Enhanced Trajectory Forecasting for Hypersonic Glide Vehicle via Physics-Embedded Neural ODE

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This article proposes a physics-embedded neural ODE model for the precise forecasting of HGV trajectories. Totally, The paper is not very innovative, and there are some problems as follows:

1. What is the basis for creating the three different datasets mentioned in Section 4.3? Why were three datasets designed instead of directly using the dataset with 1200 samples?

2. The article's elaboration on the Physics-Embedded Network is not very clear. In what specific stage does the embedding of the physical model come into play?

3. Why do LSTNet and Transformer exhibit significant instability, while according to the article, Informer and PhyTransformer, which are prediction methods based on Transformer, do not suffer from instability?

4. I am puzzled why the complex constraints faced by hypersonic vehicles during flight, such as path constraints, are not incorporated into the physical model. These constraints are not negligible during the flight process.

5. The discussion of the core technical points in the paper is overly simplistic, especially the formula derivation for the neural network construction section, which appears to be quite rudimentary. It is recommended that the authors strengthen the theoretical derivation of their innovative points.

Comments on the Quality of English Language

Author Response

We sincerely appreciate your insightful comments and suggestions on our manuscript. We have carefully considered each of your points and have made revisions to address your concerns. Below, we provide detailed responses to your questions.

Q1: What is the basis for creating the three different datasets mentioned in Section 4.3? Why were three datasets designed instead of directly using the dataset with 1200 samples?
A1: Thank you for this question. The primary reason for creating three datasets with varying sizes (100, 600, and 1200 samples) is to thoroughly assess the model's performance and generalization ability across different data scales. In real-world applications, the amount of available data for training an HGV trajectory prediction model can be quite limited. Therefore, it is crucial to evaluate the model's performance under data-scarce conditions.
By training and evaluating PhysNODE on datasets of varying sizes, we aim to demonstrate its robustness and ability to generalize effectively, even when trained on limited data. Our results show that PhysNODE consistently outperforms other methods, particularly in scenarios with smaller datasets, highlighting its practical value for real-world HGV trajectory prediction.

Q2: The article's elaboration on the Physics-Embedded Network is not very clear. In what specific stage does the embedding of the physical model come into play?
A2: We appreciate your feedback on this aspect. To enhance clarity, we have restructured the manuscript by splitting the original Section 3 into two sections: Section 3, "Background" and Section 4, "The PhysNODE Model". In Section 4.1, "Model Formulation", we provide a detailed explanation of how the physical model is embedded within PhysNODE.
Specifically, the physics-based model, denoted as "Phys" in Equation (10), is integrated into the iterative prediction process. At each time step, our model utilizes the state vector from the previous time step and the predicted aerodynamic parameters from the neural network to calculate the state evolution according to the known HGV motion equations (Equation 1). This ensures that the predicted trajectory adheres to the underlying physical laws governing the HGV's motion. This integration of data-driven predictions with physics-based calculations is a key aspect of PhysNODE's effectiveness.

Q3: Why do LSTNet and Transformer exhibit significant instability, while according to the article, Informer and PhyTransformer, which are prediction methods based on Transformer, do not suffer from instability?
A3: Thank you for bringing up this observation. The difference in stability between LSTNet and Transformer on one hand, and Informer and PhyTransformer on the other, can be attributed to the specific architectures and mechanisms employed by these models.
Informer, as a Transformer-based model designed for long sequence prediction, incorporates specialized mechanisms like the probsparse self-attention and self-distillation techniques. These enhancements enable it to capture long-range dependencies more effectively, making it better suited for handling the complexities of HGV trajectories. PhyTransformer, also based on the Transformer architecture, explicitly incorporates physics equations as regularization terms in its loss function. This integration of physics knowledge contributes to its improved stability and prediction accuracy, particularly for HGV trajectories.
In contrast, LSTNet, while incorporating a combination of CNN, LSTM, and periodic regression components, may not be as effective in capturing the specific long-range dependencies and nonlinear dynamics inherent in HGV trajectories. The standard Transformer, while powerful for sequence modeling, relies on a permutation-invariant self-attention mechanism that can lead to the loss of temporal information, particularly in long sequences. This loss of temporal information may contribute to its instability in predicting complex HGV trajectories.

