A Clustering-Based Dimensionality Reduction Method Guided by POD Structures and Its Application to Convective Flow Problems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The current paper proposed the C-POD method for constructing reduced order model with different applications. Compared with the conventional POD, the current method adopts clustering method to better identify the data similarities. Also, the new ranking method has been added to rank the different modes.

- The methods section needs more discussions especially regarding the ranking method.

- The physical meanings of the new C-POD method are not very clear as it is a data-driven method.

- The format of the references is not consistent. Please correct.

Author Response

Comments 1: The methods section needs more discussions especially regarding the ranking method.

Response 1: Thank you very much for your valuable suggestion. We appreciate your attention to the clarity and completeness of the methods section, especially the ranking method for modes. To address this concern, we have significantly expanded Section 2.2 of the revised manuscript, where we provide a comprehensive and step-by-step explanation of the Entropy-Controlled Euclidean-to-Probability Mapping (ECEPM) method. In particular, we elaborated on:

• In Section 2.2, part of the first paragraph was revised by removing some redundant descriptions regarding the motivation and overall concept of the ECEPM method, and a more concise overview was added instead：Traditional POD methods rely on the magnitude of singular values to determine the importance of each mode. However, the C-POD model does not involve singular value decomposition during its construction, making direct mode ranking infeasible. To address this, we propose the Entropy-Controlled Euclidean-to-Probability Mapping (ECEPM) method, which incorporates both the proximity between modes and snapshots and the consistency of modal contributions, thereby establishing a probabilistic ranking framework to overcome the mode-ordering limitation of clustering-based ROMs.
• In Section 2.2, a description of the physical significance of the Euclidean distance was added：The Euclidean distance quantifies the average proximity between each clustering mode and the dataset samples. A smaller distance suggests that the corresponding mode plays a significant role across a broader range of system states, thus representing a dominant dynamic feature. This measure offers clear geometric and physical interpretability.
• In Section 2.2, an explanation was added regarding the introduction of entropy control and its significance：Relying solely on distance may lead to overestimating the importance of modes with sporadic proximity. To enhance robustness, we introduce information entropy as a regulatory factor, penalizing modes with dispersed or unstable contributions. A lower entropy value indicates that the mode has a focused and consistent influence on specific samples, making it more reliable for mode ranking.

Comments 2: The physical meanings of the new C-POD method are not very clear as it is a data-driven method.

Response 2: We sincerely appreciate the reviewer’s insightful comment regarding the physical interpretability of the C-POD method. In the revised manuscript (Section 2.1), we have added a detailed explanation to clarify the physical meaning of C-POD. Specifically, we emphasize that although C-POD is fundamentally a data-driven approach, its construction process retains close physical relevance through the following mechanisms:

1. Cluster centers as representative flow structures:
   Each cluster center μi, obtained via local averaging of flow snapshots (Eq. 7), can be interpreted as a statistically dominant flow state, akin to a coherent structure in unsteady fluid dynamics. These centers reflect frequently recurring physical patterns in the system.
2. Modal coefficients as projection strengths:
   The modal coefficients are obtained by projecting original flow states onto the clustering basis via least squares (Eq. 9). This procedure minimizes residual energy and produces physically meaningful weights that represent the contribution of each mode to the overall system state.
3. Connection to nonlinear dynamics:
   Unlike POD, which enforces global orthogonality and energy ranking, C-POD captures localized nonlinear behavior by segmenting the state space. This makes the extracted modes more sensitive to dynamic transitions and multiscale flow features.
4. Statistical-to-physical mapping:
   We interpret the C-POD basis as a reduced set of physical states derived from the empirical distribution of the flow field. Hence, C-POD builds a physically grounded low-dimensional manifold that preserves dynamic and structural consistency with the full-order system.

Comments 3: The format of the references is not consistent. Please correct.

Response 3: Thank you very much for pointing this out. We have carefully reviewed and revised all references in the manuscript to ensure consistency with the journal’s formatting requirements. Specifically, we have standardized the citation order, author names, journal titles, volume and issue numbers, and page ranges. All references are now formatted according to the official style guide of Algorithms. We appreciate the reviewer’s attention to detail.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The article deals with a clustering based proper orthogonal decomposition method. The article is interesting but some issues shoudl be fixed before it can be considered for publication.

