Modeling a Typical Non-Uniform Deformation of Materials Using Physics-Informed Deep Learning: Applications to Forward and Inverse Problems
Round 1
Reviewer 1 Report
The paper presents a study that investigates the potential of physics-informed neural networks (PINN) in modeling non-uniform deformation in solid mechanics problems related to linear elasticity. The research compares the performance of PINN with the conventional finite element method (FEM) in both forward and inverse problems. The paper's topic and the writers' approach have significant importance in the field, especially in inverse problems to extract materials' properties, which is a challenging area in research and industry. There is also good verification for the the simple results with FEM in the paper.
However, there are several areas in the paper that require reconsideration. Firstly, there is a lack of referencing in some significant parts of the paper, which should be addressed to avoid unsubstantiated claims. Secondly, the paper needs to highlight the novelty of the work and justify the importance of the study by comparing statistical comparisons of the novel technique with other methods such as FEM, such as computation time or flexibility in the usage of deep learning. Another point for some concepts, the writers reply on formulation and don't explain the physical meaning of that.
Detailed comments have been provided in the attached PDF file.
Comments for author File: 
Comments.pdf
Author Response
Response to Reviewer 1’s Comments
Point 1: The paper presents a study that investigates the potential of physics-informed neural networks (PINN) in modeling non-uniform deformation in solid mechanics problems related to linear elasticity. The research compares the performance of PINN with the conventional finite element method (FEM) in both forward and inverse problems. The paper's topic and the writers' approach have significant importance in the field, especially in inverse problems to extract materials' properties, which is a challenging area in research and industry. There is also good verification for the simple results with FEM in the paper.
However, there are several areas in the paper that require reconsideration. Firstly, there is a lack of referencing in some significant parts of the paper, which should be addressed to avoid unsubstantiated claims. Secondly, the paper needs to highlight the novelty of the work and justify the importance of the study by comparing statistical comparisons of the novel technique with other methods such as FEM, such as computational time or flexibility in the usage of deep learning. Another point for some concepts, the writers reply on formulation and don't explain the physical meaning of that.
Detailed comments have been provided in the attached PDF file.
Response: Thank you very much for the valuable comments provided by the reviewers. We appreciate the constructive feedback that we have received, which will help us improve the quality of our manuscript. We agree that referencing is a crucial aspect of scientific writing, and we will address the issue of lacking references in some parts of the paper to avoid any unsubstantiated claims. We also appreciate your suggestion to emphasize the novelty of our work and its significance by providing statistical comparisons between our novel technique and other methods such as FEM. We will incorporate these suggestions into our revised manuscript to enhance the quality of our study. We acknowledge the concern raised by the reviewers about the lack of physical meaning explanation for some of the concepts presented in the paper. We will revise the manuscript to ensure that these concepts are explained in a more understandable manner, thereby increasing the clarity and coherence of our work. Thank you again for the valuable feedback and constructive criticism. We are confident that your suggestions will lead to a more thorough and compelling paper that meets the high standards of scientific research.
Detailed response have been added in the revised manuscript and attachment.
Author Response File: Author Response.pdf
Reviewer 2 Report
The paper demonstrates the use of physics-informed neural network (PINN) to predict the deformation field of a plate with a circular hole under uniaxial loading. The PINN is also used to solve the inverse problem of identifying mechanical properties based on full-field displacement data. The work is interesting. But I do have some concerns that should be addressed before it is published in Applied Sciences:
-It is common to inversely determin the modulus for the inverse problems in mechanics. However, the authors demonstrates the identification of Poisson’s ratio. Can the approach be applied to similar problems but with the modulus as unknowns to be determined, in particular for the case with non-homogeneous modulus fields? The authors should add at least one example on this to show the capability of the apporach or should clarify it cannot be done.
-The approach should be described with more details. For example, for the inverse problem, the unknown parameter, poisson’s ratio, should be the parameter to be optimized simultaneously with weights and biases of the network. However, this is never mentioned and thus needs to be elaborated.
-Some symbols are not introduced or mixed. For example, when introducing ANN structure, the authors seem to use the x and u to represent the position and displacement vectors, respectively. But later (x, y) are used to represent two components of the position. For the higher readability, the authors should express the vector or tensor in bold fonts, such as x, u, and n.
-Limited works are mentioned in literature review section. Please add more explanations by looking at other deep learning works on inverse problems in the mechancis field. The following can be helpful.
https://onlinelibrary.wiley.com/doi/abs/10.1111/str.12431
https://onlinelibrary.wiley.com/doi/10.1002/adfm.202109805
https://doi.org/10.1016/j.jmps.2023.105231
-The authors are suggested to provide the training curves (loss versus epoch) for the typical examples.
-Typos can be seen, e.g., the Poisson’s ratio is denoted by “mu” in Eq.(9)-(10) but by “nu” elsewhere.
Author Response
Please see the attachment.
Author Response File: 
Author Response.docx
Reviewer 3 Report
The manuscript proposed to extend application of PINN to linear elasticity problems. Its main focus on studying performance of conventional PINN approach to modeling non-uniform deformation with high stress concentration in relation to solid mechanics involving forward/inverse problem.
Overall the manuscript is recommended to be accepted after following comments/questions are answered:
1. Applying PINN on linear elasticity solid mechanics has been studied by many researchers. I want to know the main difference/novelty of current work compared with other published work ?
2. Can authors give more details on model training, such as how to set up training/validation dataset ? how to set learning rate, hyperparameters, etc ? and give error evolution figure during training ?
3. For technical details, how to ensure uniqueness of solution in inverse analysis based on PINN ? about this question, I recommend authors review and cite following papers:
a. Chen, Q., Xie, Y., Ao, Y., Li, T., Chen, G., Ren, S., Wang, C. and Li, S., 2021. A deep neural network inverse solution to recover pre-crash impact data of car collisions. Transportation research part C: emerging technologies, 126, p.103009.
b. Xie, Y., Wu, C.T., Li, B., Hu, X. and Li, S., 2022. A feed-forwarded neural network-based variational Bayesian learning approach for forensic analysis of traffic accident. Computer Methods in Applied Mechanics and Engineering, 397, p.115148.
4. How to incorporate B.C. into PINN so that remove the rigid body motion (i.e., singular stiffness matrix) ?
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
The writers provided sufficient information and obvious modification after the first review.
Reviewer 3 Report
All questions have been answered.
The manuscript is recommended to be published.
