Automated Rediscovery of the Maxwell Equations

Round  1

Reviewer 1 Report

This paper shows that a computer can re-discover Maxwell equations in few seconds starting from few experimental data.
It is a further example of the emerging field of data-driven discovery.
The paper is certain interesting, but I think it is not easy to read for a non-expert of the field.
Several points are rather obscure to me, and I would like the Authors to clarify them.
The title: I think that TheoSea is not something which is well-known. Actually I did not understand from this article if it is a general method that can be applied to different situations or it is just the Julia code which is given as a supplementary material. I would not mention TheoSea in the title.
The references: most of them are of very general nature (Maxwell, Borel, Dirac) some of them are unpublished works of one of the Authors, and only one [7] is a paper about the discovery of laws from data.
The method: the Authors claim that their method differs from that of previous works because thir serach is driven by "compactness and completeness". In order to explain the meaning of "compactness" a theorem by Korf is mentioned. The theorem is about the efficiency of depth-first iterative-deepening with respect to brute-force search trees. It is not clear to me what is the logical link between the "depth of the search" and the compactness defined as "the sum of the symbol weights". By the way, I think that at line 60 it should be "K=19" and in Table I the Laplacian and the second order time derivatives should be exchanged.
For what concerns "completeness" I was not able to find in the paper a clear definition.
The code: I was not able to run the Julia code which is given as supplemental material.
Probably it is my fault, because I have never run a Julia program before, but perhaps adding some instructions would allow the reader to run the program. However, reading the code I realized that the vector and scalar electrodynamics potentials are also considered, but they are not mentioned in the paper.
Finally, on a more general ground, I would like to understand what are the "initial conditions" of the search, i.e. which assumptions are made on the form of the equations. As far as I understood the only assumptions made are: 1) there are two fields, E and B, 2) the equations contain only some operators (div, curl, grad, laplacian, first and second time derivative), and 3) the maximum complexity is 26 (19+7 in the wave equation).  Are there any other assumptions?

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors discuss an interesting problem and their results are new. However the following question and remark should be answered and taken into account before the publication:
1. Eq. (1), containing all possible expressions, can be solved by fitting the coefficients c. How does the author's proposal perform compared to such a simple procedure?
2. What the authors test is the validity of certain simple expressions in describing a data set. This problem is far more trivial than "finding the truth". The struggle, involved in the latter consists mainly of the search of the language rather than the actual, explicit equations. It is better to be more modest and claim what has indeed be achieved.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf