Deep Learning for Bifurcation Detection: Extending Early Warning Signals to Dynamical Systems with Coloured Noise
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors
This manuscript explores the simple idea that deep learning architectures can be used as a black box to detect the early warning signs of an impeding bifurcation in a stochastic dynamical system, using the normal forms of classic bifurcations. The claim is that this procedure is enough, and it does not require building a training set with highly heterogeneous random systems. The authors then claim that their deep learning models can perform well even if the problem becomes more challenging by adding correlated noise instead of white noise.
I believe the premises of the study are clear, and the results look promising. However, I am not sure whether what is shown in the manuscript justifies the conclusions. This is my main concern:
- One alternative hypothesis to explain the results in tested networks with red noise is that the structure of the noise barely plays an effect in the problem. The authors claim that red noise makes the problem harder, but I don’t see whether and how this is quantified. It could be that for these types of architecture, the correlations in the noise play very little roles.
And these are some more minor concerns:
- Figure 4 shoes opposite results in the comparison between red noise and white noise for the May’s harvesting dynamics and the Neural activation. Why is it the case? Apart from that, the differences seem negligible, which would be compatible with the alternative explanation above. Also, the author mention ““Further explanations are provided in the supplementary material”, I assume about Figure 4, because the explanation is very short, but I couldn’t find it.
- I am a bit concerned that this study just shows example traces of the prediction in the results, but there is very little quantification. It would be nice to show also the predicted solution in Null trials. Also, there are no errorbars. How consistent are the solutions? What if there is a second network trained on the exact dataset?
- It is unclear what exactly is the goal function. What would a zero loss prediction correspond to? Should it only predict the bifurcation at one point in the whole input trial? Also, some details are missing: how long are the sampled trials?
- The abstract mentions “using the normal form theorem” in the study, but the theorem is not really used. The authors leverage the normal forms of three types of bifurcations, as explicited afterwards.
Author Response
Attached is our detailed response to the reviewers’ comments.
With best regards,
Yazdan Babazadeh
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsMaghsoodlo and his colleagues conducted a study on applying deep - learning techniques to forecast the dynamics of three bifurcation normal forms. Additionally, they discovered that a deep - learning model trained with white noise could be applied to coloured noise. While I consider this to be a fascinating topic, the way the results are presented in this manuscript fails to substantiate their assertions. Overall, I think a great deal more analysis is required before this manuscript can be regarded as a scientific discovery. Consequently, I recommend rejecting this manuscript in its current form.
All the captions of the figures are presented in an extremely poor format. They lack any clarification regarding what each panel represents, which is highly unprofessional. As for Table 1, it only displays Accuracy. What about metrics such as recall, precision, and other relevant indicators?
Author Response
Attached is our detailed response to the reviewers’ comments.
With best regards,
Yazdan Babazadeh
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsIn this paper, the author describes “Deep Learning for Bifurcation Detection: Extending Early Warning Signals to Dynamical Systems with Coloured Noise”. It can become an interesting paper for mathematics after major revision. Followings are my comments.
- The introduction has to be improved. The authors should focus on extending the novelty of the current study.
- The novelty of the proposed method is unclear. There are a lot of machine learning algorithms. It is unclear why CNN-LSTM is needed to provide early warning signals (EWS) for bifurcations. It requires providing more information on how the literature informed the design of this study.
- The reviewer suggested that Section 6 be integrated into Section 3 to add an explanation of relevant modeling and testing theories.
- All hyperparameters (learning rate, mini-batch size, number of epochs, optimizer) and model complexity of the proposed method should be detailed.
- What are the baseline models and benchmark results? The authors can compare the results with existing models evaluated using the dataset.
- It requires providing more information on the validation of design through proper experiment design and analysis of results of Figs. 1 to 3 by using statistical methods.
- Authors have to provide the descriptions of subfigures in the figure captions of Figs. 1 to 4 in this manuscript.
- Please first present the full form, before using an abbreviation: ROC, AUC, DL, CNN, LSTM, and so on.
- As the journal is printed in black and white, please make the different markers for the different results in Figs. 1, 2, and 3.
- The style of references is not in agreement with this Journal style, and thus, it requires a revision.
The English could be improved to more clearly express the research.
Author Response
Attached is our detailed response to the reviewers’ comments.
With best regards,
Yazdan Babazadeh
Author Response File: Author Response.pdf
Reviewer 4 Report
Comments and Suggestions for AuthorsPlease check the attached file for my review comments.
Comments for author File: Comments.pdf
Author Response
Attached is our detailed response to the reviewers’ comments.
With best regards,
Yazdan Babazadeh
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Comments and Suggestions for Authors I believe the authors have made substantial improvements to the manuscript. However, the paper still exhibits significant unprofessionalism. Furthermore, with additional results included, the problem addressed in the manuscript appears less challenging than initially presented. The ROC curve indicates an AUC approaching 1, yet the empirical data solely comprises Hopf bifurcation data, which inadequately tests the model's full capabilities. Another issue is the inconsistent use of abbreviations throughout the paper—they appear randomly without standardization (I suspect the authors may have relied on a large language model to polish the text without verifying paragraph consistency). Notably, Figures 1 and 2 share identical titles despite depicting distinct concepts. Considering all these factors, I conclude that this manuscript is unsuitable for publication in mathematics.Author Response
Please find attached our detailed point-by-point response. We have addressed all concerns raised and revised the manuscript accordingly. We appreciate the constructive feedback, which has helped us improve the clarity and rigor of the paper.
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThis paper is a revised manuscript of the previous mathematics-3674297. Almost all of the comments are revised, but some important ones are not enough, as shown below, including related ones. It is an interesting paper for mathematics after major revision. Please reconsider and revise them according to the comments.
- In Table 1, authors have to explain in detail how these two algorithms apply to the model with those parameters. In addition, authors have to explain the hyperparameters of these two algorithms. Or authors cite the paper for the results based on these two algorithms.
- What are the baseline models and benchmark results? The authors have to compare the results with existing models evaluated using the dataset.
- Please first present the full form before using an abbreviation.
The English could be improved to more clearly express the research.
Author Response
Please find attached our detailed point-by-point response. We have addressed all concerns raised and revised the manuscript accordingly. We appreciate the constructive feedback, which has helped us improve the clarity and rigor of the paper.
Author Response File: Author Response.pdf
Reviewer 4 Report
Comments and Suggestions for AuthorsIn the second version of the submitted manuscript, Section 2 is presented without numbering, which causes confusion when cross-referencing. As I was reviewing the paper, I looked for Section 5 as referenced multiple times in the response letter. However, I found that the manuscript only contains four numbered sections. As a result, I had to manually reassign the section numbers, treating the final Section 4 as Section 5 in order to align the response letter with the manuscript.
This inconsistency in section numbering should be carefully reviewed and corrected to ensure clarity and accurate correspondence between the manuscript and the response to reviewers.
Additionally, in the confusion matrices provided, the class labels are not clearly indicated. In order to properly interpret the classification results and understand how errors are distributed across different classes, the confusion matrix should explicitly show the class names for each row and column. Without this information, it is difficult to assess which classes are being misclassified and to what extent.
Author Response
Please find attached our detailed point-by-point response. We have addressed all concerns raised and revised the manuscript accordingly. We appreciate the constructive feedback, which has helped us improve the clarity and rigor of the paper.
Author Response File: Author Response.pdf
Round 3
Reviewer 3 Report
Comments and Suggestions for AuthorsReviewer recommends accepting without comments.
Comments on the Quality of English LanguageThe English could be improved to more clearly express the research.
Author Response
Thanks. We have checked and improved the English.
