[Libbum]

I'm not entirely confident in answering that directly, so perhaps you can check my reasoning here.

If F is completely unknown, perhaps you start training with a 10 dimensional polynomial basis. What is the (computational) cost of obtaining your solution? Once you have it, will this polynomial accurately represent your system in any real world manner? Perhaps higher order parameters are needed to approximate trigonometric functions - are you able to easily add such functions to your training basis? If not - then your basis could be too restrictive to provide you with a minimal implementation of your control variable.

It looks like you work with this stuff far more than I have, so perhaps that's not an adequate answer.

Another way to look at this though: If you only wanted to characterise your system with polynomials, UODEs + SINDy can do this for you - the NN is simply the optimisation method that's in place of any other optimisation algorithm.

[pidtuner]

The computational cost of "training" a polynomial would be the same as just one iteration of the training algorithm used by typical NNs. When it comes to trig functions, the story is the same as with the exp function e(). When you call the sin() or the cos() functions in your favorite language, in the end it uses taylor series (polynomials) to compute it (plus some hacks to add precision on certain ranges of the function and to overcome some floating point precision limitations).

The degree at which a polynomial model would fit the real world system has to be validated against data, just the same as with NNs. What does one do when an NN fit is not good enough or too good (overfitting)? One adds or removes layers. Same with polynomials, one increases or decreases degrees.

Sorry for the rant, I am not saying NNs are useless, because I do believe they are super useful for certain problems, specially for categorization. But it seems to me that now a days there is this trend of using NNs as a hammer, and not all problems are nails. Specially when it comes to control, and lives or big economic losses are at stake, it is the responsibility of the engineer to resist the fuzz and craze and use the right tool for the problem.

[ChrisRackauckas]

There is a saying in mathematics that the fastest way to a solution is through the complex plane. This was discovered because a lot of proofs are nicer by doing analytical continuation and analyzing the properties of the continued function. Complex-step differentiation is another example of this.

In some sense, something similar applies to neural networks in this context. Have you done a lot of fitting of classical basis methods inside of differential equations? They are very prone to local minima, so direct training of polynomials inside of a differential equation is rather hard. But through neural network magic, somehow related to [1], which essentially state that local minima are the global minima on large enough neural networks. So this lets you get pretty lazy and just do local optimization to find missing functions, and then sparsify to polynomials later, in a way where the optimization is better behaved than going directly to polynomials. The DiffEqFlux library has both approaches available, so you can try both side by side and see the difference. From years of experience doing the former, the latter is quite a breath of fresh air.

   [1] https://arxiv.org/abs/1412.0233

[freemint]

> The computational cost of "training" a polynomial would be the same as just one iteration of the training algorithm used by typical NNs.

That statement depends heavily on the dimensionality of the problem. Polynomials also have huge problems with discontinuities (even in some higher order derivative) sometimes would require an infinite number of polynomials to smooth out the errors around the discontinuities. (try to fit the Integral of |x| with polynomials)

Fear of NN in control is justified if the networks are poorly understood.

[srean]

Not just that, they tend to blow up when one extrapolates 'too' far from data. This can be controlled for using other basis functions, for example functions in a reproducing kernel Hilbert space, radial basis functions. It is best to choose the basis based upon the data (as RBFs and RKHS bases do) and not chose a basis independent of the data. This applies for polynomials too, choosing a polynomial basis that's orthogonal with respect to the data distribution makes computations much better behaved -- otherwise its common to run into ill conditioned problems that are very sensitive to noise in the data.

[Libbum]

I certainly agree with the NNs are used as hammers point. Until coming across the UODE concept I was of the opinion they were more parlour trick than anything useful. Here though, I could see some validity.

These comments are appreciated - I think a discussion like this is lacking in the SciML docs (or at least not visible enough). Will have a chat with some of the devs and see if there's something we can add.