1- The author apply their method to the parametric burgers equation involving two parameters c and beta, the space and time. How are the snapshots composed ? Usually POD method is used for one parameter the time for example, the snapshot matrix being composed of the spatial discretization of the underlying field. The variations on the parameters and the variations in time for the snapshots should not be mixed. The authors should clarify this point.

2- The bibliography on POD methods that has been developped is lacking. Some references have to be appended such as 

a - Colanera, A., Schmidt, O. T., & Chiatto, M. (2025). Robust spectral proper orthogonal decomposition. Computer Physics Communications, 307, 109432.

b- Gu, X., Xu, C., Liu, M., & Mao, Y. (2025). Frequency-domain proper orthogonal decomposition for efficient reconstruction of unsteady flows. Physics of Fluids, 37(2).

c- Volkwein, S. (2013). Proper orthogonal decomposition: Theory and reduced-order modelling. Lecture Notes, University of Konstanz, 4(4), 1-29.

d- Bui, D., Hamdaoui, M., & De Vuyst, F. (2013). POD–ISAT: An efficient POD‐based surrogate approach with adaptive tabulation and fidelity regions for parametrized steady‐state PDE discrete solutions. International journal for numerical methods in engineering, 94(7), 648-671.

3- In the article the authors use "first-order mode", usually the terminology "first mode" is used, the same for the modal coefficients.

4- The authors test their method for reconstruction purposes but they do not mention the performance of the method concerning prediction of snapshots that are not included in the snapshot matrix in other words what are the prediction capabilities of the proposed method.

5- The authors specify that the C-POD modes change over time. How this can be interpreted ? Is it necssary to perform a C-POD a t each time step ? 

Comments on the Quality of English Language

The English of the article could be improved by a review from an English native speaker.

Author Response

Comments 1: The author apply their method to the parametric burgers equation involving two parameters c and beta, the space and time. How are the snapshots composed ? Usually POD method is used for one parameter the time for example, the snapshot matrix being composed of the spatial discretization of the underlying field. The variations on the parameters and the variations in time for the snapshots should not be mixed. The authors should clarify this point.

Response 1: We sincerely thank the reviewer for pointing out this important issue regarding snapshot matrix construction. We fully agree that, according to the standard POD framework, parameter and temporal variations should not be mixed within a single snapshot.

In response to your suggestion, we have revised and clarified the description in Section 3.2 (Database Construction) of the manuscript. Specifically, for each combination of parameters (c,β)) and each time step t, we compute the spatial solution of the Burgers’ equation and treat each resulting spatial field as an independent snapshot. All such snapshots are concatenated column-wise to construct the snapshot matrix. This ensures that each snapshot corresponds to a specific parameter-time instance and that the matrix remains consistent with classical POD requirements.

Comments 2: The bibliography on POD methods that has been developped is lacking. Some references have to be appended such as 

a - Colanera, A., Schmidt, O. T., & Chiatto, M. (2025). Robust spectral proper orthogonal decomposition. Computer Physics Communications, 307, 109432.

b- Gu, X., Xu, C., Liu, M., & Mao, Y. (2025). Frequency-domain proper orthogonal decomposition for efficient reconstruction of unsteady flows. Physics of Fluids, 37(2).

c- Volkwein, S. (2013). Proper orthogonal decomposition: Theory and reduced-order modelling. Lecture Notes, University of Konstanz, 4(4), 1-29.

d- Bui, D., Hamdaoui, M., & De Vuyst, F. (2013). POD–ISAT: An efficient POD‐based surrogate approach with adaptive tabulation and fidelity regions for parametrized steady‐state PDE discrete solutions. International journal for numerical methods in engineering, 94(7), 648-671.

Response 2: We sincerely thank the reviewer for the constructive suggestion regarding the completeness of the bibliography on POD methods.

In response, we have carefully revised the Introduction section to incorporate the suggested references:

• Colanera et al. (2025) [15],
• Gu et al. (2025) [16],
• Bui et al. (2013) [17].

