The claim that the fit gets to the "physics" of a pendulum is not well supported: The physics involves the law of gravity, Newton's second law, and aerodynamic drag as a function of velocity, and their work is very far from these three basic inputs from physics.

Indeed, a good intermediate step would be just to get the ordinary differential equation initial value problem for a pendulum, that really is from the physics, that their fit solves; they didn't do that.

Or if not a pendulum, then oscillations in basic A/C circuit theory with resistors, capacitors, and inductors; they didn't do that either.

Uh, data such as they started with ain't necessarily from a pendulum, and to assume that it is is a serious mistake. In particular there is no way that a computer program, or anything else, starting with just the data, can conclude a pendulum instead of an A/C circuit or something else. Sorry guys.

The role of 'complexity' in the expressions very much needs to be made clear. This is especially the case since otherwise can fit such data with just Lagrange interpolation (a polynomial) or splines.

The whole idea of the program encounters one of the biggest old problems in science: Starting with the observational data on the motions of the planets, Ptolemy had some complicated mechanical contraptions that fit the data fairly well (we will set aside some of the accusations that he, uh, 'smoothed' his data, that is, had an early use of Kelly's variable constant and Finkel's fudge factor).

The big problem in Ptolemy's work was his complicated contraptions didn't seem to be from any 'physical laws'. Of course, he didn't know that any suitable physical laws existed.

Big, huge steps later were from Copernicus, Kepler, etc. by which time it was clear that, looked at in the right coordinate system, the planets were moving in ellipses with the sun at one focus of each ellipse.

Then Newton assumed his second law and law of gravity, invented calculus, and used the three to derive the ellipses. That's why we think that Newton was one of the greatest of all scientists. The computer program is suggesting that it is hoping to automate such work; ROFL.

The claim that the program can find function f so that x = f(t, x) is not so amazing: Just set f(t,x) = x. Then perfectly x = x. Semi-amazing.

The example was for only one independent variable. More important cases are for several independent variables and several dependent variables. That is, for the real numbers R and positive integers m and n, find function f: R^m --> R^n so that for independent variables x in R^m and dependent variables y in R^n we get f(x) close to y. Pull this off with any generality, execution efficiency, and usefulness, and I'll start to be impressed.

To be more impressive than now, they might just look for linear f: R^m --> R and show why their work is better than old step-wise regression. There they need to address the conclusions now over 40 years old that step-wise regression didn't work very well, and doing all possible regressions took too long and gave a mess.

Generally they have the cart before the horse: The researcher is supposed to bring some understanding or at least some conjectures to the work and let the numerical work just fill in the details. Doing the numerical work first is bass ackwards.

The idea of a 'complexity measure' to select among different formulas is close to some old moon worship, black magic, or superstition about the magical properties of selected frogs and mushrooms.

Setting aside all the hype and nonsense, where could such work be used? Well, the answer is the same as has been the case for decades: It's a curve fitting program and, thus, could be used where curve fitting programs have been useful:

First, it can do the fitting arithmetic when the researcher brings to the computer some 'form' of the formula and is just looking for some constants and NOT the formula.

Second, if there is some data and want a simple expression that 'compresses' the data. That can be useful, say, in parts of stochastic dynamic programming; see, e.g., D. Bertsekas at MIT where he used neural nets for such 'compression'. That is, just storing the data might be 10^n bytes for n = 20, 30, 40, ..., but a few formulas might do just as well for big savings. Generally splines and multivariate splines have been among the leading approaches, but I didn't see splines mentioned in their video.

The scary, upsetting, objectionable parts are the claims that they are "detecting equations and hidden mathematical relationships in data" and, as in the video, getting at the "physics". No they are not. Here their hype reeks of the usual in 'artificial intelligence' and 'machine learning' that their software can 'learn' and be 'intelligent', and that's so far, and likely for a long time, nonsense.

Another objectionable part is their theme of setting aside all the assumptions, definitions, theorems, and proofs of statistical estimation of parameters in statistical models, hypothesis tests on the parameters, confidence intervals on the parameters and the predicted values from the fits, etc. How do they do that? They just leave it all out!

They have an option of 'smoothing' the data: That's a big subject. So, the implication is that researchers might just click on the button that says "Smooth" and report that in their work. Then to 'smooth' becomes some unlabeled bottle of snake oil.

The theme here is to omit all the rational and deductive framework of applied math and replace it with GUIs: This is a future of irrationality with no hope.

This work is a special case of the more general situation that 'computer science' is getting a C- in applied math and is out'a gas.

The work fills a much needed gap in the literature and would be illuminating if ignited.