Polynomial regression
A polynomial regression wants to model a non-linear, polynomial relationship within the data. It can be just reduced to a linear regression via variable substitution, reducing the features to linear ones. A generic polynomial model is
$y(x) = \sum_{i=0}^{i=m}a_i x^i = a_0 + a_1x + a_2x^2 + \ldots + a_mx^m$
The problem can be reduced to that of a multiple linear regression where the features are the polynomials of
$x$
:
$\begin{cases} x_1 = x \\ x_2 = x^2 \\ \ldots \\ x_m = x^m \end{cases}$
so that the model is linear in all of these new features.
The same applies to non-polynomial features, say exponentials or other functions.