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.