two factors significantly limit its engineering applications [16].

2.2. State Estimation Models

Methods based on state estimation models employ state space theory to mathematically represent physical systems, transforming dynamic processes into probabilistic models that handle sensor noise and motion uncertainties [17]. Typical models include KF [18], PF [19], and others. Taking KF as an example, it estimates the system state by combining the system model and observation data using the minimum mean square error criterion. Through continuous prediction and updating, it gradually reduces the state estimation error and provides the optimal state estimation result. Lin et al. [38] proposed using the KF to solve the continuous state estimation problem of aircraft and combined probabilistic methods to determine the aircraft’s intent model, thereby achieving high-precision trajectory prediction. Yan et al. [39] introduced the unscented KF algorithm to achieve two-dimensional trajectory tracking and prediction of nonlinear maneuvering targets in camera coordinates. Ioannis Lymperopoulos et al. [40] proposed a stochastic nonlinear hybrid system model based on particle filtering to address uncertainty issues in aircraft trajectory prediction. To address the issue of insufficient estimation performance of a single model in complex environments, Li et al. [41] proposed the IMM algorithm, which employs two or more motion models to match the possible states of the target during maneuvering. And then it estimates the target state by weighted fusion of the filtering results from multiple models. Typically, this model combines non-maneuvering and maneuvering models, using a Markov chain to govern the switching between models, thereby achieving comprehensive adaptive tracking of the target. Its modular characteristics enable the model to be designed with relatively flexible combinations, and it has been widely applied in the field of target tracking. Overall, state estimation models can effectively predict aircraft trajectories and have achieved significant success in practical applications. However, these methods are susceptible to environmental uncertainties, prone to accumulation of multi-step errors [11], which ultimately affects the precision of aircraft trajectory prediction.

2.3. Machine Learning Models

Inspired by the great success of machine learning algorithms in fields such as computer vision, natural language processing [42], autonomous driving [43], dynamic traffic control [44], and automatic speech recognition [20], a large number of researchers have innovatively applied machine learning algorithms to the field of aircraft trajectory prediction. To address short-term and medium-term aircraft trajectory prediction problems, Tastambekov et al. [21] proposed a local linear functional regression method based on Sobolev spaces. The relative prediction error for their 40 tested trajectories ranged between 2% and 16%, demonstrating the method’s effectiveness. This approach was ultimately successfully implemented in several airports in France. Rodríguez-Sanz et al. [45] introduced a method for aircraft trajectory and traffic flow prediction during the pre-tactical phase. They developed a trajectory prediction framework centered on three core principles: data-driven modeling, dynamic adaptability, and airspace user orientation. Validation using actual operational data from Spanish air traffic control showed significant improvements in the model’s predictive performance for flights with high volatility. Jeung et al. [46] introduced the Hidden Markov Model (HMM) to uncover the underlying movement patterns of flight trajectories. Jiang et al. [47] used Support Vector Machine (SVM) for aircraft conflict detection; Wiest and colleagues [48] utilized Gaussian Mixture Model (GMM) for aircraft trajectory prediction and achieved good results. Choi et al. [22] proposed a terminal airspace trajectory prediction framework combining machine learning and estimation methods, which reduced horizontal positioning errors by 46.8% and 76.0% compared to GMM and LSTM models, respectively, demonstrating its superior performance. Furthermore, BP neural network [26], Bayesian neural network [27,28], LSTM [29,30], CNN-LSTM [31], cLSTM [49], Transformer, and other machine learning algorithms and their variants have been widely applied in aircraft trajectory prediction and have achieved great success. However, as mentioned above, purely data-driven machine learning models are black-box models that lack strong interpretability [50] and may produce prediction points that violate physical laws [33]. As researchers such as Hashemi [51] and Neretin [52] have demonstrated through comparative evaluations of traditional models and deep neural network (DNN) models, while DNN models outperform traditional approaches in predictive accuracy and learning speed, they also exhibit the highest sensitivity to unfamiliar information—such as abnormal operations by human experts or sudden weather changes—leading to significant deviations from actual conditions in prediction outcomes. If this issue is not effectively resolved, it will still limit the practical application of machine learning models in aircraft trajectory prediction.