These works have been added to provide a more comprehensive overview of recent developments and classical foundations in POD-based modeling. In addition, we have substantially restructured the Introduction to improve the logical flow and clarity, ensuring that the context and motivation of our work are better aligned with existing literature.

We greatly appreciate the reviewer’s input, which has helped us improve the completeness and academic rigor of the manuscript.

Comments 3: In the article the authors use "first-order mode", usually the terminology "first mode" is used, the same for the modal coefficients.

Response 3: We sincerely thank the reviewer for pointing out the inconsistency in terminology. We fully accept the suggestion and have carefully reviewed and revised all related expressions concerning mode order (e.g., "first-order mode", "modal order") throughout the manuscript.

We have replaced all such terms with standard and professional expressions such as “first mode”, “second mode”, etc., and consistently used “modal coefficients” to describe the coefficients, in line with accepted terminology in the field. The modifications are summarized in the table below, indicating the original expressions, their locations, and the revised terms. We believe these revisions significantly improve the technical clarity and professionalism of the manuscript. Thank you again for your valuable feedback.

Location	Original Expression	Revised Expression
Section 3.3	first-order mode	first mode
Figures 5 and 6 in Section 3.3	In the figure caption: ‘the first-order mode’	the first mode
The first paragraph of Section 4.4.3	“first six modal orders”	first six modes
The second paragraph of Section 4.4.3	“1st, 2nd, and 5th modal orders”	1st, 2nd, and 5th modes
Figures 11 and 12	Expressions such as "first-order mode" used in the figures	first mode
Section 4.4.2	“modal order increases”	mode number increases

 

Comments 4: The authors test their method for reconstruction purposes but they do not mention the performance of the method concerning prediction of snapshots that are not included in the snapshot matrix in other words what are the prediction capabilities of the proposed method.

Response 4: reduced-order models—especially for unseen or out-of-sample snapshots—is essential for evaluating generalizability. While the current manuscript focuses primarily on reconstruction performance, we acknowledge the importance of predictive validation. Due to space constraints and the already broad scope of this work, we were not able to include additional experiments related to predictive performance in this version. However, we have explicitly incorporated a discussion and outlook on this issue in the concluding section, where we highlight the need to investigate generalization and forecasting capabilities in future studies.

Comments 5: The authors specify that the C-POD modes change over time. How this can be interpreted ? Is it necssary to perform a C-POD a t each time step ?

Response 5: We thank the reviewer for raising this important point regarding the time-varying behavior of the C-POD modes. What we mean by “modes change over time” does not imply that a new C-POD is computed at each time step. Rather, the C-POD basis functions are extracted once from the full dataset, which includes multiple time steps and parameter combinations. However, we acknowledge that there was some misleading representation in our original figures—specifically Figures 5, 6, and 16 to 18—which may have led readers to mistakenly infer that the modal structures themselves evolve over time. To eliminate this ambiguity, we have revised the figure captions and related explanations to clarify that the temporal evolution refers to the time-varying waveforms dominated by the first mode, rather than changes in the mode shapes themselves.

We have added a detailed clarification of this point in Section 4.5 and thank the reviewer for helping us improve the clarity and completeness of this aspect.

 

Note: We have made substantial revisions to the structure, content, and expression of the manuscript to enhance its logical clarity and improve the fluency of the presentation.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

Comments on algorithms-3636834

This study proposes a Clustering-Based Reduced-Order Model (C-POD) that integrates Proper Orthogonal Decomposition (POD) with clustering algorithms to improve reduced-order modeling (ROM) of nonlinear convective flows.

Moderate revisions are required:

Abstract

1. Avoid redundancy (e.g., "C-POD achieves superior dimensionality reduction accuracy compared to POD" is repeated in different forms).
2. Clarify acronyms earlier (define ECEPM in full once before acronym).
3. Consider simplifying sentence structures for clarity.

1. Introduction

4. Split long paragraphs for readability (e.g., the initial 100+ line intro could be broken down into thematic sections).
5. Include a concise diagram to summarize the classification of ROM methods.
6. More clearly state the research gap and novelty

2. C-POD ROM and Modal Sorting Method

7. Improve figure quality and clarity in Figure 1; label steps more intuitively.
8. Consider using an algorithm box or pseudocode for C-POD steps.
9. The ECEPM section could benefit from a simpler example or visual to explain entropy-temperature effects.

