A Stable Generalized Finite Element Method Coupled with Deep Neural Network for Interface Problems with Discontinuities

Round 1

Reviewer 1 Report

1)     In the introduction section, please mention the application of the XFEM and GFEM. For example. M.M. Shahzamanian has published a systematic literature review about the application of the XFEM to predict failure in pipeline materials.

2)     Also in the introduction section, please mention the application of DNN to predict failure.

3)     Please discuss whether such an approach is efficient in terms of time consumption or not?

 

4)     Please make more discussion for the presentation of the results.

Author Response

Please see the attched PDF file for the responses.

Author Response File: Author Response.pdf

Reviewer 2 Report

The review is in the attached file.

The article is not recommended for publication in a journal.

Comments for author File: Comments.pdf

Author Response

Please see the attched PDF file for the responses.

Author Response File: Author Response.pdf

Reviewer 3 Report

This is a very interesting work regarding the SGFEM method coupled with a DNN for solving elliptic interface problems with a discontinuous interface condition, free from penalty terms.

Some minor changes that need to be addressed are:

-Can this method be extended to solve parabolic interface problems with discontinuities?

-Based on what criteria is the ResNet deep neural network selected? The authors should discuss this.

-What is the qualitative meaning of the contrast c in the numerical experiments?

-The last term on the RHS of eq. 4.9 is minus or this is a mistake? The same for eq. 4.14.

-The authors should change the numbering of the references so that the first ones that appear in the manuscript are not [5,6,23,24] but [1-4]. The references should start from number 1 going to the last number, which is 63 in this manuscript.

 

Minor errors

-At the start of the second paragraph correct “It is realized” to become “It was realized”.

-In the end of the text legend of figure 6 there is a missing parenthesis.

-Below equation (3.3) correct the phrase “…and mimic the non-smooth the exact solutions…” to become “…and mimic the non-smooth exact solutions…”.

-On page 17 correct the phrase “can also be obtain…” to become “can also be obtained”.

Author Response

Please see the attched PDF file for the responses.

Round 2

Reviewer 1 Report

The paper is acceptable in this format.

Author Response

Thank you very much!

Reviewer 2 Report

See attached file.

 

Comments for author File: Comments.pdf

Author Response

Thank you for the comments. We revised the manuscript according to this reviewer's comment, see attached PDF file (Aug 3...). The changes are in page 7, line 199 and (4.6), and page page 10, lines 263-264. Due to line 199 and (4.6), the solution of (4.4)-(4.8) is continuous across the interface, so the solution can belong to H^{^1}. Since the solution of (2.1)-(2.4) is discontinuous, u cannot belong to H^{^2} in the whole domain Omega. We always mean that u belongs to H^{^2}(Omega_r), r=0, 1, namely, in the sub-domian Omega_r, the solution u is in H^{^2}(Omega_r).

Author Response File: Author Response.pdf