Enhancing polynomial approximation of continuous functions by composition with homeomorphisms

TL;DR

This paper introduces a novel method of improving polynomial approximation of continuous functions by composing polynomials with homeomorphisms, enhancing accuracy especially in multivariate cases.

Contribution

It proves the density of composed functions in continuous function spaces and demonstrates practical applications using neural network parametrizations.

Findings

Enhanced approximation accuracy demonstrated in numerical experiments

Theoretical proof of density of composed functions in continuous spaces

Effective application to molecular potential-energy surface modeling

Abstract

We enhance the approximation capabilities of algebraic polynomials by composing them with homeomorphisms. This composition yields families of functions that remain dense in the space of continuous functions, while enabling more accurate approximations. For univariate continuous functions exhibiting a finite number of local extrema, we prove that there exist a polynomial of finite degree and a homeomorphism whose composition approximates the target function to arbitrary accuracy. The construction is especially relevant for multivariate approximation problems, where standard numerical methods often suffer from the curse of dimensionality. To support our theoretical results, we investigate both regression tasks and the construction of molecular potential-energy surfaces, parametrizing the underlying homeomorphism using invertible neural networks. The numerical experiments show strong agreement with our theoretical analysis.