Q4: I am puzzled why the complex constraints faced by hypersonic vehicles during flight, such as path constraints, are not incorporated into the physical model. These constraints are not negligible during the flight process.
A4: Thank you for raising this important point. You are correct that path constraints play a significant role in HGV flight. However, in our approach, we address these constraints indirectly by focusing on the prediction of aerodynamic parameters, which are directly related to the HGV's maneuvering capabilities.
The HGV adjusts its bank angle to manage energy and reach its target accurately. This bank angle adjustment directly influences the lift and drag forces experienced by the vehicle, which further affect its velocity, flight path angle, and heading angle. By accurately predicting the aerodynamic parameters, such as lift and drag coefficients, PhysNODE can indirectly account for the path constraints imposed by the HGV's energy management and maneuvering strategies.
Directly incorporating path constraints into the physical model is challenging due to the highly coupled nature of the governing equations. For instance, the altitude is influenced by the flight path angle and angle of attack, while the heading angle is affected by the bank angle and lift coefficient. Explicitly modeling these coupled relationships as path constraints would require complex decoupling strategies and could introduce additional computational burden. Our approach, by focusing on the accurate prediction of aerodynamic parameters, provides an effective and computationally efficient way to indirectly address these path constraints.

Q5: The discussion of the core technical points in the paper is overly simplistic, especially the formula derivation for the neural network construction section, which appears to be quite rudimentary. It is recommended that the authors strengthen the theoretical derivation of their innovative points.
A5: We appreciate your feedback and have made revisions to strengthen the theoretical grounding of our method. We have expanded Section 4.1, "Model Formulation" to provide a more detailed explanation of the principles and workflow of PhysNODE. Additionally, Section 4.3, "Neural Network Architecture" now includes a comprehensive description of the hybrid neural network architecture, including the mathematical formulation of the 1D CNN and MLP components.
While the mathematical derivations of the neural network components themselves are based on standard deep learning principles, the innovation of PhysNODE lies in its unique integration of these components with the physics-based model. This integration, as detailed in Section 4.1 and illustrated in Figure 2, enables the model to leverage both data-driven learning and physical knowledge for accurate and physically consistent HGV trajectory prediction.

We believe that these revisions provide a more comprehensive and rigorous presentation of our method.

We thank you again for your valuable feedback, which has significantly improved the clarity and quality of our manuscript.

 

Reviewer 2 Report

Comments and Suggestions for Authors

Most of the information presented in the document pertains to algorithms rather than drones. This suggests that the article proposes an approach to improve trajectory prediction for hypersonic vehicles. However, the contribution of this approach is questionable due to insufficient scientific soundness.

The section on the model vehicle is weak; it lacks diagrams, schematics, and vehicle parameters. Equations (3) are not well explained. Section 3 needs rewriting to include the mission of the vehicle, schematics, equations, and their parameters.

The authors mention the following sentence: “The matrix G incorporates the predicted aerodynamic parameters into the appropriate positions within HGV motion equations, resulting in a physically realistic predicted state derivative ds(t)/dt.” However, this argument lacks demonstration. Equation (15) does not include aerodynamic parameters; thus, matrix G only incorporates the flight path angle and glider velocity.

Section 5 covers numerical simulations, but there are no real-time experiments. The authors do not explain how to implement the proposed algorithm in the HGV.

Author Response

Thank you for your valuable feedback on our manuscript. We appreciate your insightful comments and have carefully considered each point. Below, we address your concerns and outline the revisions made to the manuscript.

Q1: The section on the model vehicle is weak; it lacks diagrams, schematics, and vehicle parameters. Equations (3) are not well explained. Section 3 needs rewriting to include the mission of the vehicle, schematics, equations, and their parameters.
A1: Thank you for pointing this out. We acknowledge that the original presentation of the model could be improved. To address this, we have restructured Section 3 into two separate sections: Section 3, now titled "Background" and Section 4, titled "The PhysNODE Model".
In Section 3.1, we provide a comprehensive description of the HGV, including its mission profile, typical launch sequence, unique flight characteristics during the hypersonic glide phase, and typical maneuvering mode (Figure 1). This section also elaborates on the governing equations of motion for the HGV, providing clear definitions of each variable and explaining the aerodynamic forces involved.
Section 4.1 focuses on the principles of our proposed PhysNODE method, providing a schematic overview (Figure 2) that illustrates the model's architecture and the integration of data-driven and physics-based components.
Furthermore, we have included a dedicated table (Table 1) in Section 5, "Data" which details the specific parameters of the HGV model utilized in our numerical simulations.
We believe these revisions provide a clearer and more comprehensive presentation of the HGV model and the underlying principles of our methodology.