3. Methodology

3.1. Problem Formulation

The ATP task is generally defined as a time series forecasting (TSF) problem. The last N trajectory points ${\{\mathit{P}}_t\in {\mathit{R}}^d|t=1, 2,\dots ,N\}$ of the previous time period, arranged in chronological order, from the air traffic control system are used to predict the future aircraft’s trajectory point ${\mathit{P}}_{N+1}$. Here is the definition of ${\mathit{P}}_t$.

$${\mathit{P}}_t\triangleq {\left[{Lon}_t,{Lat}_t,{Alt}_t\right]}^T$$

(1)

where Lon, Lat, Alt, correspond to the longitude, latitude and altitude, respectively. $t$ represents the time of the trajectory point. It should be noted that $t+1$ means that it is 1000 ms later than $t$ in terms of time. Utilizing the input data ${\mathit{P}}_{1:N}\in R^{3\times N}$, the ATP task can be expressed as:

$${\mathit{P}}_{N+1}=f({\mathit{P}}_{1:N},\theta )$$

(2)

$${\mathit{P}}_{1:N}=\left[{\mathit{P}}_1,{\mathit{P}}_2,\dots ,{\mathit{P}}_N\right]$$

(3)

where $\theta$ denotes the parameters of the model (e.g., weights and biases), f denotes the prediction model. For clarity in subsequent discussions, let us denote ${\mathit{P}}_{N+1}$ as Y, i.e., $\mathit{Y}= {\mathit{P}}_{N+1}$.

3.2. The Proposed Framework

The proposed model’s structure, as illustrated in Figure 1, primarily comprises two components: an encoder and the HPC-MLP. The details of the architecture and main blocks are presented as follows.

3.2.1. Encoder

To more effectively leverage the input data and reduce redundancy, an encoder is utilized to extract degradation information while reducing the dimensionality of the input. The encoder is composed of three blocks, which are the input embedding block, attention block and feature fusion block.

The embedding block is used to unify the dimensions of the input and make preparation for the feature fusion process, and its internal block diagram is shown in input embedding block in Figure 1. The embedding block consists of one Reshape layer, three fully connected layers (Fully Connected), and three activation layers (Relu). The primary function of the Reshape layer is to modify the data organization of the output tensor from the first fully connected layer, enabling the temporal attention block to focus on the time dimension of the input data. The input to the input embedding block is a Data Matrix in the form of a three-dimensional tensor, organized as Channel × Batch × Time (abbreviated as C × B × T). In this paper, Channel corresponds to the three channels: longitude, latitude, and height, which are the dimensions that the spatial attention block needs to focus on. Batch refers to the sample batch size during a single training step. Time represents the time series dimension, which is the dimension that the temporal attention block focuses on, and its value is denoted as N. The specific workflow of the embedding block can be described as follows:

The Data Matrix (C (3) × B (1) × T (N), where the batch size is written as 1 for simplicity) passes through the first fully connected layer and outputs a tensor of shape C (32) × B (1) × T (N). For the left branch in embedding block, this tensor then passes through another fully connected layer and an activation layer (resulting in a tensor of shape C (16) × B (1) × T (N)), and is output to the spatial attention block. For the right branch in embedding block, the tensor (C (32) × B (1) × T (N)) from the first fully connected layer is processed by the Reshape layer, which reorganizes the tensor into the form C (N) × B (1) × T (32). It then passes through another fully connected layer and an activation layer, and is output as a tensor of shape C (16) × B (1) × T (32) to the temporal attention block.

The attention block consists of spatial attention block and temporal attention block. The self-attention mechanism is employed to calculate attention scores for various trajectory’s features across different time steps. The spatial attention block is utilized to extract features from the spatial dimension of aircraft trajectories. The temporal attention block is responsible for extracting features from the time step dimension of aircraft trajectories. The attention blocks for spatial and temporal attention block share the same structure. As shown in Figure 1, the attention block consists of three sublayers, which are Self-Attention 1 (lower-level self-attention layer), Self-Attention 2 (upper-level self-attention layer), and Add & Norm (residual connection and layer normalization). Self-attention 1 and Self-attention 2 have the same attention structure. The data flow begins with the input passing through Self-Attention 1 to capture local dependencies, followed by Self-Attention 2 to model global long-range associations. Finally, Add & Norm combines the self-attention output with the original input via residual addition and applies normalization to ensure feature stability and training convergence. By stacking dual self-attention layers, the Attention block achieves hierarchical feature abstraction for aircraft trajectory data (from local to global patterns). The residual connection mitigates vanishing gradient issues, while layer normalization suppresses feature distribution shifts. The design is visualized with clean blue blocks and directional arrows against a plain background, emphasizing intuitive data flow and modularity.

The input of the self-attention [34] layer contains query ($Q$), key ($K$) of dimension $d_k$, and value ($V$) of dimension $d_v$. The Q, K and V are all obtained by passing the input data through a linear layer, which can be calculated separately by the following equations.

$$=P_{emb}{W}^Q, K=P_{emb}{W}^K, V=P_{emb}{W}^V$$

(4)

where $P_{emb}$ is the embedding representation obtained by applying the embedding operation to the input data matrix P. $W^Q$, $W^K$ and $W^V$ are all learned weight matrices.

