[princeofwands]

PR is quite similar to a neural network with a single extremely wide layer. Unless you are hashing the polynomials, you will quickly run into an intractible problem, while properly setup NNs automatically learn which polynomials are needed for solving the problem, giving the budget of the network architecture (number of layers, number of neurons, etc.).

Nobody in industry will abandon NNs over PRs if they are looking to making it easier to handle. I doubt on most industrial problems, that PR even comes close to NNs in performance.

[zozbot123]

I think what you literally need is a network that is "narrow" but also has many layers. This is because what PR features over simple linear regression is interaction terms, and multiple layers are an alternate way to get those. This also hints at one problem with true PR - in a multi-variate context, the amount of coefficients you'll need to fit is going to scale very badly given the number of variables and the degree of your polynomial. NN layers have natural dimenionality-reduction properties that make them more flexible.