Optimization through the Levenberg—Marquardt Backpropagation Method for a Magnetohydrodynamic Squeezing Flow System

Round 1

Reviewer 1 Report

Neural networks are being used for several decades with considerable success in a large range of applications. Levenberg-Marquard back propagation neural network is well established and with sucessful application in a number of different situations being effective and reliable with a fair level o complexity.

In the present work the authors explore its application to a particular problem: the measurement of flow of nano particles between two adjacent disks. Authors considered a sound group of parameters that are probably sufficient to establish most conditions of magnetohydrodynamic squeezing flow, although this is a rather complex problem. The mathematical formulation and the computational method are simple yet seem adequate. The analysis of the LMB-NN application was done in a correct way to justify the conclusions.

Please consider revising some of the figures like for instance “Figure 9. Regression views of the LMB-NN paradigm for (MHDSF)” in order to not over burden the text with too much data or data that can not be assessed in the paper. You may also expand a bit the explanation in the text of the results in graphs.

Please adjust the terminology avoiding terms like "error" in "mean square error (MSE)" and following ISO standards and GUM.

Please revise carefully the English writing. Some are just simple misspellings but some time understanding the message conveyed is difficult by using wrong terms, for instance at the beginning of chapter 3.

Author Response

Response to the comments on the submitted manuscript:

Submission ID coatings-1242872

Title: Optimization through Levenberg–Marquardt Backpropagation Method for Magnetohydrodynamic Squeezing Flow System. The authors would like to thank the reviewers for their valuable, detailed and constructive comments that allow us to further improve the quality of the submitted paper. All the comments and suggestions are carefully addressed in this document and the paper is accordingly revised. In this response to the reviewer’s comments.

Reviewer#1, General Concerns:

In the present work the authors explore its application to a particular problem: the measurement of flow of nano particles between two adjacent disks. Authors considered a sound group of parameters that are probably sufficient to establish most conditions of magnetohydrodynamic squeezing flow, although this is a rather complex problem. The mathematical formulation and the computational method are simple yet seem adequate. The analysis of the LMB-NN application was done in a correct way to justify the conclusions

Author response: Many thanks for your valuable remarks “The paper is written quite well, and hence the publication of this paper is strongly recommended” and nice recommendation on our submitted manuscript.

Reviewer#1, Concern # 1:

Please consider revising some of the figures like for instance “Figure 9. Regression views of the LMB-NN paradigm for (MHDSF)” in order to not over burden the text with too much data or data that cannot be assessed in the paper. You may also expand a bit the explanation in the text of the results in graphs.

Author response:

Agreed. The authors revised the whole manuscript to modify the figures as suggested. Also, figure 9 is regenerated for better visibility. Additionally, all the figures are described more elaborately for better understanding of the readers.

Reviewer#1, Concern # 2:

Please adjust the terminology avoiding terms like "error" in "mean square error (MSE)" and following ISO standards and GUM.

Please revise carefully the English writing. Some are just simple misspellings but some time understanding the message conveyed is difficult by using wrong terms, for instance at the beginning of chapter 3

Author response:

Agreed The authors revisited the whole manuscript to avoid typos and grammatical errors for better understanding of the readers. Additionally, the linguistic quality of the manuscript is further improved as suggested by the anonymous reviewers.

With best Regards

Maryam Al-Malki

Corresponding author 

Reviewer 2 Report

In this paper, we investigate an unsteady compressive flow between two disks using a Levenberg-Marquard neural network. A technique for solving the problem using similarity transformation and the Runge-Kutta method is proposed. The solver was investigated on various scenarios, the root-mean-square error was estimated, which gives the reliability of the method.

Remarks.

1. In the mathematical formulation of the problem, not all parameters were deciphered. For example, parameters a and γ, as well as other parameters in equations (1) - (6). This is very important, as it gives a better understanding of the problem being solved.
2. In formulas (8) and (9), dimensionless constants Le and PR should not be italicized, they should also be explained.
3. In conditions (10), the first line after the first condition must contain a comma, similarly in the second line.
4. In the work, the authors show the reliability of the method by estimating the mean square error and present several scenarios. The paper [31] investigates the same problem but using the HAM method. Does the question arise as to how the results obtained in the reviewed article are compared with the results in [31]? Which method works best?
5. All abbreviations should be placed on the table at the end of the article.

In general, the work can be recommended for publication, after correction.

Comments for author File: Comments.pdf

Author Response

Response to the comments on the submitted manuscript:

Submission ID coatings-1242872

Title: Optimization through Levenberg–Marquardt Backpropagation Method for Magnetohydrodynamic Squeezing Flow System. The authors would like to thank the reviewers for their valuable, detailed and constructive comments that allow us to further improve the quality of the submitted paper. All the comments and suggestions are carefully addressed in this document and the paper is accordingly revised. In this response to the reviewer’s comments.

Reviewer#2, General Concerns:

In this paper, we investigate an unsteady compressive flow between two disks using a Levenberg-Marquard neural network. A technique for solving the problem using similarity transformation and the Runge-Kutta method is proposed. The solver was investigated on various scenarios, the root-mean-square error was estimated, which gives the reliability of the method.

