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A formatting suggestion: Summarize results for test systems pertaining to final model (the (m,p) parameters, number of free parameters, computational time & number of iterations required) in a table.
On that note, there doesn't seem to be much analysis on the differing convergence rates for different systems - what conditions cause this-or-that test system to converge quicker?
Would also like to see some comments on how the computation time scales for 2-dimensional or higher dimensional systems (since every test system presented has been 1-dimensional) - is this approach for finding the Lagrangian computationally feasible for larger systems?
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The idea of using the principle of least action seems novel and is interesting. The paper is well written, structured and complicated physics and algorithms are explained well albeit at a high level of abstraction.
I was left wondering a little about the scalability of the first approach using the Nelder-Mead optimisation. The examples, and the excellent results even with noise, seem to be dealing with situations with very small values of m and p. Could you say something about how this method would scale to more complicated examples? Also, what sorts of values of m and p might be found in typical real-world examples?
In the second, GA, approach the use of syntactic expressions as the genetic material is appropriate but I do not think that that is novel, as such. Perhaps some more references to related previous work is needed here.
Minor points... Some of the specifics of the modifications to EL_D to construct the score function seem a little arbitrary. For the non-physicist reader the discounting of certain terms from the Lagrangian because they can have "no physical significance" could be explained.
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