Deep Embedding Koopman Neural Operator-Based Nonlinear Flight Training Trajectory Prediction Approach

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Here are my comments:

1. Although the introduction gives a decent summary, it is devoid of specific background information about earlier approaches and their drawbacks. To clearly position the originality of this work versus existing models, a more complete literature analysis would be beneficial. As a result, think about adding more concrete instances of the present issues in flight trajectory prediction that this model solves.

2. A more detailed description of the Koopman operator's integration with the neural network should be included in the methodology section. Understanding the algorithm in detail might improve understanding.

3. Give more details on the neural network architecture and procedure chosen. Why did specific hyperparameters get selected? The repeatability of this data depends on it.

4. The Koopman operator theory and its application in this model are not completely explained in the mathematical formulations. Make sure every equation has a clear definition and that all of the variables and parameters are explained.

5. There should be a more thorough description of the datasets (such as the CAFUC dataset) that were utilized. Provide details on the size of the training and testing divides, feature selection, and data pre-processing procedures.

6. A thorough comparison of the suggested model with baseline models ought to be included in the findings section. Tests for statistical significance should be included to verify performance gains.

7. The model's physical interpretability via operator visualization is mentioned in the manuscript. Nevertheless, there aren't enough specifics or illustrations to fully convey this point. Provide additional visual aids (charts, graphs, etc.) to show how the model's predictions correspond with actual occurrences.

8. The research's wider significance should be emphasized in the conclusion in addition to summarizing the results. Talk about how this model could affect advancements in FOQA and flight trajectory prediction in the future.

9. There are several instances of complex and convoluted sentences. Simplify the language to improve readability and ensure that technical jargon is adequately explained.

Comments on the Quality of English Language

Moderate editing of English language required

Author Response

Comments 1:Although the introduction gives a decent summary, it is devoid of specific background information about earlier approaches and their drawbacks. To clearly position the originality of this work versus existing models, a more complete literature analysis would be beneficial. As a result, think about adding more concrete instances of the present issues in flight trajectory prediction that this model solves.

Response 1:We have added specific information about early methods and their shortcomings in lines 67-80. Additionally, we included some literature analysis on flight trajectory prediction in lines 216-228. The actual problems solved by this model are discussed in the conclusion.

Comments 2:A more detailed description of the Koopman operator's integration with the neural network should be included in the methodology section. Understanding the algorithm in detail might improve understanding.

Response 2:In response, we added a description of the algorithm in line 283.

Comments 3:Give more details on the neural network architecture and procedure chosen. Why did specific hyperparameters get selected? The repeatability of this data depends on it.

Response 3:Thank you for your suggestion. We added a detailed description of the neural network architecture in line 368. We have already explained the choice of specific hyperparameters in section 3.

Comments 4:The Koopman operator theory and its application in this model are not completely explained in the mathematical formulations. Make sure every equation has a clear definition and that all of the variables and parameters are explained.

Response 4:We reviewed all mathematical formulas and provided explanations for the unexplained variables in equations (16) and (32), added in lines 330-332 and 372-375, respectively.

Comments 5:There should be a more thorough description of the datasets (such as the CAFUC dataset) that were utilized. Provide details on the size of the training and testing divides, feature selection, and data pre-processing procedures.

Response 5:We greatly appreciate your comments. In lines 392-402, we added a more detailed description of the CAFUC dataset and included the division of training and test sets. Data preprocessing and feature selection are already explained in section 4.2.

Comments 6:A thorough comparison of the suggested model with baseline models ought to be included in the findings section. Tests for statistical significance should be included to verify performance gains.

Response 6:Thank you for the suggestion. We added an explanation of the significance tests between models in lines 459-464.

Comments 7:The model's physical interpretability via operator visualization is mentioned in the manuscript. Nevertheless, there aren't enough specifics or illustrations to fully convey this point. Provide additional visual aids (charts, graphs, etc.) to show how the model's predictions correspond with actual occurrences.

Response 7:We have already included figures in Table 2, but without explanation. We added explanations for the figures in lines 484-486.

Comments 8:The research's wider significance should be emphasized in the conclusion in addition to summarizing the results. Talk about how this model could affect advancements in FOQA and flight trajectory prediction in the future.

Response 8:In response, we added some prospects and discussions on FOQA and flight trajectories in lines 510-514.

Comments 9:There are several instances of complex and convoluted sentences. Simplify the language to improve readability and ensure that technical jargon is adequately explained.

Response 9:Thank you for this comment. We had our British colleagues review the manuscript, and we have made an effort to revise the complex and difficult-to-understand sentences.

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript presents a novel deep learning model for predicting flight training trajectories using a Koopman Neural Operator approach. This method addresses the challenges posed by nonlinear chaos, unconstrained airspace maps, and randomized driving patterns by leveraging modern Koopman operator theory and dynamical system identification.

1. The integration of Koopman operator theory with deep learning for trajectory prediction is novel and presents a fresh perspective on handling nonlinear chaotic systems.

2. The use of stacked neural networks to create a scalable depth approximator is well-explained and promising.

3. The model's ability to gain physical interpretability through operator visualization and generative dictionary functions adds significant value for practical applications.