Author response: The authors are doing their best to treat the manuscript on solid grounds.

Reviewer#2, Concern # 1:

In the mathematical formulation of the problem, not all parameters were deciphered. For example, parameters a and γ, as well as other parameters in equations (1) - (6). This is very important, as it gives a better understanding of the problem being solved.

Author response: 

Agreed. Now all the equations are checked/reviewed, and all the variables are well defined with elaborative description for better understanding of the readers. Additionally, all the parameters and symbols are mentioned more clearly.

Reviewer#2, Concern # 2:

In formulas (8) and (9), dimensionless constants Le and PR should not be italicized, they should also be explained.

Author response: 

Now dimensionless constants Le and PR in formulas (8) and (9) are corrected as suggested by the anonymous reviewer. Additionally, the necessary information regarding the parameters are provided now in the revised version of the manuscript.

Reviewer#2, Concern # 3:

In conditions (10), the first line after the first condition must contain a comma, similarly in the second line.

Author response: 

The whole manuscript is revisited to avoid grammatical errors and punctuations. Moreover, Eq. 10 is modified as suggested by the anonymous reviewer.

Reviewer#2, Concern # 4:

In the work, the authors show the reliability of the method by estimating the mean square error and present several scenarios. The paper [31] investigates the same problem but using the HAM method. Does the question arise as to how the results obtained in the reviewed article are compared with the results in [31]? Which method works best?

Author response: 

The performance of stochastic method cannot be compared with deterministic approach due to different nature of the algorithms, however, the salient features of proposed scheme are provided as follows:

• A soft computing technique-based Levenberg-Marquard algorithm is used to solve the fluid flow problem MHDSF.
• Mathematical simplification is presented for MHDSF in terms of partial differential equations to be easier in dealing with the proposed model (LMB-NN).
• Creation of the data set for suggested (LMB-NN) based on squeezing parameter, Prandtl number, Brownian motion parameter, thermophoresis parameter is used in solution (MHDSF) by employing the Runge-Kutta technique for various scenarios and cases.
• The processes of training, testing, and validation that are creating with (LMB-NN) are implemented for every scenario and case of (MHDSF) to find the approximate solution and comparison with standard results.
• The performance of NN-BLMS is established through convergence plots of mean squared error-based fitness/merit function, state transition, regression metrics, and histogram error.

The validation part is provided in more elaborative way in the revised version of the manuscript. The close agreement of both proposed and reference results with matching of level E-07 to E-12 verify the validity of the method. Moreover, graphical and numerical illustrations of convergence plots on mean square errors, error-histograms and regression dynamics further validate the worth of the scheme. The presented study is limited to design the two-layer feed-forward artificial neural networks (ANNs) backpropagated with LMM, i.e., LMB-NN for finding the solution presented fluidic system, however an improved performance can be obtained by exploiting the evolutionary/swarming computing techniques including GA, PSO, GSA, QPSO, etc. for optimization of ANN instead of LMM. This will be a promising recommended, relevant future research directions to be investigated by the interested readers.

Reviewer#2, Concern # 5:

All abbreviations should be placed on the table at the end of the article.

Author response: 

Agreed. All the abbreviations are now placed on the table at the end of the article as suggested.

With best Regards

Maryam Al-Malki

Corresponding author

Reviewer 3 Report

This paper presents the optimization using the backpropagation method for the magnetohydrodynamic flow system. The authors analyze the mathematical model based on the Levenberg-Marquard backpropagated neural network (LMB-NN). The abstract does not clearly indicate the contribution of the authors to the problem addressed. It would be advisable to highlight this contribution, as well as to anticipate a brief summary of the results obtained.

The introduction contains the state of the art of the problem addressed. The number of references cited is adequate. Although the contribution of the authors to the problem developed is not clearly appreciated. It would be advisable to indicate the differences with respect to some of the research works mentioned in the introduction. It is recommended that these differences appear in the text.

The document contains an analysis of the state of the art of the problem addressed. It also incorporates different mathematical expressions associated with the development of the model. In sections 1 and 2 the authors show the different mathematical models used, as well as the assumption raised. The simple neural network (neuron) diagram is hard to understand, see figure 2. There are many figures in the literature that are clearer. The overall diagram of the network used (multiple neurons and backpropagation system) does not appear either.

It would be interesting to indicate what the Levenberg-Marquard neural-network algorithm is applied to non-linear training, albeit briefly. In this way, it would be achieved that many readers do not get lost during their understanding. It is intuited in the text that the number of neurons and layers has been selected experimentally according to the results obtained. This information should appear in the text. There is no information related to the data set used for the training of the neural network.

Some elements that appear in figure 4 are redundant, as they have been previously shown in the document. It is recommended to develop a flow chart according to the proposed development. Table 2 shows some numerical results, among which the MSE (mean square error) stands out, it would be advisable to increase the size of the data. In Table 2 replace "main square error" with "mean square error".