3. Burgers’ Equation

10. Define the Cr metric earlier and provide intuition behind its choice.
11. Figures 3–6: add color bars, legends, or more descriptive captions to aid interpretation.
12. State explicitly how noise or perturbations (if any) are handled in the test.

4. Cylinder Wake Flow Case

13. The sub-sections (especially 4.4 and 4.5) are too text-heavy — break into smaller parts or bullet points.
14. Figures 11–14 could benefit from annotated vortex structures to help readers distinguish mode characteristics.
15. Discuss limitations (e.g., computational cost, sensitivity to sensor layout) in modal reconstruction.

5. Application to Inverse Problems

16. Clarify how sensor positions were optimized (e.g., specifics of the "correlation coefficient filtering method").
17. Compare to at least one additional baseline (e.g., machine learning-based field reconstruction).
18. Include uncertainty quantification or sensitivity analysis if possible.

6. Conclusions

19. Add a short statement on future work (e.g., extending to 3D flows or adaptive clustering).
20. Avoid repeating numerical improvements already emphasized in prior sections unless adding context.

 

Author Response

Comments 1: Abstract

- Avoid redundancy (e.g., "C-POD achieves superior dimensionality reduction accuracy compared to POD" is repeated in different forms).

- Clarify acronyms earlier (define ECEPM in full once before acronym).

- Consider simplifying sentence structures for clarity.

Response 1: Thank you very much for your thoughtful and constructive feedback. In response to your suggestions, we have made the following revisions to Chapter 1: Introduction:

1. To improve readability, we have divided the long introductory paragraph into several shorter paragraphs, each centered on a specific theme:
   (i) the motivation and challenges in high-dimensional flow modeling,
   (ii) classical linear ROM methods,
   (iii) advances in data-driven approaches, and
   (iv) the emergence of clustering-based ROMs.
2. To provide a clearer overview of existing ROM approaches, we added Figure 1, a taxonomy chart that categorizes ROM methods into traditional and data-driven branches, and further into linear and nonlinear structures. This visual aid helps readers grasp the methodological landscape more intuitively.
3. Regarding the clarity of the research gap and innovation, we revised the final paragraphs of the introduction to:
   • Emphasize the limitations of current clustering-based ROMs (e.g., unstable initialization and lack of effective mode ranking),
   • Highlight our solution (the proposed C-POD framework and ECEPM method),
   • Explicitly state the novelty and technical contributions of the work.

Thank you for the insightful comments. We have carefully revised the abstract as follows:

1. We have removed repeated expressions such as “C-POD achieves better accuracy than POD” to avoid redundancy.
2. The abbreviation ECEPM has now been defined in full at its first occurrence.
3. Several long and complex sentences have been restructured for improved clarity and readability.

We believe these revisions improve the clarity and conciseness of the abstract, and we sincerely thank the reviewer for the constructive suggestions.

Comments 2: Introduction

- Split long paragraphs for readability (e.g., the initial 100+ line intro could be broken down into thematic sections).

- Include a concise diagram to summarize the classification of ROM methods.

- More clearly state the research gap and novelty

Response 2: We sincerely thank the reviewer for the valuable suggestions regarding the Introduction. In response, we have substantially revised this section as follows:

• The original lengthy paragraphs have been reorganized into smaller, more structured parts to improve readability and logical flow;
• Redundant or overly complex expressions have been streamlined for greater clarity and conciseness;
• Figure 1 has been added to provide a systematic summary of existing reduced-order modeling methods and their classifications;
• Additionally, the end of the Introduction has been revised to more clearly articulate the research gap and highlight the main contributions of this study.

We are grateful for your constructive feedback, which has greatly helped us enhance the clarity and coherence of the Introduction section.

Comments 3: C-POD ROM and Modal Sorting Method

• Improve figure quality and clarity in Figure 1; label steps more intuitively.
• Consider using an algorithm box or pseudocode for C-POD steps.
• The ECEPM section could benefit from a simpler example or visual to explain entropy-temperature effects.

