A Formal Approach to Optimally Configure a Fully Connected Multilayer Hybrid Neural Network

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This work advances a new and mathematically sound framework of hybrid multilayer neural networks using optimal control theory. While the authors make important theoretical achievements, the manuscript has a glaring drawback in not providing empirical validations and practical relevance. It is essential to address the issues concerning evaluations in the context of well-defined benchmark problems to convincingly establish the real-world relevance and utility of the proposed methodology. Including experimental results and detailed comparisons with existing optimization techniques will not only strengthen the paper's contributions but also make it more compelling for acceptance!

1.       Provide an appendix or a list for symbols, and operators.

2.       Line 118 and Line 150. Line 210. Missed references.

3.       Line 99. It seems that the regression problems is intended. Can the developed OCP be applied to the classification problems? Or a hybrid one in which some outputs are real values, and others are classes. Describe the limitations of the method.

4.       Is the developed OCP is applicable to networks that are not fully-connected? What about cascade networks? Describe the limitations of the new formulation.

5.       The theoretical derivations are solid, but there is no experimental validation or simulation to demonstrate the efficacy of the proposed method. Including numerical experiments with benchmarks would significantly enhance the paper's impact.

6.       Add practical aspects to the manuscript. Add novel practical references using Multilayer Hybrid Neural Network for real applications to Introduction, including https://doi.org/10.1080/19942060.2022.2046167

7.       The method is presented as a practical solution, but there is limited discussion on its computational efficiency.

8.       The paper focuses on a fixed number of layers and nodes. Extending the analysis to include dynamically adjusting network sizes during training would address broader scenarios in neural network optimization.

9.       The paper would benefit from visual aids, such as flowcharts of the optimization process, graphs, etc.

1.   Line 35. “Two basic optimization criteria are faster convergence and better generalization of the mapping function.” I cannot follow neither in the manuscript. How the faster convergence and better generalization is warranted. How does the proposed optimal control framework specifically address generalization in practical terms?

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper addresses an important problem in data processing and provides a solution that has both theoretical and practical implications. The methodology employed in this study is robust and well-designed. The author utilizes state-of-the-art techniques from machine learning and artificial intelligence, which have been successfully applied in similar contexts. By integrating these methods into their framework, they achieve improved performance compared to traditional approaches.

The following minor issues are currently present:

1. In the section of Introduction, it is recommended to further refine the innovative points and clarify the motivation behind the proposed method.

2. The presented approach lacks a comparison with existing methods/algorithms.

3. All symbols used in the formulas require explanations.

4. There are some formatting errors, such as on line 118 where the references fail to display.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The author devoted to a formal analysis and thereby optimizing the learning models for the feed forward multilayer neural networks with hybrid structure. Some questions are as follows.

1. The selection of keywords is not very reasonable. The keywords of the article should be selected based on literature analysis, so that the selected keywords can most accurately reflect the theme and research content of the paper. For example, "deep learning" does not highlight the importance of this keyword in the abstract or title.

2. There are errors in the citation of references in the article, and the author should carefully verify and revise them. For example, on page 4, the reference cited in the first line.

3. The reference on line 210 is missing, please supplement.

4. What does “Definition 1” refer to in line 222?

5. What does “tIt” refer to in line 237?

6. The reference on line 298 is missing, please supplement.

7. The author mentioned in Section 4: "We refer to [12,16] for the necessary technical details, examples, and analytical results." Similarly, I believe that the article should also supplement some simulation examples to demonstrate the effectiveness of the proposed method, which can make the structure of the article more complete.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

As mentioned in comments, the manuscript has a glaring drawback in not providing validations by benchmark problems and practical relevance. Due to lack to address the issues concerning evaluations in the context of well-defined benchmark problems, I have to reject the paper. The authors do not improve the paper in comparison to the previous version.

Author Response

I checked the reviewers' comments. Reviewer 1 rejected for not simulating the proposed theory. As I have told time and again, this is the reason we have submitted the present work to "Mathematics". Reviewer 1 wanted to include a paper which is completely irrelevant to this work. He misunderstood the concept of Hybrid referred in our work. He wanted us to include an application work using two soft-computing tools - not at all related to the theory we proposed. 

Reviewer 2 Report

Comments and Suggestions for Authors

The authors answered all the questions I was concerned about. I don't have any more questions.

Author Response

Thanks

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript has solved all my concerns, which I think could be accepted.

Author Response

Thanks