Deep Learning Method Based on Physics-Informed Neural Network for 3D Anisotropic Steady-State Heat Conduction Problems

Round 1

Reviewer 1 Report

The manuscript introduces a novel approach for solving elliptic problems using deep learning, specifically through the Physics-Informed Neural Network method. The method's effectiveness is thoroughly demonstrated through multiple examples featuring analytic solutions. The manuscript's content is well-structured and organized. Nevertheless, there is still room for refinement in terms of writing quality. For this, please see the attached PDF.

Comments for author File: Comments.pdf

The manuscript is well-written, and there are no major grammar issues. However, I can suggest a few minor improvements for clarity as those in the attached PDF.

Author Response

Please see the attachment. Thank you.

Author Response File: Author Response.doc

Reviewer 2 Report

The results reported are interesting. The paper can be accepted as its.

Author Response

Please see the attachment. Thank you.

Author Response File: Author Response.docx

Reviewer 3 Report

The manuscript reported that the authors used the physical information neural network (PINN) model to solve a heat conduction problem which is anisotropic and steady-state. Heat conduction equations and the PINN model are clearly presented in the manuscript. Three examples were used to validate the effectiveness of the PINN model, the results are clearly presented and they showed that the solutions from the PINN model are very close to the analytical solutions. The presented PINN model is interesting, apart from the heat conduction it could be used in other engineering problems.

 

Since the manuscript is clearly presented, the following comments could be addressed by the authors:

 

Line 116, “represents the rate at which internal heat source is generated” needs to specify what represents the rate.

Line 215, Epsilon is a very small positive number, the value of this number should be specified.

Line 285 and 286,  “The error is minimized for initial value of learning rate as 0.03 and the decay rate as 0.005” From Figure 7, it seems that the decay rate could be greater than 0.005 and the error is further reduced.  Could more decay rates be studied?

Author Response

Please see the attachment. Thank you.

Author Response File: Author Response.docx

Reviewer 4 Report

The article entitled "Deep learning method based on physics informed neural network for 3D anisotropic steady-state heat conduction problems" is well-written and, from my point of view, would be of interest for the readers of Mathematics. In spite of this, I would recommend to perform some changes before its publication.

- Line 140 it is said: "The DNN utilizes a feed-forward, fully connected neural network": if the model is feed-forward NN it is not required to speak about DNN.

- Figure 1: please, justify the number of hidden layers included in the net.

- Figure 3: why the figures of activation functions that are not employed are included?

- Line 175: Sobol sequence: please introduce a reference.

Line 208: Adam optization please introduce a reference.

- Lines 270 to 279: is it possible to introduce any kind of theorical background here?

- Figure 8: there is a lack of regularity in this results. Please, explain.

- Figure 14, 15, 19, 20: labels size must be increased.

 

 

Author Response

Please see the attachment. Thank you.

Author Response File: Author Response.docx

Round 2

Reviewer 4 Report

After the changes performed by the author, the article is ready for its publication. Congratulations.