The self-attention mechanism can be expressed as:

$$Attention\left(Q,K,V\right)=softmax\left({\displaystyle \frac{Q{K}^T}{\sqrt{d_k}}}\right)V$$

(5)

$Attention\left(Q,K,V\right)$ represents the attention score matrix. $\sqrt{d_k}$ is used to scale the dot products to prevent $Q{K}^T$ falls into the area where the softmax function has extremely small gradients [34].

The feature fusion block is responsible for combining the features of spatial attention and temporal attention and transforming the new feature into a low-dimensional representation, which contains the compressed information from the original aircraft trajectory data. The structure of the feature fusion block is shown in Figure 1. First, the output of the spatial attention block ${\mathit{F}}_f\in {\mathit{R}}^{3\times D}$ and the output of the temporal attention block ${\mathit{F}}_t\in {\mathit{R}}^{T\times D}$ are concatenated to get a new feature map ${\mathit{F}}_r\in {\mathit{R}}^{(T+3)\times D}$. Subsequently, a convolutional layer is employed to extract the features from ${\mathit{F}}_r$ and fuse the information from spatial attention block and temporal attention block. Finally, the Indexing 1-D Layer selectively filters critical feature positions along specific dimensions, compressing redundant information to output the low-dimensional fused result (H).

3.2.2. HPC-MLP

By abstracting the aircraft as a point mass, based on Newton’s second law, the vector form differential equation of the point mass can be described by the following equation.

$${\displaystyle \frac{{\partial }^2\mathit{r}}{\partial t^2}}-{\displaystyle \frac{\mathit{F}\left(\mathit{r},t\right)}{m}}=0$$

(6)

where $m$ is the mass of aircraft. $\mathit{r}$ represents the position coordinates of the aircraft in airspace, which corresponds to $\mathit{Y}$ in Figure 1. $\mathit{F}\left(\mathit{r},t\right)$ represents the vector sum of the forces acting on the aircraft, which is a function of the aircraft’s position and time.

Equation (6) clearly shows that the second derivative of the aircraft position $\mathit{r}$ with respect to time is proportional to the vector sum of the forces exerted on the aircraft. In this way, the prediction of the trajectory of the aircraft will be transformed into the prediction of the vector sum of the current force on the aircraft and the mass m of the aircraft. Theoretically, the current state of force on an aircraft is unpredictable, because the force on an object can change its magnitude and direction instantaneously. However, fortunately, the size and direction of the force on the aircraft in the real environment will not change instantaneously. For example, the size of the thrust of the aircraft is the change of time, and the closing and opening of the engine cannot be completed instantaneously. Similarly, other forces such as the air flow on the aircraft cannot change instantaneously, and they all change with time. This provides a realistic basis for the prediction of forces. Here, we use the second order nonlinear partial differential equation (PDE) N to describe the change of the ratio ${\displaystyle \frac{\mathit{F}\left(\mathit{r},\mathit{t}\right)}{\mathit{m}}}$ of force to mass with time. N can be expressed in detail as:

$$\begin{array}{c}N\left(u,u_t,u_{tt}\right)={\alpha }_{\mathrm{0,0}}+{\alpha }_{\mathrm{1,0}}u+{\alpha }_{\mathrm{2,0}}{u}^2+{\alpha }_{\mathrm{3,0}}{u}^3+\\ {\alpha }_{\mathrm{0,1}}{u}_t+{\alpha }_{\mathrm{1,1}}u{u}_t+{\alpha }_{\mathrm{2,1}}{u}^2{u}_t+{\alpha }_{\mathrm{3,1}}{u}^3{u}_t+\\ {\alpha }_{\mathrm{0,2}}{u}_{tt}+{\alpha }_{\mathrm{1,2}}u{u}_{tt}+{\alpha }_{\mathrm{2,2}}{u}^2{u}_{tt}+{\alpha }_{\mathrm{3,2}}{u}^3{u}_{tt} \end{array}$$

(7)

where α is hyper parameter. The solution $u=(u_1\dots \dots u_n)$ could be n dimensional in which case $u_t$ denotes the collection of all element-wise first order derivatives ${\displaystyle \frac{\partial u_1}{\partial t}},\dots ,{\displaystyle \frac{\partial u_n}{\partial t}}$. Similarly, $u_{tt}$ includes all element-wise second order derivatives ${\displaystyle \frac{{\partial }^2{u}_1}{\partial t^2}},\dots \dots ,{\displaystyle \frac{{\partial }^2{u}_n}{\partial t^2}}$. In this paper, the solution $u$ corresponds to $\mathit{r}$, and the $n=3$. Because the position coordinate $\mathit{r}$ is a vector, containing x, y, and z directions, the system of partial differential equations that fully describe the ratio of force to mass ${\displaystyle \frac{\mathit{F}\left(\mathit{r},t\right)}{m}}$ over time is shown below.

