Physics-Informed Dynamics Modeling: Accurate Long-Term Prediction of Underwater Vehicles with Hamiltonian Neural ODEs

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript presents a Hamiltonian Neural Network (HNN) framework for modeling underwater vehicle dynamics, validated on the REMUS 100 AUV. The integration of physical priors with neural ODEs is promising and addresses key limitations of both purely mechanistic and purely data-driven methods. However, several aspects of the methodology and explanation of results could be improved for clarity, reproducibility, and scientific rigor.

- The second term of the loss function in Eq. 9 should be |H_q + p_t|
- In section 3.1 the variable p has been used for both position and velocity. Moreover, a schematic to show the kinematic model with the assumed variables would be useful to the reader with a limited expertise in the subject and help to improve the clarity of the model. 
- In eq. 13, the variable u (here the control input) is already used in velocity vector of eq 12. In general, I would recommend to add a nomenclature of all variables with their definition at the beginning of the manuscript. 

- (dynamic model, Port Hamiltonian formulation): I find the derivations too condensed for readers without deep expertise. maybe a schematic or flow diagram of how the classical hydrodynamic model transitions into the Hamiltonian formulation? and extending more the mathematical derivations in an appendix. 

 
- The eq.23 is not correct and is missing R(0) factor if assuming a special case of constant w. 

- (Training strategy) In the loss function, the relative weighting of the loss terms is not discussed, are they equally weighted or tuned empirically? because in  such a complex problem I assume the optimization is difficult and the loss terms would have different gradient flow which causes instability during optimization. 
- Would including an explicit energy conserving term in the loss improve stability over long horizons? 
- The Leapfrog scheme is well motivated, but it would be useful to justify the chosen time stem delta_t, how was it determined and how sensitive are results to this choice? 
- The model is trained using simulation data (ideal without any noise, or maybe far fraom real world scenarios). There is a risk of overfitting to a particular model. How to do you comment on this when applying your model on real world data or the methods to mitigate this? 
- The training data are cropped into 0.1 second samples. does such short sampling bias the model toward short term dynamics? 10 s prediction is very short, what about the longer time horizons, could the model be useful there? and how and what to improve in this model for such cases? 
- please proved information on the training time and strategy, computational architecture and so on. 

- In table 2, please list the name of the input and output variables to the neural networks
- In the figures presented in the results section, e.g. Fig 15, the roll ang, pitch ang,... are corresponding to which variables in the model?

- How sensitive is the model to the simplifying assumption (e.g. ignoring certain off-diagonal hydrodynamic terms) in the dynamic model? could there assumptions limit generalizability to high-speed AUVs?

- Are the results in figure 7 (M22=M33) are the trivial consequences of the network architecture or meaningful demonstartions of physical consistency? 
- The TCN slightly outperforms HNN in simple cases. This is important but not dicussed. Why HNN is underperform eventhough is physics informed. could this indicate over-parametrization?
-  Figures (13-18), the robustness of HNN uder complex inouts lacks quantitative metrics such as (RMSE, energy conservation, trajectory divergence rate or ...). Some error metrics across scenarios are needed to support claims beyond qualitative plots. 
- in the conclusion, maybe the authors acknowledge the limitation of the model for real world AUS trajectroies. How the model handle environmental disturbances (currents, waves) not present in the simulation? 

 

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The prediction of the long-term behavior of complex dynamical systems has been investigated in this study. The physics-informed framework has been built through the continuous-time dynamics of an Autonomous Underwater Vehicle (AUV) by discovering its underlying energy landscape. There are some collected results which have provided from this proposal. However, there are some drawbacks that the authors should address them to improve this study.

1. In the abstract, the certain results of this study should be provided clearly though the proposed model.
2. In the introduction, the authors must state the novelty points of this study. Not only providing the proposed methodologies which have been based on the previous studies through the statements: “In this study, we establish the neural ordinary differential equation (neural ODE) [19] as a fundamental framework to modeling underwater vehicle dynamics. This framework combines ordinary differential equations with neural networks starting only from the initial state of the system and using non-uniform observation data to model continuous dynamics. We then express the motion model of underwater vehicles in the form of Hamiltonian mechanics, using the Hamiltonian neural network (HNN)…”.
3. The abbreviation and nomenclature should be implemented to explain the equations from the proposed methods.
4. In table 1, the parameters of REMUS 100 AUV should be cited the original sources of these specific parameters.
5. The priority features of comparative results should be made clear to state the significance of this study.
6. The discussion should be expanded to verify with the experimental works.

Comments for author File: Comments.pdf

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed all comments of this reviewer. The current version is fine to be accepted for publishing on Journal of Marine Science and Engineering.