Machine Learning Topological Invariants in Disordered Systems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the manuscript, the authors presented “Machine Learning Topological Invariants In Disordered Systems”. The study is good; however, several areas need further attention and clarification. Please see my detailed comments below:

1. Add a dedicated abbreviations section in which all acronyms and abbreviations appearing in the manuscript are defined at first use and listed comprehensively (e.g., FNN, SHH, etc.).
2. Add a dedicated section introducing all parameters used in the manuscript (e.g., Φ, T, etc.).
3. The authors should carefully recheck all annotations on the figures and implement any necessary corrections (e.g., Fig. 3).
4. For better clarification, the authors should introduce a real example of a disordered system, solve the problem using their proposed model, and compare the results with previously studied cases to demonstrate the model’s capabilities.
5. For completeness, include a comparative table that benchmarks the current study against other related works.
6. For accuracy, please reexamine key equations (e.g., Equation 5) and implement any needed corrections.
7. Conduct a thorough review of the manuscript, including all figures and tables, to correct typographical errors and improve overall quality to meet the journal’s standards (e.g., correct “expection”, etc.).
8. Please consider using a non-linear neural network instead of a linear model, using ReLU as well as Sigmoid as activation functions, and present/discuss the results (for Fig. 3a).

Comments on the Quality of English Language

Conduct a thorough review of the manuscript, including all figures and tables, to correct typographical errors and improve overall quality to meet the journal’s standards (e.g., correct “expection”, etc.).

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

 

The authors present a data-driven approach to identify topological phases in disordered SSH models using LSTM networks trained on population dynamics. Their work is technically sound, clearly motivated, and demonstrates a convincing advantage of recurrent architectures over feedforward networks in capturing disorder-induced global correlations. The inclusion of a deployable web-based prediction tool enhances the practical impact of the study. However, there are a few concerns that in my view should be addressed as follows:

1- The LSTM shows excellent classification accuracy; however, authors should sufficiently address what physical features the network is learning from the population dynamics. In my view, an analysis of interpretability, such as sensitivity or saliency maps along the lattice/time dimension, comparison between LSTM outputs and known physical quantities (e.g., chiral displacement, edge localization length), or ablation studies removing long-time or long-range correlations to demonstrate their necessity can give the reader the physics insight beyond black-box prediction.

2-The authors do not clearly state whether disorder realizations used in testing are fully independent from those used in training, particularly when concatenating multiple disorder sequences. So, I suggest clarifying: How disorder realizations are split between training, validation, and test sets, whether the model generalizes to unseen disorder distributions (e.g., different disorder strengths or distributions).

3-The justification for using LSTM focuses on limitations of FNNs, but does not compare against other modern sequence models. Thus consideration of a benchmarking comparison with: GRU-based RNNs, 1D temporal/spatial CNNs with large receptive fields, or (Optionally) transformer-style attention models, can contextualize the LSTM choice and strengthen the machine-learning contribution.

4-All simulations are performed for a fixed system size (200 unit cells). It is unclear how the trained model scales with lattice length. I recommend adding a brief analysis showing: Performance when testing on larger or smaller chains than those used in training, or a discussion explaining whether retraining is required for different system sizes.

5- The winding number has been computed for labeling but the relationship between the LSTM output and the disorder-averaged winding number has not been quantitatively analyzed. Thus, correlation plots between predicted outputs and time-averaged chiral displacement values and error analysis near phase boundaries where the winding number fluctuates are required. 

 

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript "Machine Learning Topological Invariants In Disordered Systems" proposes a framework based on LSTM (Long Short-Term Memory) neural networks to classify topological phases in the disordered SSH (Su-Schrieffer–Heeger) model. The approach uses site population dynamics as sequential input data, allowing the model to learn long-range spatiotemporal correlations, which are essential for characterizing disordered systems where translational symmetry is broken. The results demonstrate that the LSTM significantly outperforms feedforward neural networks (FNN) in classification accuracy, especially as the amount of data increases. Furthermore, the authors develop an interactive web tool (front-end in Vue.js, back-end in Flask) for real-time prediction of topological phases based on user-defined parameters. The work suggests that LSTM is a robust and generalizable architecture for the study of disordered topological systems.

The manuscript proposes the use of LSTM networks for the analysis of disordered topological systems. The justification is solid: since disorder breaks translational symmetry, conventional topological invariants fail, and recurrent networks can capture long-range spatiotemporal correlations that feedforward networks ignore. The choice of LSTM is well-grounded in the sequential machine learning literature.

The development of an interactive web tool (front-end in Vue.js + back-end in Flask) is a relevant differentiator. This not only demonstrates the model's feasibility but also makes it accessible to other researchers, educators, or students, promoting reproducibility and scientific dissemination. The manuscript properly situates the problem within the field of condensed matter physics and reviews key concepts (SSH model, topological invariants, effects of disorder). The explanation of the mathematical model and data generation is sufficiently detailed to allow for the reproduction of the study.

