An Improved PINN Algorithm for Shallow Water Equations Driven by Deep Learning
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe Manuscript entitled: "An Improved PINN Algorithm for Shallow Water Equations Driven by Deep Learning" deals with solving shallow water equations is crucial in science and engineering for understanding and predicting natural phenomena. To address the limitations of Physics-Informed Neural Network (PINN) in solving shallow water equations, we propose an improved PINN algorithm integrated with a deep learning framework. The algorithm introduces a regularization term as a penalty in the loss function, based on the PINN and Long Short-Term Memory (LSTM) models, and incorporates an attention mechanism to solve the original equation across the entire domain. Simulation experiments were conducted on one-dimensional and two-dimensional shallow water equations. The improved algorithm shows significant advantages in handling discontinuities, such as sparse waves, in one-dimensional problems.
I have the following comments.
1. The paper lacks sufficient justification for the use of LSTM and attention mechanisms, relying heavily on L_{1} regularization without exploring alternatives.
2. Connect the improved algorithm with real world problems.
3. compare other modern PDE-solving methods with classical PINN.
4. Computational overhead introduced by LSTM and attention mechanisms is not discussed.
5. Weight selection in the loss function is manually handled, with only a brief mention of adaptive mechanisms.
6. Include a comparison with alternative algorithms like DGM or cPINN to highlight the improved method's competitiveness.
Comments on the Quality of English LanguageMinor editing required in english language.
Author Response
Please see the attachment.
Author Response File: 
Author Response.docx
Reviewer 2 Report
Comments and Suggestions for Authors1. The introduction provides a decent overview of the context and motivation for the research, but it could be enhanced by providing a more structured and comprehensive literature review. Specifically, the authors could consider organizing the related work into thematic categories, highlighting the key contributions and limitations of each approach, and explicitly stating how their proposed method addresses the existing gaps. Additionally, the authors could ensure that all relevant references are included and cited.
2. The reference list in this paper appears reasonably up-to-date, with several references from 2021, 2022, and even a few from 2023. However, given the rapid advancements in the field of machine learning and its applications to solving partial differential equations, it is always advisable for the authors to conduct a final literature search before submission to ensure they haven't missed any very recent and highly relevant work.
Specifically, the authors could consider checking for any new developments in physics-informed neural networks (PINNs), long short-term memory (LSTM) networks, attention mechanisms, and their applications to shallow water equations. They could also look for any recent benchmark datasets or evaluation metrics that could be used to further validate their proposed method.
3. The research design seems appropriate for the stated objectives. The authors clearly define the problem, propose a novel solution, and conduct experiments to evaluate its effectiveness. The use of both one-dimensional and two-dimensional shallow water equations as test cases is also reasonable, as it allows for assessing the performance of the proposed method under different scenarios.
4. The methods section provides a reasonable description of the proposed algorithm, but some aspects could be clarified further. For instance, the authors could provide more details on the specific architecture of the LSTM network, the implementation of the attention mechanism, and the choice of hyperparameters. Additionally, the authors could consider including a flowchart or pseudocode to illustrate the overall workflow of the algorithm.
5. The results are presented clearly and concisely, with the use of figures and tables to illustrate the key findings. The authors also provide a detailed discussion of the results, highlighting the strengths and limitations of the proposed method.
6. The conclusions are well-supported by the results presented in the paper. The authors provide evidence that the proposed method outperforms the classical PINN algorithm in terms of handling discontinuities, reducing oscillations, and capturing fine details. The limitations of the method are also acknowledged, suggesting directions for future research.
7. The image quality in some figures needs to be enhanced, as the legends and fonts are too small.
I also give some suggestions for improvements in the attachment.
Comments for author File: 
Comments.pdf
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Reviewer 3 Report
Comments and Suggestions for AuthorsThe introduction of LSTM and attention mechanisms into the PINN framework is a novel contribution. The combination of neural networks with the physics of PDEs allows for robust solutions that avoid some pitfalls of classical PINN approaches, such as spurious oscillations and poor handling of discontinuities.
The authors validate their proposed algorithm through experiments on one-dimensional and two-dimensional shallow water equations. The results convincingly demonstrate the superiority of the improved PINN model over classical approaches, particularly in handling complex wave phenomena with higher accuracy and stability.
The improved model, while more accurate, may introduce additional computational complexity. The integration of LSTM networks, attention mechanisms, and regularization terms increases the number of parameters and layers to be optimized, which may result in longer training times or higher resource requirements. However, the paper does not provide a detailed comparison of computational costs between the classical and improved PINN models.
While the improved PINN model is compared against classical PINNs, it would have been valuable to benchmark it against other contemporary deep learning methods or hybrid algorithms used in solving PDEs, such as convolutional neural networks (CNNs) or other meta-learning approaches. This would provide further context for the improvement margins.
Although the experiments focus on idealized dam-break problems, more discussion on how this improved algorithm can be applied to real-world cases—such as flood simulations, coastal engineering, or other fluid dynamic scenarios—would enhance the practical relevance of the work.
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThe author response all the comments. From the response of the author, I a satisfied. So, I accept the manuscript for publication.
