Neural-Network-Assisted Finite Difference Discretization for Numerical Solution of Partial Differential Equations

Round 1

Reviewer 1 Report

Using neural network method to solve the related problems of differential equations is a new direction in the field of scientific computing, which has great influence on the upgrading of traditional methods.This paper caters to this hot research trend. In this paper, a neural network-based numerical method is proposed for the solution of Laplace and Poisson problems. Then it is applied to the solution of PDEs problem. Main idea of the present work is to approximate the Laplacian by an expansion of basic functions (see (3)), then using ANN and optimization methods to determine the combination coefficients. The procedure is aided by traditional difference scheme. In this way, for any geometric scenario of the neighboring grid points, the responding coefficients are obtained efficiently. The method was experimentally demonstrated on several numerical tests.  It delivered a fast and reliable algorithm for solving Poisson problems.

The method has potential application in solving PDEs. At least, we have sawn its ability to solve linear PDEs.

The expectation is the method can be generalized to nonlinear PDE cases.

Author Response

Dear Reviewer,

In the revised version, we have performed a new proofread of the manuscript and corrected a few minor mistakes.

Also, following the advice of the other reviewers, we have competed the text in several points. All changes have been highlighted.

Thank you for your efforts.

Ferenc Izsák (corresponding author)

Reviewer 2 Report

See the enclosed file.

Comments for author File: Comments.pdf

Author Response

Dear Reviewer,

We have revised the manuscript following your advice.

Indeed, this is an algorithmic approach to the numerical solution of the problem in (2).

Since a major part of the results are presented in Figures, during the revision, we took care to explain them more in details and cite them more.

In concrete terms, the caption of Fig.  1 has been extended and also, an extra explanation has been inserted in line 133.

For Figure 2, we have a short extra link in line 154.

Both for Figure 3 and Figure 4, we have again extended the captions.

The present Figure 6 is new; here the entire algorithm is depicted.

Also, Figure 8 is new, where the error between the training and the validation loss is shown.

Finally, we have added Table 1 to point out the speed-up of our ANN-based approach vs. a conventional pointwise optimization procedure to get the finite difference coefficients.

Thank you very much for your efforts.

With best regards,

Ferenc Izsák (corresponding author)

Reviewer 3 Report

In this paper, the authors presented the neural network-assisted numerical method for the solution of Laplace and Poisson problems. Depending on the position of a given gridpoint x0 and its neighbors,  a nonlinear optimization problem was obtained to get the finite difference coefficients in x0. Finally, some examples were given. In my opinion, this research is interesting. Therefore, I recommend it accpeted if the authors can make some revisions as follows.

1) The mian motivations and contributions should be clarified in the introduciton part.

2) The authors should give some explanations about figures. 

3) Some remarks should be added to show the effectiveness of the methods comparing with the existing results.

4) The block diagram should be presented for the algorithm.

5) The error between testing and the training loss should be given.

Author Response

Dear Reviewer,

We have revised the manuscript following your remarks.

First of all, we found all of your points to be justified and tried our best to follow them. To streamline the revision procedure, we have highlighted all changes including the new details. (In Latex, the reference numbers could not have been highlighted.)

1) The main motivations and contributions should be clarified in the introduction part.

Reply: We have extended the introduction with a new paragraph summarizing these issues. See lines 80-85.

2) The authors should give some explanations about figures. 

Indeed, this is an algorithmic approach to the numerical solution of the problem in (2).

The caption of Fig.  1 has been extended and also, regarding this, an extra explanation has been inserted in line 133. Also, lines 112-114 refers to this figure.

For Figure 2, we have a short extra link in line 154.

Both for Figure 3 and Figure 4, we have again extended the captions.

3) Some remarks should be added to show the effectiveness of the methods comparing with the existing results.

Regarding this, we have inserted a short paragraph in line 348-352. Also, we have extended the Conclusion in line 356-359. Finally, we have added Table 1 to point out the speed-up of our ANN-based approach vs. a conventional pointwise optimization procedure to get the finite difference coefficients.

4) The block diagram should be presented for the algorithm.

The present Figure 6 is new; here the entire algorithm is depicted (this is also a reply to point 4 raised by You).

5) The error between testing and the training loss should be given.

Accordingly, we have generated Figure 8 is now, where the error between the training and the validation loss is shown.

On the top of these modifications and completion, we have performed a full revision and corrected some minor mistakes.

Thank you very much for your efforts.

With best regards,

Ferenc Izsák (corresponding author)

Round 2

Reviewer 3 Report

The revision version is ok. Therefore, I recommend it accepted.