M-WDRNNs: Mixed-Weighted Deep Residual Neural Networks for Forward and Inverse PDE Problems
Round 1
Reviewer 1 Report
I am impressed by the quality of work in an area that is starting to generate relatively high interest to researchers such as me. The result obtained suggest that the method is a good one.
I will suggest that you include the equation you solved in example 1. Also, have spaces between example and the number for all examples.
Also, I noticed that, in several instances in the introduction, spaces were not left after a word and the abbreviation. Kindly fix.
Leave spaces before the citations also.
At the end of paragraph 2 of the introduction, you have " Based on the Galerkin method, variational physics-informed neural networks(VPINNs)[15,16] algorithm were proposed". I suggest you change to " Based on the Galerkin method, a variational physics-informed neural networks (VPINNs) [15,16] algorithm was proposed".
In example 2, replace the first sentence with, "Here, we apply our proposed algorithm to the problem of the non-homogeneous Klein-Gordon equation:"
On page 10, it should be "secondly" and not "second".
The first sentence of the summary and discussion section should be " Although PINNs are widely used ..."
Keep the references consistent. Some have initial and surname while some have surname and initial.
Author Response
Please see the attachment.
Author Response File: 
Author Response.docx
Reviewer 2 Report
I suggest to accept the paper.
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Reviewer 3 Report
In this paper the authors proposed a novel approach: the introduction of a self-optimized mixed-weighted residual block. This innovative block incorporates autonomously determined weighted coefficients through an optimization algorithm. Moreover, to enhance performance, they replace one of the transformer networks with a skip connection. Finally, they test their algorithms by some partial differential equations, such as the non-homogeneous Klein-Gordon equation, the (1+1) advection diffusion equation and the Helmholtz equation. Experimental results show that the proposed algorithm greatly improves the numerical accuracy.
Remarks:
1- I ask the authors why did not discuss The most interesting question which is how they proved that the proposed algorithm is valid for the inverse problem
2- Numerical results are not compared with rigorous methods
3- The figure s do not reflect the results very well
4- The title should be changed to more suitable with the scope of the paper.
English is bad written.
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Round 2
Reviewer 3 Report
I think the authors have made all the requested corrections
The authors have carefully polished and revised the English structure of the full text.