4. The manuscript lacks a detailed comparative analysis with existing state-of-the-art trajectory prediction models. Consider and compare the following Deep Embedding methods presented by now, 
Elastic deep autoencoder for text embedding clustering by an improved graph regularization
A novel link prediction algorithm for protein-protein interaction networks by attributed graph embedding

5. There is insufficient discussion on the validation of the proposed model in real-world scenarios beyond the presented dataset.

6. The manuscript does not thoroughly address the computational efficiency and resource requirements of the proposed method.

7. Include a detailed comparison of the proposed method with other leading models in trajectory prediction. This should highlight the strengths and limitations of each approach.

8. Expand on the discussion of the model's applicability in real-world scenarios. Provide insights into how the model can be integrated into existing flight training and aviation safety systems.

9. Discuss the computational efficiency of the proposed model. Provide information on the training time, resource requirements, and potential scalability issues.

10. Enhance the literature review by including more recent works on trajectory prediction and Koopman operator applications in other domains.

11. Ensure that all figures and tables are clearly labeled, and their relevance to the text is explicitly stated. More detailed captions could enhance comprehension.

12. Include pseudocode or a more detailed description of the algorithm to improve reproducibility and understanding.

13. Provide more details about the datasets used for experiments, including the size, source, and any preprocessing steps.

14. Address the major and minor comments to improve the manuscript's clarity, depth, and applicability.

15. Ensure adherence to the journal's formatting and submission guidelines.

Overall, the manuscript presents a promising and innovative approach to flight training trajectory prediction. With additional comparative analysis, discussion on real-world applications, and details on computational efficiency, the manuscript could significantly contribute to the field of aviation safety and training.

Author Response

Comments 1:The integration of Koopman operator theory with deep learning for trajectory prediction is novel and presents a fresh perspective on handling nonlinear chaotic systems.

Response 1:Thank you for recognizing our approach.

Comments 2: The use of stacked neural networks to create a scalable depth approximator is well-explained and promising.

Response 2:Thank you for recognizing our approach.

Comments 3:The model's ability to gain physical interpretability through operator visualization and generative dictionary functions adds significant value for practical applications.

Response 3:Thank you for recognizing our approach.

Comments 4:The manuscript lacks a detailed comparative analysis with existing state-of-the-art trajectory prediction models. Consider and compare the following Deep Embedding methods presented by now,

Elastic deep autoencoder for text embedding clustering by an improved graph regularization A novel link prediction algorithm for protein-protein interaction networks by attributed graph embedding

Response 4:Thank you for your suggestion. We carefully reviewed the literature you recommended and agreed that adding an analysis of these references would make our paper more comprehensive. We added this analysis in lines 162-168. We also included a comparative analysis with state-of-the-art trajectory predictions in lines 216-228.

Comments 5: There is insufficient discussion on the validation of the proposed model in real-world scenarios beyond the presented dataset.

Response 5:We added a discussion on the model's validation in real-world scenarios in lines 497-506.

Comments 6:The manuscript does not thoroughly address the computational efficiency and resource requirements of the proposed method.

Response 6:Thank you for your comment. We added an analysis of the model's computational resources and speed in lines 473-478.

Comments 7: Include a detailed comparison of the proposed method with other leading models in trajectory prediction. This should highlight the strengths and limitations of each approach.

Response 7:We included a comparative analysis with state-of-the-art trajectory predictions in lines 216-228.

Comments 8:Expand on the discussion of the model's applicability in real-world scenarios. Provide insights into how the model can be integrated into existing flight training and aviation safety systems.

Response 8:We added insights into how the model could be integrated into flight training and aviation safety systems in lines 510-514.

Comments 9:Discuss the computational efficiency of the proposed model. Provide information on the training time, resource requirements, and potential scalability issues.

Response 9:We added an analysis of the model's training time, resource requirements, and scalability in lines 473-478.

Comments 10:Enhance the literature review by including more recent works on trajectory prediction and Koopman operator applications in other domains.

Response 10:We added a literature analysis of the applications of Koopman operators in other fields in lines 162-168.

Comments 11: Ensure that all figures and tables are clearly labeled, and their relevance to the text is explicitly stated. More detailed captions could enhance comprehension.

Response 11:We revised the captions for Table 2 and Figure 7, adding explanatory notes to make them easier to understand.

Comments 12: Include pseudocode or a more detailed description of the algorithm to improve reproducibility and understanding.

Response 12:We greatly appreciate your comment. We added a description of the algorithm in line 283.

Comments 13:Provide more details about the datasets used for experiments, including the size, source, and any preprocessing steps.

Response 13:We included a more detailed description of the CAFUC dataset in lines 392-402. Data preprocessing is already explained in section 4.2.

Comments 14:Address the major and minor comments to improve the manuscript's clarity, depth, and applicability.

Response 14:Thank you for your comment. We have addressed the primary and secondary comments to the best of our ability.

Comments 15:Ensure adherence to the journal's formatting and submission guidelines.

Response 15:Thank you for your comment. We will ensure compliance with the journal's formatting and submission guidelines.