“Polynomial regression” implies to me that the basis functions are polynomials. I‘ll assume you meant “good basis for a simple fit, maybe by least squares”. More local functions like “radial basis functions” can work well. Or use splines or sigmoidal functions, which saturate to a flat line or linear trend. In some applications Fourier or wavelet bases might be appropriate.

Gaussian process regression is a Bayesian treatment of some basis function models, potentially with an infinite number of basis functions. Artificial neural nets usually use local or sigmoidal basis functions, potentially in a more complicated way.