Interpolation for Neural Network Operators Activated by Smooth Ramp Functions^†

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors extended the results of the neural network interpolation operators activated by smooth ramp functions proposed by Yu and discussed the higher-order approximation. Overall, this work seems interesting. However, it lacks the clear explanation of the novelty. Therefore, the reviewer suggests the authors to revise the manuscript by addressing the following concerns:

 

1 The abstract does not clearly explain the novelty of this manuscript. Specifically, it lacks details on which aspects are extended.

 

2 The title of this manuscript is similar to that of Ref. 27. What is the difference between “Interpolation of neural network operators” and “neural network interpolation operators”? The reviewer suggests that the authors clearly explain these definitions and highlight the difference between these two papers.

 

3 The equations in the introduction are not numbered.

 

4 The equation in line 78 contains errors. It should be corrected to

 

5 Section 1, “Main results” includes some theorems and proofs. Are these theorems and proofs new and obtained for the first time? The reviewer suggests that the authors clearly indicate which results are original contribution of this manuscript.

 

6 In Figure 1(a), the colors of the lines for ‘n=5’ and ‘Function’ are too similar, making them difficult to distinguish.

 

7 The authors stated this manuscript extended the results of Ref.[26]. But the two examples do not show the comparison with previous methods. The reviewer suggests adding at least one comparison with other methods.

 

8 Some errors in the approximation for n=100 are zero in Tables 1 and 2. The reviewer suggests that the authors further check their results for accuracy.

 

 

 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript studies the construction and approximation results of interpolation NN operators based on ramp function. With theoretical proof and numerical tests, it discusses and verifies the convergence rate and consistency. The followings are my comments and suggestions.

 

1. Can the author explain more about how they choose test points on the example 1 and 2? It seems they are not randomly or uniformly chosen. Also, how about training samples?

2. In order to discuss convergence, I suggest adding another figure which plots the correlation between N and error.

3. The author mentioned they also verified the computational efficiency. But from the numerical example, I didn’t find discussion about that.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have addressed my concerns.