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mathit{F}}_x\left(\mathit{r},t\right)}{m}}=\mathit{N}\left(x\right)\\ {\displaystyle \frac{{\mathit{F}}_y\left(\mathit{r},t\right)}{m}}=\mathit{N}\left(y\right)\\ {\displaystyle \frac{{\mathit{F}}_z\left(\mathit{r},t\right)}{m}}=\mathit{N}\left(z\right)\end{array}\right.$$

(8)

Although for the ATP problem, $\mathit{N}$ is unclear, one can still embed the hidden physics relationship between output data Y and the input data P into the neural network. Here we introduce HPC-MLP. The HPC-MLP is adapted from the original work of DeepHPM [35] with the self-attention mechanism to enhance its capability. As shown in Figure 1. The HPC-MLP is composed of two same multilayer perceptron (MLP) networks. The first MLP network is responsible for predicting the Y with H and t as inputs. And the second MLP network is used to distil the hidden mechanisms (nonlinear function, $\mathit{N}$ in Equation (7)) between the compressed input data P and output data Y. The inputs of second MLP are $\left[P, {\displaystyle \frac{\partial \mathit{P}}{\partial t}}, {\displaystyle \frac{{\partial }^2\mathit{P}}{\partial t^2}},\dots \right]$, which are the derivatives of P with respect to time. In fact, the process of training the second MLP is precisely to search for the hyper parameters in Equation (7), and to dynamically adjust these hyper parameters according to the different trajectory characteristics of the aircraft.

The loss function of the whole model is formed by four terms, which are the data driven loss term, physics-driven loss term, acceleration constraint loss term and curvature constraint loss term, respectively. The loss function can be expressed as:

$${\mathit{L}}_{total}={\lambda }_1{\mathit{L}}_{data}+{\lambda }_2{\mathit{L}}_{physics}+{{\lambda }_3\mathit{L}}_{acc}+{\lambda }_4{\mathit{L}}_{curve}$$

(9)

where λ is the weight factor, which is used to balance different terms. In this paper, gradient descent is used to adjust the value of $\lambda$. ${\mathit{L}}_{data}$ is data driven loss term, which can be expressed as:

$${\mathit{L}}_{data}= {\displaystyle \frac{1}{N}}\sum _{t=1}^N{\left(Y_t-\widehat{Y_t}\right)}^2$$

(10)

where $\widehat{Y}$ denotes the predicted $Y$, and $Y$ and denotes the real $Y$. ${\mathit{L}}_{physics}$ is the physics-driven loss term:

$${\mathit{L}}_{physics}= {\displaystyle \frac{1}{N}}\sum _{t=1}^{N}f(\mathit{r},t)$$

(11)

where f is:

$$f= {\displaystyle \frac{{\partial }^2\mathit{r}}{\partial t^2}}- {\displaystyle \frac{\mathit{F}\left(\mathit{r},t\right)}{m}}$$

(12)

In practice, in addition to the physics-driven loss term introduced to account for dynamical constraints, it is essential to augment the loss function with acceleration and curvature constraint terms. These enforce adherence to kinematic principles such as the maximum acceleration limits and minimum turning radius for aircraft. This necessity arises from the physical reality that no existing aircraft can operate with arbitrarily large accelerations or infinitesimally small turning radii. In this work, acceleration constraint loss term ${\mathit{L}}_{acc}$ is shown below.

$${\mathit{L}}_{acc}={\sum }_t{\mathit{max}\left(0,{\left|\left|{\mathit{a}}_{\mathit{t}}\right|\right|}_2-a_{max}\right)}^2$$

(13)

$${\left|\left|{\mathit{a}}_{\mathit{t}}\right|\right|}_2=\sqrt{({{\displaystyle \frac{d^{2}x}{d{t}^2}})}^2+{\left({\displaystyle \frac{d^{2}y}{d{t}^2}}\right)}^2+{\left({\displaystyle \frac{d^{2}z}{d{t}^2}}\right)}^2}$$

