Coupling of Lagrangian Mechanics and Physics-Informed Neural Networks for the Identification of Migration Dynamics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper presents a novel approach by integrating Lagrangian mechanics and physics-informed neural networks (PINNs) to model migration dynamics. It offers fresh insights into migration research, but there are areas that need improvement.

1. The authors should explain why such datasets were chosen for the experiments and what the specific selection criteria were.

2. The architecture of the neural network is unclear. For example, how many neurons per layer and the specific hardware environment in which the experiments were performed are not described.

Comments on the Quality of English Language

The manuscript is generally well-written, with a strong command of terminology in the fields of physics, machine learning, and migration studies.

Author Response

Dear reviewer,

Thank you for your valuable feedback on our article. We appreciate your comments and suggestions, which have helped us to improve the quality of the content.

Comment 1: The authors should explain why such datasets were chosen for the experiments and what the specific selection criteria were.

Response 1: Thank you for your suggestion. On page 18, we have added a couple of points regarding the reasons for selecting the data. The primary reason is linked to the openness and completeness of information in the data.

Comment 2: The architecture of the neural network is unclear. For example, how many neurons per layer and the specific hardware environment in which the experiments were performed are not described.

Response 2: On page 13, we have added detailed information regarding the network architecture as well as the training process. Additionally, we have provided information on software and hardware requirements.

Reviewer 2 Report

Comments and Suggestions for Authors

I think the paper can benefit to a wider range of scholars if the authors properly cite other relevant applications where the analysis can be similar, such as BT models, and other settings of NN and functions that can be used for the numerical counter part. Yust as an example:
Abbas, Mudassar, et al. "PDE models for vegetation biomass and autotoxicity." Mathematics and Computers in Simulation 228 (2025): 386-401.
Auricchio, Ferdinando, et al. "On the accuracy of interpolation based on single-layer artificial neural networks with a focus on defeating the Runge phenomenon." Soft Computing (2024): 1-19.
Calabrò, Francesco, et al. "Refinable functions, functionals, and iterated function systems." Applied Mathematics and Computation 272 (2016): 199-207.

Author Response

Dear reviewer,

Thank you for your valuable feedback on our article. We appreciate your comments and suggestions, which have helped us to improve the quality of the content.

Comment 1: I think the paper can benefit to a wider range of scholars if the authors properly cite other relevant applications where the analysis can be similar, such as BT models, and other settings of NN and functions that can be used for the numerical counter part. 

Response 1: Thank you for the suggestion. On page 4, we have added an article to our Related Works "Abbas, Mudassar, et al. "PDE models for vegetation biomass and autotoxicity." Mathematics and Computers in Simulation 228 (2025): 386-401".

Reviewer 3 Report

Comments and Suggestions for Authors

This paper introduces a Lagrangian mechanics-based framework for modeling migration dynamics, integrating Physics-Informed Neural Networks (PINNs) to learn governing equations from data while incorporating physical constraints. The authors derive and interpret the potential energy of a migration network by introducing specific functions that define migration patterns, allowing for a physics-based understanding of mobility trends. The study highlights how economic factors and external force factors shape migration trends, offering a novel approach to computational migration modeling with physics-informed learning. I have the following comments for improvement:

1. To better assess the model’s effectiveness, it should be compared with both traditional migration models and machine learning approaches such as simple neural networks. A performance comparison experiment with appropriate metrics would help clarify the advantages of the PINN framework over both general ML models and conventional approaches.
2. To help determine which components are essential and critical for improving model performance, it’s recommended to conduct an ablation analysis that removes specific terms in the Lagrangian formulation or PINN constraints to assess their impact on accuracy.
3. A more detailed discussion of hyperparameter choices (e.g., hidden layer size, hidden layer number, activation functions, …) is needed. It would be helpful to include a sensitivity analysis to show how these parameters affect model accuracy and stability.
4. Figures, tables, and equations should explicitly indicate units and scales where applicable to prevent ambiguity.
5. It's recommended to improve the presentation of results using more types of visualization types, e.g. flow maps, vector fields, or heatmaps. May also consider overlaying predicted migration flows with historical data on the same spatial-temporal map..

Comments on the Quality of English Language

The manuscript is clear and well-structured. Minor grammar refinements are needed.

Author Response

Dear reviewer,

Thank you for your valuable feedback on our article. We appreciate your comments and suggestions, which have helped us to improve the quality of the content.

Comment 1: To better assess the model’s effectiveness, it should be compared with both traditional migration models and machine learning approaches such as simple neural networks. A performance comparison experiment with appropriate metrics would help clarify the advantages of the PINN framework over both general ML models and conventional approaches.

Response 1: Thank you for your comment. The aim of this research is to develop a theoretical framework and identify macroeconomic and social variables that influence the heterogeneity function, and consequently, can impact the dynamics of migration flows. Your suggestion regarding the addition of comparisons with other machine learning techniques was not part of the original scope of this project, as we have already addressed this aspect in a previous publication "Zakharov, K.; Aghajanyan,A.; Kovantsev, A.; Boukhanovsky, A. Forecasting Population Migration in Small Settlements Using Generative Models under Conditions of Data Scarcity. Smart Cities 2024, 7,2495–2513. https://doi.org/10.3390/smartcities7050097".

Comment 2: To help determine which components are essential and critical for improving model performance, it’s recommended to conduct an ablation analysis that removes specific terms in the Lagrangian formulation or PINN constraints to assess their impact on accuracy.

Response 2: Thank you for your suggestion regarding ablation analysis. At this time, we are planning to write a separate article on this topic, as the current work has already covered the theoretical aspects of the model in detail. In our next project, we intend to explore different forms of potential energy and investigate their effect on migration dynamics.

Comment 3: A more detailed discussion of hyperparameter choices (e.g., hidden layer size, hidden layer number, activation functions, …) is needed. It would be helpful to include a sensitivity analysis to show how these parameters affect model accuracy and stability.

Response 3: On page 13, we have added detailed information regarding the network architecture as well as the training process. Additionally, we have provided information on software and hardware requirements.

Comment 4: Figures, tables, and equations should explicitly indicate units and scales where applicable to prevent ambiguity.

Response 4: Thank you for the suggestion, we have added units of measurement in FIgures 8, 9, 11, 15 and we updated the text for Table 1 and description for potential energy.

Comment 5: It's recommended to improve the presentation of results using more types of visualization types, e.g. flow maps, vector fields, or heatmaps. May also consider overlaying predicted migration flows with historical data on the same spatial-temporal map.

Response 5: Unfortunately, due to the large number of variables in our system, specifically, the number of regions in Russia (16) and the USA (51), as well as the number of velocities, it would be impossible to represent the phase portrait of the system in the form of a vector field of flow maps. At the same time, reducing the number of dimensions may not provide sufficient information about the original vector field.