Response 3: Author Response: We sincerely thank the reviewer for this insightful suggestion. We have redrawn the original Figure 1 (now updated as Figure 2) with improved resolution, refined structure, and a revised step sequence and caption to ensure a more intuitive and readable presentation of the workflow. A standardized pseudocode box (Algorithm 1) has been added to systematically describe the construction process of the C-POD ROM, including inputs, procedures, and outputs, which improves the clarity and accessibility of the method.

Comments 4: Burgers’ Equation

• Define the Cr metric earlier and provide intuition behind its choice.
• Figures 3–6: add color bars, legends, or more descriptive captions to aid interpretation.
• State explicitly how noise or perturbations (if any) are handled in the test.

Response 4: We sincerely thank the reviewer for this insightful suggestion. In response, we have substantially revised the first paragraph of Section 3.3. The definition of the Cr (Compression Ratio) metric has been moved forward and its formulation is explicitly presented (Equation 22). We have also elaborated on the rationale for selecting Cr as an integrated evaluation metric.

（1）Specifically, while the correlation coefficient and RMSE respectively measure trend similarity and numerical deviation, using both together can be unintuitive and somewhat cumbersome for comparative analysis. To address this, we introduce the Cr metric, which combines both aspects into a concise and unified indicator. This helps facilitate clearer and more efficient assessment of the reconstruction performance of different reduced-order models.

（2）Figure captions have been revised to be more descriptive, clearly indicating the physical meaning and comparative context of each visualization.

（3）We have added a clarification in Section 3.1 regarding disturbance and noise handling.Although no artificial noise was added explicitly to the dataset, numerical disturbances may inherently arise during simulation and data sampling. Since the C-POD method incorporates Proper Orthogonal Decomposition (POD), which inherently performs denoising by extracting dominant energy modes, the overall framework benefits from a natural robustness to noise and high-frequency perturbations.

Comments 5: Cylinder Wake Flow Case

• The sub-sections (especially 4.4 and 4.5) are too text-heavy — break into smaller parts or bullet points.
• Figures 11–14 could benefit from annotated vortex structures to help readers distinguish mode characteristics.
• Discuss limitations (e.g., computational cost, sensitivity to sensor layout) in modal reconstruction.

Response 5: We sincerely thank the reviewer for the constructive suggestion. In response, we have revised Sections 4.4 and 4.5 to improve clarity and readability.

In particular, Section 4.5 has undergone significant restructuring. We now present the content in sub-segmented format with smaller paragraphs and structured lists where appropriate. This allows readers to more clearly follow the modal analysis and evolution process of C-POD, especially the interpretation of key transition moments and vortex structure changes. We believe this modification substantially enhances the presentation and understanding of our results.

In the revised manuscript, we have added a new subsection (Section 5.1) to explicitly discuss the limitations of the proposed C-POD ROM method. Specifically, we analyzed three key aspects:

• The increased computational cost due to clustering and entropy-based mode ranking;
• The sensitivity to sensor layout, particularly in sparse reconstruction tasks;
• The dependence on hyperparameters, which may affect robustness and accuracy.

We also included a brief outlook on possible improvement directions, such as lightweight clustering algorithms and adaptive sensor placement strategies. This addition helps provide a more comprehensive and balanced evaluation of the method.

 

Comments 6: Application to Inverse Problems

• Clarify how sensor positions were optimized (e.g., specifics of the "correlation coefficient filtering method").
• Compare to at least one additional baseline (e.g., machine learning-based field reconstruction).
• Include uncertainty quantification or sensitivity analysis if possible.

Response 6: Thank you for this insightful suggestion. In the revised manuscript, we have incorporated a detailed explanation of the correlation coefficient filtering method at the beginning of Section 5, where sensor layout and inverse reconstruction are introduced.

Specifically, we clarified that this method calculates the Pearson correlation coefficient between each candidate sensor point and the low-order POD modal coefficients derived from the training dataset. Sensor locations with the highest absolute correlation values are then selected, ensuring high sensitivity to dominant flow features. This addition helps the reader better understand the rationale and effectiveness of the chosen sensor placement strategy in sparse reconstruction scenarios.

