Enhancing polynomial approximation of continuous functions by composition with homeomorphisms

TL;DR

The paper develops a theory showing that composing polynomials with a homeomorphism induces dense sets in C(Omega) and enables finite-degree polynomials to approximate broad classes of continuous functions, especially those with finitely many local extrema. It proves existence (and minimal degree) results for univariate approximations p ∘ h with degree M+1 and demonstrates that h can be represented by invertible neural networks in practice. Numerical experiments validate the theory across univariate cases with one or multiple extrema and extend to higher-dimensional problems, including 2-D fitting and potential energy surface modeling, where the induced dense sets yield large accuracy gains with far fewer basis functions. The framework offers a principled approach to overcoming dimensionality challenges in polynomial approximation, with concrete benefits for molecular PES computations and related applications.

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.

Figures (4)

• Figure 1: Approximation of \ref{['eq:f1']} using polynomials composed with a homeomorphism.Left: Plotted are the target function and the fitted second-order power series expansion $p_2(x) = x^2 +2$ composed with an invertible bi-Lipschitz function $h_\theta$, parametrized by an iResNet with parameters $\theta$. Right: The invertible function that resulted as the solution of the optimization and its comparison with the expected function $h(x) = \text{sign}(x) \cdot \sqrt{f(x) - 2}$.
• Figure 2: Approximation of \ref{['eq:f2']} using polynomials composed with a homeomorphism. Left: The target function and the fitted second-order expansion composed with an invertible function. Right: The invertible function that resulted as the solution of the optimization.
• Figure 3: Approximation of \ref{['eq:f3']} using polynomials composed with a homeomorphism. Left: The target function (solid red) and the fitted second-order expansion composed with an invertible function (dashed black). The error of the fit is also plotted (dotted blue) for a visualization of the points where the fitting achieves higher and lower accuracy. Right: The obtained invertible transformation $h$ for the fit.
• Figure 4: Approximation of \ref{['eq:f4']} using polynomials and polynomials composed with a homeomorphism. Top left: Plotted are the target function and the fitted second-order expansion composed with a homeomorphism. Bottom left: Plotted is the difference between the target function and the induced polynomial on the validation set. Top right: Plotted are the target function and the fitted $13$-degree polynomial. Bottom right: Plotted is the difference between the target function and the $13$-degree polynomial on the validation set.

Theorems & Definitions (15)

• Theorem 3.1
• proof
• Remark 3.1
• Remark 3.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.2
• proof