It is advisable to increase the dimensions of the graphs shown in Figure 9. Readers cannot correctly appreciate the different details of the regressions. The conclusions are somewhat brief. It is advisable to delve into the conclusions according to the results obtained throughout the paper. The number of bibliographic references provided is adequate. Furthermore, most of them are directly linked to the subject of the document and have been published in the last decade.

Author Response

Response to the comments on the submitted manuscript:

Submission ID coatings-1242872

Title: Optimization through Levenberg–Marquardt Backpropagation Method for Magnetohydrodynamic Squeezing Flow System. The authors would like to thank the reviewers for their valuable, detailed and constructive comments that allow us to further improve the quality of the submitted paper. All the comments and suggestions are carefully addressed in this document and the paper is accordingly revised. In this response to the reviewer’s comments.

Reviewer#3, Concern # 1:

The abstract does not clearly indicate the contribution of the authors to the problem addressed. It would be advisable to highlight this contribution, as well as to anticipate a brief summary of the results obtained.

The introduction contains the state of the art of the problem addressed. The number of references cited is adequate. Although the contribution of the authors to the problem developed is not clearly appreciated. It would be advisable to indicate the differences with respect to some of the research works mentioned in the introduction. It is recommended that these differences appear in the text.

Author response: 

Agreed. We have updated the manuscript after critical review of the abstract so that sufficient information without too cumbersome and is hard to catch the key points is listed as suggested.

Please see the abstract of the revised manuscript as follows:

The present study introduced the unsteady squeezing flow of two-dimensional viscous fluid with nanoparticles between two disks by using the Levenberg-Marquard backpropagated neural network (LMB-NN). Conversion of the partial differential equations (PDEs) into equivalent ordinary differential equations (ODEs) is performed by suitable similarity transformation. The data collection for suggested (LMB-NN) is made for various magnetohydrodynamic squeezing flow (MHDSF) scenarios in terms of the squeezing parameter, Prandtl number, Brownian motion parameter, thermophoresis parameter by employing the Runge-Kutta technique with the help of Mathematica software. The worth of the proposed methodology has been established for the proposed solver (LMB-NN) with different scenarios and cases, and the outcomes are compared through the effectiveness and reliability of mean square error (MSE) for the squeezing flow problem MHDSF. Moreover, the state transition, Fitness outline, histogram error, and regression presentation also endorse the strength and reliability of the solver LMB-NN.

The high convergence between the reference solutions and the solutions obtained by incorporating the efficacy of designed solver LMB-NN indicates the strength of the proposed methodology, where the accuracy level is achieved in the ranges from 1E−06 to 1E−12.

Also, the introduction section is updated with elaborative description of problem state for better understanding of the readers.

In the presented study, a novel implementation of the integrated stochastic computational intelligent solver SNN is presented with following salient features:

Reviewer#3, Concern # 2:

In sections 1 and 2 the authors show the different mathematical models used, as well as the assumption raised. The simple neural network (neuron) diagram is hard to understand, see figure 2. There are many figures in the literature that are clearer. The overall diagram of the network used (multiple neurons and backpropagation system) does not appear either.

It would be interesting to indicate what the Levenberg-Marquard neural-network algorithm is applied to non-linear training, albeit briefly. In this way, it would be achieved that many readers do not get lost during their understanding. It is intuited in the text that the number of neurons and layers has been selected experimentally according to the results obtained. This information should appear in the text. There is no information related to the data set used for the training of the neural network.

Author response: 

Agreed. We have updated the manuscript by providing the necessary description of methodology base on neural networks including the layer structure, hidden neurons, topology of the networks and arbitrary selection of input and target data set for training, testing and validation samples.

Reviewer#3, Concern # 3:

Some elements that appear in figure 4 are redundant, as they have been previously shown in the document. It is recommended to develop a flow chart according to the proposed development. Table 2 shows some numerical results, among which the MSE (mean square error) stands out, it would be advisable to increase the size of the data. In Table 2 replace "main square error" with "mean square error".

Author response: 

Agreed. The authors reconstructed the flow chart according to the proposed development as suggested by the anonymous reviewer. Table 2 in the revised manuscript is readjusted in more clear way with good readability for better understanding of the readers. Also, error in table 2 (main square error) is corrected as mean square error.

Reviewer#3, Concern # 4:

It is advisable to increase the dimensions of the graphs shown in Figure 9. Readers cannot correctly appreciate the different details of the regressions. The conclusions are somewhat brief. It is advisable to delve into the conclusions according to the results

obtained throughout the paper. The number of bibliographic references provided is adequate. Furthermore, most of them are directly linked to the subject of the document and have been published in the last decade.

Author response: 

Agreed. The dimensions of the graphs shown in Figure 9 are increased as suggested by the anonymous reviewer. The necessary detail regarding the regression's plots are provided in clear and precise manner for better understanding of the readers.

Special emphasis is exerted to improve the conclusion sections of the revised manuscript as suggested. Moreover, the necessary inferences from graphical illustration (figures) are now provided in the conclusion section of the revised manuscript as suggested

With best Regards

Maryam Al-Malki

Corresponding author

Round 2

Reviewer 1 Report

Authors toke into consideration the comments of the referees. Not being a stricktly a metrology work, I believe it can considered for acceptance.