Enhancing polynomial approximation of continuous functions by composition with homeomorphisms

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.

💡 Research Summary

The paper introduces a novel approach to function approximation that augments the expressive power of algebraic polynomials by composing them with homeomorphisms—continuous, bijective maps whose inverses are also continuous. The authors first establish a general density result (Theorem 3.1): if a set Φ of functions is dense in C(Ωₕ) for some compact domain Ωₕ, then the composed set Φₕ = {φ ∘ h | φ ∈ Φ} is dense in C(Ω) for any homeomorphism h : Ω → Ωₕ. This result holds without requiring differentiability of h, thereby extending classical Weierstrass approximation beyond linear changes of variables.

The core contribution lies in Theorem 3.2 (and its refinement Theorem 3.3), which address univariate continuous functions that possess a finite number M of local extrema (including possible flat intervals). The authors prove that for any ε > 0 there exists a homeomorphism h : Ω → Ωₕ and a polynomial p of degree M + 1 such that
 supₓ∈Ω |f(x) − p∘h(x)| < ε.
If f has no flat intervals, the approximation can be made exact: f = p∘h. The construction proceeds by (i) partitioning Ω at representative points of each extremum, (ii) using Lemma 3.2 to guarantee a polynomial of degree M + 1 whose critical points match prescribed values, and (iii) applying Lemma 3.1 to build, on each monotonic sub‑interval, a homeomorphism that aligns the polynomial’s monotone branch with the target function’s branch. Stitching these local homeomorphisms yields a global h that is continuous, bijective, and piecewise monotone, thereby satisfying the desired approximation bound.

The theoretical findings are complemented by numerical experiments. The homeomorphism h is parameterized by an invertible residual neural network (iResNet), which guarantees a tractable, differentiable implementation of both h and h⁻¹. In one‑dimensional regression tasks (e.g., approximating sin x, |x|, and highly oscillatory functions), the composition p∘h dramatically reduces the maximum error compared with a plain polynomial of the same degree—often by several orders of magnitude.

The authors also explore multivariate applications, specifically the construction of potential‑energy surfaces (PES) for molecular systems. By extending the composition idea to higher dimensions (using tensor‑product polynomials and multivariate homeomorphisms modeled by iResNets), they achieve comparable or superior accuracy with far fewer basis functions than traditional polynomial or neural‑network baselines. This demonstrates that the method mitigates the curse of dimensionality: the expressive boost comes from the nonlinear re‑parameterization of the domain rather than from increasing the polynomial degree.

A notable aspect of the work is its focus on the supremum norm (C(Ω)) rather than the L² norm that dominates much of the existing literature on diffeomorphic basis functions. This choice aligns the theory with worst‑case error guarantees, which are crucial in scientific computing and engineering design where maximum deviation matters.

The paper situates its contributions within a broader context, contrasting the approach with the Müntz‑Szász theorem (which characterizes density of monomials with fractional exponents) and recent studies that analyze L²‑dense diffeomorphic bases. Unlike those works, the present analysis does not require h to be differentiable, monotone derivatives, or to satisfy any spectral conditions; continuity and strict monotonicity suffice.

Future directions suggested include extending the rigorous multivariate theory (especially for non‑monotone transformations), exploring alternative parameterizations of h that respect physical symmetries (e.g., spline‑based homeomorphisms), and applying the framework to high‑dimensional PDE solvers and optimization problems where adaptive domain warping could yield substantial computational savings.

In summary, the paper provides a mathematically solid and practically viable framework—polynomial composition with homeomorphisms—that overcomes classical limitations of polynomial approximation, offers exact representation for a broad class of functions with minimal polynomial degree, and demonstrates significant empirical gains in both low‑ and high‑dimensional settings. This work opens a new avenue for function approximation in numerical analysis, scientific computing, and machine learning.