# What is the number of neurons required to approximate a polynomial of degree n?

I learned about the universal approximation theorem from this guide. It states that a network even with a single hidden layer can approximate any function within some bound, given a sufficient number of neurons. Or mathematically, $${|g(x)−f(x)|< \epsilon}$$, where $${g(x)}$$ is the approximation, $${f(x)}$$ is the target function and is $$\epsilon$$ is an arbitrary bound.

A polynomial of degree $$n$$ has at maximum $$n-1$$ turning points (where the derivative of the polynomial changes sign). With each new turning point, the approximation seems to become more complex.

I'm not necessarily looking for a formula, but I'd like to get a general idea on how to figure out the sufficient number of neurons is for a reasonable approximation of a polynomial with a single layer of the neural network (you may consider "reasonable" to be $$\epsilon = 0.0001$$). To ask in other words, how would adding one more neuron affect the model's ability to express a polynomial?

• If the network has one hidden layer, there are papers like A closer look at the approximation capabilities of neural networks (Kai Fong Ernest Chong) saying "... given approximation threshold $\epsilon$ ... if $X$ in $\mathbb R^n$ is compact, then a neural network with $n$ input units, $m$ output units, and a single hidden layer with $\binom{n+d}{d}$ hidden units (independent of $m$ and $\epsilon$), can uniformly approximate any polynomial function $f:X \to \mathbb R^m$ whose total degree is at most $d$ for each of its $m$ coordinate functions." Jan 4, 2021 at 15:38
• @Vepir - that sounds much more like an answer than a comment. Nov 16, 2022 at 19:24
• Not an answer, but I find that simple neural networks are quite expressive. I suspect that any complete answer will have to address the network topology in some way. Mar 15 at 1:05