Comment by 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.