Regarding the figures; they are informative and well-designed, especially those showing the population dynamics (Figure 2) and the bidirectional LSTM architecture (Figure 3e). This facilitates understanding even for readers less familiar with machine learning.

Regarding the references, some corrections are necessary. Reference 18 (Meier et al., 2018) appears to be duplicated with reference 16. References 44 (Rahaman et al., 2022) and 45 (Goh et al., 2023) are mentioned in the text (pages 2 and 10) but are not listed in the references. The citations "[44]" and "[45]" in the text (pages 2 and 10) seem to lack corresponding entries in the final list. There are also some formatting errors and inconsistencies in the references (e.g., incomplete authors, truncated titles).

In summary, the manuscript is technically solid and contributes significantly to the intersection of machine learning and theoretical physics. I therefore recommend the publication of the manuscript after the references are corrected.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript presents a machine learning based framework using LSTM networks to classify topological phases in the disordered SSH model. By processing population dynamics as sequential data, the authors demonstrate that LSTM models outperform FNNs in recognizing topological phases in the presence of disorder, but it currently lacks sufficient validation depth and physical interpretability. I think, by addressing the following comments will make this a valuable and citable contribution to computational condensed matter physics.

• The manuscript would benefit from a clearer articulation of the novelty relative to prior works such as Zhang et al. (Phys. Rev. Lett. 2018, Ref. 24 in the submitted manuscript) and Sun et al. (Phys. Rev. B 2018, Ref. 23 in the submitted manuscript), both of which employed neural networks for topological invariant prediction. The distinction between using population dynamics as sequential inputs and using Hamiltonian data should be emphasized more concretely.
• Quantitative metrics beyond accuracy (e.g., confusion matrices, precision/recall, or ROC curves) are missing. Since classification accuracy reaches 99.9%, additional validation or cross-testing on unseen data is necessary to ensure robustness and avoid overfitting. I suggest the authors discuss computational cost and scalability by how model performance scales with system size (number of sites), and longer time evolutions are not addressed.
• The physical interpretability of the learned features is not discussed; for example, are there identifiable temporal signatures corresponding to phase transitions? Also, a discussion connecting these learned representations to known topological markers or symmetry properties would significantly enhance the physics insight.
• The captions of the figures are overly descriptive; it would help to highlight the physical insights these figures convey. Moreover, the color scale of the figure 4 is not annotated clearly.
• In the introduction, the authors should highlight the gap that this work fills. Also, several typographical errors exist (e.g., “feedforward neural netnetworks,” “streghth,” “Fun(W,p)”). Finally, the authors mention “experimentally measurable population dynamics,” but no discussion of experimental feasibility or noise robustness is provided.

Author Response

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Reviewer 5 Report

Comments and Suggestions for Authors

The core idea—using LSTM networks to classify topological phases in disordered systems via population dynamics—is novel and scientifically sound. The demonstrated superiority of LSTM over FNN for capturing long-range correlations is a valid and valuable result. However, the manuscript is unpublishable in its current state due to severe presentational flaws:

1. Structure & Narrative: Poorly organized, with abrupt transitions, misplaced sections (e.g., data generation), and incorrect/inconsistent figure references, disrupting the logical flow.

2. Figures & Visualization: The graphics are unprofessional and unacceptable for publication. They appear as unrefined, low-resolution screenshots or hastily generated plots with poor color schemes and labeling. This reflects a lack of basic academic rigor and severely undermines credibility.

3. Technical Gaps: Key details are missing, particularly regarding the implementation and integration of the web-based tool (Flask backend). The discussion on why FNN fails remains superficial.

4. Language & Proofreading: Contains typos, grammatical errors, and awkward phrasing, indicating insufficient revision.

5. Some recent related works on topological invariance in disorders and with AI were missing, e.g. Light Sci Appl 13, 314 (2024), eLight 5, 26 (2025), arXiv:2509.05727, and so on

While the scientific contribution is promising, the manuscript reads like an unpolished draft, requiring a complete restructuring, replacement of all figures with publication-quality vector graphics, expansion of methodological details, and thorough language editing before it can be considered for peer review.

Author Response

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Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have addressed most of the comments and concerns; however, there are still minor issues that warrant attention. For example, in three instances, the manuscript uses the term ‘expection value,’ which should be corrected to ‘expectation values.’

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed most of my comments and the manuscript can be accepted for publication.

Author Response

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Reviewer 4 Report

Comments and Suggestions for Authors

There is no further comment; the authors addressed all the comments well.

Author Response

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Reviewer 5 Report

Comments and Suggestions for Authors

The authors have addressed most of the comments. But the figures are still poor, for instance, it is not good to use same colormap to plot different physical value distributions, that very induce confusions, also the schematic diagram is very unprofessional, not in good design. I don‘t agree the author regard the suggested papers not related, which are indeed related, for instance, the LSA 13, 314 (2024) showed skyrmion topology of light protected in disordered media, which is a very important and novel disorder-induced topology of light. Also the AI with structured light elight review also discussed how to use AI to identify topology of light. It is disappointing that the authors didn't take action in their paper.

Author Response

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