(14)

where ${\left|\left|{\mathit{a}}_{\mathit{t}}\right|\right|}_2$ represents the second-order norm of acceleration ${\mathit{a}}_{\mathit{t}}$, and $a_{max}$ is the maximum acceleration. And curvature constraint loss term ${\mathit{L}}_{curve}$ can be expressed as:

$${\mathit{L}}_{curve}={\sum }_t{\mathit{max}\left(0,{\displaystyle \frac{d{\mathit{\theta }}_t}{dt}}-{\displaystyle \frac{{\mathit{v}}_t}{R_{min}}}\right)}^2$$

(15)

$${\mathit{v}}_t={\displaystyle \frac{d\mathit{r}}{dt}}$$

(16)

$${\mathit{\theta }}_t=\mathit{arctan}2\left({\displaystyle \frac{dy}{dt}},{\displaystyle \frac{dx}{dt}}\right)$$

(17)

where ${\mathit{\theta }}_t$ represents heading Angle, and ${\mathit{v}}_t$ is velocity vector. $R_{min}$ is minimum turning radius for aircraft. The $arctan2$ function is the two-argument arctangent, which calculates the angle from the positive x-axis to a point (x, y) in the Cartesian plane.

4. Experiments

4.1. Data Processing

In this work, raw aircraft trajectories are sourced from the OpenSky Network [53], comprising 2671 aircraft over European airspace within 2 h period. These trajectories were collected via Automatic Dependent Surveillance-Broadcast (ADS-B) systems from operational air traffic control (ATC) infrastructure. The dataset encapsulates complete flight state representations including: temporal stamps, geospatial coordinates (longitude, latitude, altitude), and velocity vectors (horizontal/vertical components). In this paper, we specifically leverage the temporal dynamics and three-dimensional positional data (timestamps and spatial coordinates).

The complete set of aircraft trajectories in the dataset is visualized in Figure 2a. Based on the latitude and longitude, Figure 2a shows that the dataset covers most of Europe (longitude from 15° E to 30° E, spanning 1365 km; latitude from 35° N to 65° N, spanning 3336 km), indicating a wide distribution and rich sampling of aircraft trajectory data across these coordinates. Regarding altitude, the dataset includes flight data ranging from 0 to 14,000 m (with trajectory colors representing altitude, transitioning from blue to yellow as shown in the color bar at the bottom of Figure 2a), further demonstrating a broad and diverse distribution of altitude data, which is crucial for model training.

Figure 2b,c fully illustrate a single aircraft trajectory from the dataset, encompassing the complete process of climb, high-altitude cruise, and descent. Figure 2b highlights the horizontal aircraft path, with longitude ranging from 9° E to 11° E and latitude from 46° N to 50° N. Figure 2c provides a three-dimensional view of the complete trajectory, clearly showing the aircraft’s gradual climb from an altitude of about 1000 m to around 4000 m, followed by a period of level flight, a quicker ascent to a cruising altitude of 10,000 m, a duration of high-altitude cruising, and then a gradual descent back to approximately 1000 m.

Due to various complex airspace conditions, the ADS-B trajectory data received by ATC system for aircraft is not completely continuous. Occasionally, there will be breakpoints in the trajectory data of the aircraft. Therefore, the original trajectory data needs to be preprocessed. Here, Fourier interpolation is employed to address missing points in the original trajectory data while concurrently achieving uniform time intervals (as the original data exhibited inconsistent timing between trajectory points). It should be noted that targets exhibiting extensive missing data will be directly discarded. Following preprocessing, min-max normalization is applied to scale the new trajectory data. The normalization formula is shown below.

$${\mathit{P}}_{nor}={\displaystyle \frac{{\mathit{P}}_{pre}-min}{max-min}}$$

(18)

where ${\mathit{P}}_{pre}$ is preprocessed data, max and min indicate the maximum and the minimum value of the sample, respectively, and ${\mathit{P}}_{nor}$ is the normalized data.

The sliding window process, which is a commonly used data segmentation process to get the time information from the time series data, is used to segment the aircraft trajectory data. In this method, the window sizes are set as 30, and the sliding stride is 1.

4.2. Experimental Setting

In this work, 80% of trajectory data are randomly chosen as the training dataset, the remaining 20% is set aside as the test dataset. The batch size is set as 64. In this model, the MLP in HPC-MLP contains 2 hidden layers with 16 neurons per layer. Before each activation function, batch normalization is used to normalize the output of the previous layer. Tanh is adopted as the activation function. Adam optimizer in MATLB 2024b is used to update the model. The learning rate will be tested as a hyper-parameter, with the range being from 0.1 to 0.001. The training is conducted on a single NVIDIA GeForce RTX 4060 GPU.

