[freemint]

Polynomials (or rather multinomials) suffer from the curse of dimensionality badly when needing more terms (look how taylor series terms explode). Neural networks do better. The fact that a neural network is computed using polynomials is irrelevant since the way the NN is parametrized is different from a sum of a basis of polynomials. You can inspect the vector field to proof certain properties of the neural network. SINDy is already mentioned in another reply.

[Libbum]

Yeah, this is the crux. Here's a comment from one of the devs when I asked about the polynomial vs NN basis:

The answer is quite simple really. Classical basis functions suffer from the curse of dimensionality because if you tensor product polynomial basis functions or things like Fourier basis, with N basis functions in each direction, then you have N^d parameters that are required in order to handle every combination `sin(x) + sin(2x) + ... + sin(y) + sin(2y) + ... + sin(x)sin(y) + sin(2x)sin(y) + ....`

Neural networks only grow polynomially with dimensional, so at around 8 dimensional objects it becomes more efficient. In fact, this is why we have https://diffeqflux.sciml.ai/dev/layers/BasisLayers/

[pidtuner]

Polynomials are just an example, the easiest one. The point is that there are many more universal approximators (as some other user commented here), many of them much more suitable for control applications than NNs.