Q2: The authors mention the following sentence: “The matrix G incorporates the predicted aerodynamic parameters into the appropriate positions within HGV motion equations, resulting in a physically realistic predicted state derivative ds(t)/dt.” However, this argument lacks demonstration. Equation (15) does not include aerodynamic parameters; thus, matrix G only incorporates the flight path angle and glider velocity.
A2: Thank you for raising this concern. We acknowledge that our initial explanation of the matrix G was unclear. We have revised the relevant section (line 270) to provide a more accurate and detailed description of its function.
The matrix G does not directly incorporate the aerodynamic parameters themselves but acts as a mapping matrix that links the predicted aerodynamic parameters, contained in the vector $p(t)$ (Equation 15), to their corresponding terms within the state derivative vector $\frac{ds(t)}{dt}$ (Equation 14). As depicted in Figure 2/3, this mapping ensures that the predicted state derivative encompasses all variables required for a complete and physically consistent representation of the HGV motion, as defined by Equation 1. The final integrated state derivative is then used by the ODE solver (Equation 16) to predict the HGV trajectory.
We have rephrased the sentence on line 270 as follows:
"The matrix $G$ maps the predicted aerodynamic parameters to their corresponding terms within the HGV motion equations, producing a physically realistic predicted state derivative $\frac{ds(t)}{dt}$ that encompasses all variables required for the complete HGV motion equations."
We believe this revised explanation provides a clearer understanding of the role of matrix G within the PhysNODE model.

Q3: Section 5 covers numerical simulations, but there are no real-time experiments. The authors do not explain how to implement the proposed algorithm in the HGV.
A3: We appreciate your feedback regarding real-time experiments. However, conducting real-time experiments with hypersonic glide vehicles is extremely costly and, due to legal restrictions and security concerns in our region, such experiments are classified and not feasible for our research.
Therefore, our study focuses on validating the feasibility and effectiveness of our proposed method through numerical simulations, demonstrating its potential advantages in predicting HGV trajectories. We have expanded Section 4.1 to provide a more detailed explanation of the underlying principles of PhysNODE, including a schematic diagram (Figure 2) and a discussion of the iterative prediction process. We also compare our approach to PINNs, highlighting the key differences and advantages of our method.

We believe that our numerical simulations, combined with the detailed explanation of the model's principles and its comparison to existing methods, provide compelling evidence for the potential of PhysNODE as a valuable tool for HGV trajectory prediction.
We hope that these revisions address your concerns. Thank you again for your valuable feedback.

 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The author's response has largely addressed my concerns, but I would like to offer an additional suggestion. It would be beneficial if the author could present simulation results for the aircraft's path constraints to observe whether the predicted trajectory aligns with practical expectations.

Author Response

Thank you for the valuable suggestion. In response to your suggestion for presenting simulation results with the additional path constraints, we have incorporated these constraints into our updated model, PhysNODE-Path, an extension of PhysNODE that incorporates path constraints derived from intention inference. Specifically, we have added azimuth angle and distance from the target as path constraints, which are integrated with the original observed state vector to form an augmented state vector. This enhancement aims to improve prediction accuracy.

We have updated sections 6.1.2, 6.4, and 6.5 to include a detailed description of PhysNODE-Path and its performance analysis. Additionally, Figures 5, 6, 7, 8, and 9 have been updated to incorporate the results of the PhysNODE-Path model. While PhysNODE-Path consistently outperforms the standard PhysNODE, the magnitude of improvement is generally small. We believe this is primarily because, as discussed in our previous response, the standard PhysNODE model, by design, already indirectly incorporates path information through its physics-based modeling approach. Despite the modest improvement, this constraint-based improvement highlights the potential of integrating additional domain-specific knowledge within the PhysNODE framework. Building upon the foundation of physics-informed learning, exploring such integrations presents a promising avenue for further improving predictive performance, and we intend to explore this direction further in our future work.

The specific modifications are as follows:

(1) Section 6.1.2 Compared Methods: 
We have detailed the inclusion of additional path constraints and their integration into the PhysNODE framework. 

(2)Section 6.4 Model Performance Comparison Analysis:
We observed a small but consistent improvement in accuracy with PhysNODE-Path across varying training sample sizes, highlighting the benefit of integrating additional domain-specific information.

(3) Section 6.5 Forecast Accuracy over Time: 
We have noted that PhysNODE-Path, incorporating additional path constraints, exhibits slightly better accuracy, particularly in the first half of the flight. This improvement underscores the advantage of including path information for enhancing prediction reliability.

Thank you again for this insightful suggestion. We believe these revisions address your concerns and enhance the quality of the paper.