Comments 7: Conclusions

• Add a short statement on future work (e.g., extending to 3D flows or adaptive clustering).
• Avoid repeating numerical improvements already emphasized in prior sections unless adding context.

Response 7: Thank you sincerely for your valuable and constructive suggestions. We have revised the Conclusion section accordingly:

At the end of the conclusion, we added a brief outlook on future work, focusing on two aspects: (1) extending the C-POD framework to three-dimensional unsteady and turbulent flow problems, in order to validate its generalizability and robustness in high-dimensional contexts; and (2) incorporating adaptive clustering strategies, allowing the number and position of clusters to be adjusted dynamically during training, which will enhance the model's capability in handling time-varying and multi-scale systems.

Furthermore, we carefully avoided repeating numerical results that have already appeared in the main text. Instead, we rephrased key findings from a higher-level perspective, emphasizing their implications for physical interpretability, real-time inverse reconstruction, and sparse-sensing robustness. This improves the conclusion’s informativeness without redundancy.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Ok

Author Response

(Note: To distinguish from the first revision, the second-round modifications are marked in blue font.)

Comments 1: The English could be improved to more clearly express the research.

Response 1: Thank you for your suggestion. In response, we have conducted a thorough language revision throughout the manuscript to improve clarity, fluency, and overall readability. Specifically, we have:

Abstract: Simplified long sentences to enhance readability and deliver key messages more directly.

Introduction: Improved transitions between paragraphs and compressed overly dense technical sections for smoother reading.

Section 2: Refined the description of the mode sorting algorithm and streamlined the explanatory language around equations.

Section 3: Reduced subjective language and ensured consistent terminology usage.

Section 4: Strengthened logical connections between paragraphs and broke down overly long sentences.

Section 5: Removed redundant expressions and improved the objectivity in data presentation.

Conclusion: Condensed repeated content and sharpened the forward-looking statements for future work.

All changes have been carefully made to better communicate the research contribution while maintaining technical accuracy. Revised parts are highlighted in the revised manuscript.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have improved the manuscript by adressing several issues. However some issues have to be fixed before publication : 

• Line 163, please change Where in where, same in line 238
• Line 248, please correct the typo for Wij
• Lines 190-193 : the authors state that they do not compute singular values while using their method, however they compute eigenvalues of the correlation matrix that related to the singular values of the snapshot matrix and these eigenvalues could be used to rank the modes, but as the authors change the modes by adopting cluster centers, this ranking could not be used anymore as it is inconsistent for the new modes. The authors should put it like that so it is more accurate, to introduce the proposed probabilistic mode sorting method.
• In section 2.2, the authors should include for clarity an algorithm as done between lines 182 and 183 to describe the sorting algorithm
• For the mode sorting method, they should provide guidelines to choose the parameter tau, tau ->0 is an indication but not accurate !
• The authors should change the title by something like "POD based clustering based dimensionality reduction method ..." as initialized by POD is too specific to figure in the title
• Fig.6 and Fig.7 the word "dominated" is unclear, i guess that it is the reconstructed solution using the first mode only, so the author can say "First mode reconstructed solution" or clarify.
• The author should prefer using "dimensionality reduction" rather than "reduced order model" since there is no time varying model obtained by Galerkin or Petrov-Galerkin projection of the equations on the modes.

Author Response

(Note: To distinguish from the first revision, the second-round modifications are marked in blue font.)

Comments 1: Line 163, please change Where in where, same in line 238

Response 1: Thank you for pointing out this typographical error. We have corrected the capitalization of Where to where in both Line 163 and Line 238 as suggested.

Comments 2: Line 248, please correct the typo for W_ij

Response 2: Thank you for spotting this mistake. We have corrected the typo in Line 248 and ensured that W_ij is now consistently and correctly formatted throughout the manuscript.

Comments 3: Lines 190-193 : the authors state that they do not compute singular values while using their method, however they compute eigenvalues of the correlation matrix that related to the singular values of the snapshot matrix and these eigenvalues could be used to rank the modes, but as the authors change the modes by adopting cluster centers, this ranking could not be used anymore as it is inconsistent for the new modes. The authors should put it like that so it is more accurate, to introduce the proposed probabilistic mode sorting method.