In order to comprehensively evaluate the performance of our model, three evaluation indexes, root mean absolute error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE), were used to evaluate the built model to ensure that the built model has strong generalization and high robustness. The RMSE can be displayed as:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}\sum _{t=1}^N{\left(Y_t-\widehat{Y_t}\right)}^2}$$

(19)

The MAE can be displayed as:

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|Y_t-\widehat{Y_t}\right|$$

(20)

The MAPE can be displayed as:

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\displaystyle \frac{\left|Y_t-\widehat{Y_t}\right|}{\widehat{Y_t}}}\times 100\%$$

(21)

4.3. Maximum Acceleration and Minimum Turn Radius

In the model loss term, the maximum acceleration and minimum turn radius are important physical constraint parameters. By setting the value of the maximum acceleration to penalize predicted values that exceed the maximum acceleration, the model weights are thereby updated through the backpropagation algorithm; similarly, by setting the minimum turn radius to penalize predicted values below the minimum turn radius, the model weights are updated through the backpropagation algorithm.

Table 1. Maximum acceleration and minimum turn radius of various typical aircraft.

Aircraft Type	Representative Aircraft	Min Airspeed $\left({\mathit{V}}_{\mathit{m}\mathit{i}\mathit{n}}\right)$ (m/s)	Max Load Factor $\left({\mathit{n}}_{\mathit{m}\mathit{a}\mathit{x}}\right)$ (g)	Max Acceleration $({\mathit{a}}_{\mathit{m}\mathit{a}\mathit{x}})$ (g)	Min Turn Radius $\left({\mathit{R}}_{\mathit{m}\mathit{i}\mathit{n}}\right)$ (m)
Civil Aviation	Boeing 737	77	2.5	2.5	264
Fighter Jets	F-16	67	9	9	51
Bombers	B-52	70	3	3	177
Reconnaissance Aircraft	U-2	51	2.5	2.5	116
The model adopts values				10	50

details the minimum airspeed ${\mathit{V}}_{\mathit{m}\mathit{i}\mathit{n}}$, maximum load factor ${\mathit{n}}_{\mathit{m}\mathit{a}\mathit{x}}$, maximum acceleration $a_{max}$, and minimum turn radius ${\mathit{R}}_{\mathit{m}\mathit{i}\mathit{n}}$ for four typical aircraft categories: civil aviation (Boeing 737) [54], fighter jet (F-16) [55,56], bomber (B-52) [57], and reconnaissance aircraft (U-2) [58]. Among them, the minimum turn radius is calculated based on the aircraft’s minimum airspeed ${\mathit{V}}_{\mathit{m}\mathit{i}\mathit{n}}$ and maximum load factor ${\mathit{n}}_{\mathit{m}\mathit{a}\mathit{x}}$. The calculation formula is:

$${\mathit{R}}_{\mathit{m}\mathit{i}\mathit{n}}= {\displaystyle \frac{{{\mathit{V}}_{\mathit{m}\mathit{i}\mathit{n}}}^2}{g\sqrt{{{\mathit{n}}_{\mathit{m}\mathit{a}\mathit{x}}}^2-1}}}$$

(22)

According to Table 1, the maximum acceleration for the four typical aircraft is 9 g (g is the gravitational acceleration, with a value of 9.81 m²/s), and the calculated minimum turn radius is 51 m. In the model, to account for exceptional cases of individual aircraft and to retain appropriate margins, the maximum acceleration is determined to be 10 g, and the minimum turn radius to be 50 m.

4.4. Hyperparameter Configuration

To get the best performance, grid search is conducted to determine the learning rate and the number of self-attention heads in attention blocks. The learning rate is configured in the range of 0.1 to 0.001, and the number of self-attention heads is set to 1 to 5. We compare the outcomes of the average RMSE and MAE of the aircraft trajectory dataset under different settings. Each RMSE and MAE is averaged from 3 independent trials.

Table 2. The average RMSE and MAE of under different settings.