Response3: We appreciate the reviewer’s insightful observation. In the revised manuscript, we have clarified this point by explicitly stating that although the C-POD method does not compute singular values directly, it does solve the eigenvalue problem of the snapshot correlation matrix, whose eigenvalues are mathematically related to the singular values of the snapshot matrix. While these eigenvalues could be theoretically used for mode ranking, the replacement of orthogonal POD modes with clustering centroids—being neither orthogonal nor naturally ordered—renders this ranking inapplicable. Accordingly, we have revised the corresponding paragraph to reflect this relationship and to provide a more accurate motivation for introducing the Entropy-Controlled Euclidean-to-Probability Mapping (ECEPM) method. The updated explanation appears on lines 195–207 of the revised manuscript.

Comments 4: In section 2.2, the authors should include for clarity an algorithm as done between lines 182 and 183 to describe the sorting algorithm

Response 4: We appreciate the reviewer’s suggestion. Following the format used in Section 2.1, we have added a detailed algorithm (Algorithm 2) in Section 2.2 to describe the complete implementation of the Entropy-Controlled Euclidean-to-Probability Mapping (ECEPM) method for probabilistic mode sorting. This addition enhances the clarity and reproducibility of the proposed method. The new algorithm summarizes the key computational steps, including distance matrix calculation, entropy-adjusted transformation, and probability normalization.

Comments 5: For the mode sorting method, they should provide guidelines to choose the parameter tau, tau ->0 is an indication but not accurate !

Response 5: We thank the reviewer for pointing out the lack of practical guidance on the selection of the temperature parameter τ in the ECEPM method. In response, we have added a dedicated paragraph in Section 2.2 to clarify how τ affects the entropy of the resulting probability distribution. We also provide practical recommendations for selecting τ: a smaller τ yields sharper, more confident mode assignments, while a larger τ yields smoother, more robust distributions. In our experiments, τ was selected via grid search in the range [0.1, 5], and the optimal value was determined based on reconstruction performance using the Cr index. This addition now appears on lines 249–259 of the revised manuscript.

Comments 6: The authors should change the title by something like "POD based clustering based dimensionality reduction method ..." as initialized by POD is too specific to figure in the title

Response 6: We appreciate the reviewer’s helpful suggestion regarding the manuscript title. As recommended, we have revised the title to better reflect the methodology without relying on the overly specific phrase “initialized by POD.” The new title — “A POD-Guided Clustering Method for Dimensionality Reduction and Its Application to Convective Flow Problems” — preserves the core idea while improving generality and clarity. This change has been updated in both the manuscript

Comments 7: Fig.6 and Fig.7 the word "dominated" is unclear, i guess that it is the reconstructed solution using the first mode only, so the author can say "First mode reconstructed solution" or clarify.

Response 7: We appreciate the reviewer’s careful observation. We agree that the term “dominated” may lead to ambiguity in this context. In response, we have revised both the figure captions and the corresponding descriptions in the text to use the clearer and more precise term “first-mode reconstructed solution.” This change better reflects the nature of the content—i.e., the solution reconstructed using only the first mode extracted by C-POD or POD. The updated wording appears in Figures 6 and 7 and in Section 3.3 of the revised manuscript.

Comments 8: The author should prefer using "dimensionality reduction" rather than "reduced order model" since there is no time varying model obtained by Galerkin or Petrov-Galerkin projection of the equations on the modes.

Response 8: We appreciate the reviewer’s insightful comment. In response, we have revised the terminology throughout the manuscript to consistently use “dimensionality reduction” instead of “reduced-order model” where appropriate. This change reflects the nature of our method more accurately, as it does not involve a Galerkin-type projection to derive time-evolving low-dimensional equations.

Accordingly, we have also updated the title of the paper to:

“A Clustering-Based Dimensionality Reduction Method Guided by POD Structures and Its Application to Convective Flow Problems”

We have carefully reviewed and revised the manuscript text (e.g., abstract, introduction, conclusion) to ensure alignment with this terminology.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

Done.