Learning Rate		Number of Self-Attention Heads
		1		2		3		4		5
		RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE
0.1	longitude	0.0137	0.0132	0.0955	0.0776	2.8490	2.7859	0.0250	0.0200	0.5042	0.5033
	latitude	0.0013	0.0010	0.0352	0.0237	0.3013	0.2969	0.0052	0.0045	0.0118	0.0112
	altitude	1.9363	0.8684	66.0704	40.8829	116.3533	98.7107	67.6720	67.2449	291.2350	275.5930
0.01	longitude	0.0074	0.0060	0.0062	0.0047	0.0054	0.0049	0.0077	0.0067	0.0074	0.0057
	latitude	0.0056	0.0054	0.0053	0.0050	0.0028	0.0023	0.0020	0.0016	0.0054	0.0050
	altitude	1.5560	0.9227	2.5098	2.2539	3.0472	2.6376	3.0526	2.6909	1.7229	0.9786
0.001	longitude	0.0023	0.0021	0.0047	0.0040	0.0058	0.0052	0.0056	0.0050	0.0055	0.0049
	latitude	0.0011	0.0008	0.0042	0.0038	0.0040	0.0036	0.0031	0.0025	0.0029	0.0024
	altitude	1.5441	0.5643	1.8980	1.4056	1.6178	1.0255	2.5090	2.2496	3.4439	3.2511

shows the average RMSE and average MAE. It can be observed that the model gives the best results when the learning rate is 0.001 and the number of self-attention heads is 1.

4.5. Result and Comparison with Other Methods

In this work, we selected 8 representative models—namely Kalman filter, Vanilla LSTM, PCA LSTM, PSO LSTM, CNN LSTM, Transformer, Informer and Temporal fusion transformer (TFT)—as our baseline approaches.

Table 3. Comparison with several typical methods on trajectory dataset.

Approach	RMSE				MAE				MAPE (%)
	Long	Lat	Alt	Average	Long	Lat	Alt	Average	Long	Lat	Alt	Average
Kalman Filter	0.0122	0.0038	2.7944	0.9368	0.0111	0.0035	1.8449	0.6198	0.0723	0.0076	0.0169	0.0323
Vanilla LSTM	0.0109	0.0036	2.1128	0.7091	0.0098	0.0033	1.3403	0.4511	0.0637	0.0046	0.0124	0.0269
PCA LSTM	0.0180	0.0060	4.3536	1.4592	0.0163	0.0054	3.6384	1.2200	0.1047	0.0117	0.0337	0.0500
PSO LSTM	0.0056	0.0021	2.4918	0.8332	0.0043	0.0017	1.7833	0.5964	0.0280	0.0041	0.0164	0.0162
CNN LSTM	0.0133	0.0045	2.9542	0.9907	0.0118	0.0039	2.0237	0.6798	0.0765	0.0085	0.0188	0.0346
Transformer	0.0093	0.0016	3.3321	1.1143	0.0082	0.0013	3.0632	1.0242	0.0511	0.0028	0.0287	0.0275
Informer	0.0037	0.0015	1.6610	0.5554	0.0031	0.0008	0.9478	0.3172	0.0202	0.0018	0.0087	0.0102
TFT	0.0040	0.0020	1.6892	0.5651	0.0036	0.0017	0.9818	0.3290	0.0221	0.0037	0.0092	0.0117
STPC-MLP (Proposed)	0.0023	0.0011	1.5441	0.5158	0.0021	0.0008	0.5643	0.1891	0.0131	0.0018	0.0053	0.0067

presents a comparison between the proposed model and these baseline models. It is known from the table that the proposed model’s average RMSE is only 0.5158, which represents a 7.13% reduction compared to the optimal Informer model (average RMSE of 0.5554). Similarly, on the average MAE metric, the proposed model’s MAE is reduced by 40.38% compared to the optimal Informer model (average MAE of 0.3172). On the average MAPE metric, the proposed model’s MAPE is reduced by 34.31% compared to the optimal Informer model (average MAPE of 0.0102). The above results show that, whether in longitude, latitude or altitude, the performances of our method show significant improvement compared with those existing approaches. It indicates that with the aid of physics constraint and attention mechanism, our model has superior ability for the ATP task.

Figure 3a displays the test results of the proposed model and other common models on one of the aircraft trajectory datasets. On the one hand, the STPC-MLP model proposed in this paper shows closer alignment to the target values compared to other models. On the other hand, when the target altitude undergoes significant changes, especially at elevations of 10,650 m and 11,000 m, the STPC-MLP model exhibits smaller oscillations and deviations than other models. This demonstrates its effective tracking capability of the target. Figure 3b,d,f further elaborates the temporal variations of the different model’s predicted trajectory in longitude, latitude, and altitude, providing a granular validation of its consistency in multi-dimensional space. Figure 3c,e,g demonstrates the absolute error trends of an aircraft trajectory prediction model across three dimensions—longitude (0–0.03 degrees), latitude (0–0.01 degrees), and altitude (0–15 m)—over 0–35 time steps, revealing that horizontal errors are significantly smaller than vertical ones while maintaining overall stability.

5. Ablation Study

5.1. Ablation of HPC-MLP

Figure 4a illustrates the predicted aircraft trajectory without the HPC-MLP module. In contrast to Figure 3 (with HPC-MLP), the incorporation of HPC-MLP clearly demonstrates enhanced predictive capability for the target trajectory. Similarly, Figure 4b,d,f more clearly delineates this improvement through separate analyses of the trajectory’s longitude, latitude, and altitude dimensions. Figure 4c,e,g further present the absolute error of trajectory prediction in longitudinal, latitudinal, and altitudinal dimensions, respectively. The model with HPC-MLP shows significant improvement. This indicates that the proposed method, which integrates physics constraints, is more effective. Figure 5 exhibits the dynamic evolution of loss weights ${\lambda }_i$ for the data-driven loss term, physics-driven loss term, acceleration constraint loss term, and curvature constraint loss term during training process. The initial values of ${\lambda }_1$ to ${\lambda }_4$ are set to 1.0, 0.75, 0.5, and 0.5, respectively. The trajectories of the four loss weights in Figure 5 demonstrate that the model rapidly establishes foundational capabilities through heavy reliance on data fitting and physics regularization during early training stages, while acceleration and curvature constraints drive further refinement in later iterations.

5.2. Ablation of Attention Mechanism

Table 4. Ablation study of attention mechanism.

Approach	RMSE				MAE
	Long	Lat	Alt	Average	Long	Lat	Alt	Average
M1	0.0022	0.0013	9.9187	3.3074	0.0022	0.0012	6.8139	2.2722
M2	0.0022	0.0038	30.421	10.142	0.0022	0.0038	29.812	9.9395
STPC-MLP (Proposed)	0.0023	0.0011	1.5441	0.5158	0.0021	0.0008	0.5643	0.1891

shows the results of the ablation study of temporal attention (M1), spatial attention (M2). Notice that for M1, the model shows better performance on longitude and altitude, but shows worse performance on altitude. As for M2, the results are similar to those of M1; that is, M2 achieves better accuracy on longitude and latitude, but worse accuracy on altitude. Compared to M1 and M2, our proposed STPC-MLP demonstrated smaller errors in predicting both latitude and altitude.

5.3. Ablation of Loss Balancing Scheme

Table 5. Comparison of the prediction performance of the model with loss balancing.

Approach	RMSE				MAE
	Long	Lat	Alt	Average	Long	Lat	Alt	Average
L1	0.0023	0.0040	31.029	10.3452	0.0023	0.0039	30.904	10.3037
L2	0.0028	0.0016	12.442	4.1488	0.0027	0.0016	12.267	4.0907
L3	0.0028	0.0015	11.109	3.7044	0.0028	0.0015	11.042	3.6822
Loss balancing scheme	0.0023	0.0011	1.5441	0.5158	0.0021	0.0008	0.5643	0.1891

shows the results of the ablation study of loss balancing scheme, which were ablation experiments of physics-driven loss term (L1), acceleration constraint loss term (L2) and curvature constraint loss term (L3), respectively. The results demonstrate that the loss balancing scheme significantly enhances model accuracy, particularly when integrated with the physics-driven loss term, highlighting its critical role in the training process.

6. Conclusions

In this work, a spatial–temporal physics-constrained multilayer perceptron model was proposed to address the ATP problem. A spatial–temporal attention mechanism was proposed to emphasize the spatial and time steps. The extracted features were fused to remove redundant features and get the hidden vectors that contain useful information. HPC-MLP is used to make predictions and capture the hidden physics between the input data and output data. In HPC-MLP, dynamic equations are integrated as physical constraints to ensure predicted trajectories strictly adhere to aircraft kinematic principles. Furthermore, the loss function is designed to penalize violations of the aircraft’s maximum acceleration and minimum turning radius, thereby preventing physically implausible predictions beyond the aircraft’s maneuverability limits. These innovations collectively enable outstanding performance (avg. RMSE 0.5158, MAE 0.1891, MAPE 0.0067%) and strong interpretability, particularly due to the physics-guided constraints. Compared to the optimal Informer models, our model reduces prediction errors by 7.13% (RMSE), 40.38% (MAE), and 34.31% (MAPE), demonstrating superior accuracy in modeling complex spatial–temporal dynamics. This underscores its significant potential for real-world applications demanding high reliability and precision.

However, the model we proposed still has the following limitations: (I) The proposed model currently can only perform single-step prediction, and multi-step prediction will be added in future work; (II) The model has not yet considered information about small unmanned aircraft, and the prediction results for small unmanned aircraft may have certain limitations; (III) The proposed model employs many physical constraints, which increases the computational load of the loss function, leading to excessively long model training times. In addition, the larger scale of model parameters will limit its practical application.