Convergence theory for Hermite approximations under adaptive coordinate transformations

TL;DR

This paper provides theoretical error estimates for Hermite function approximations under adaptive coordinate transformations, showing how nonlinear maps can improve convergence rates.

Contribution

It introduces the first error bounds for Hermite expansions composed with adaptive transformations, linking approximation quality to the pullback function's regularity.

Findings

Error estimates relate Hermite approximation accuracy to pullback regularity.

A constructed monotone transport map aligns decay properties, ensuring spectral convergence.

Analysis offers theoretical support for adaptive Hermite methods using normalizing flows.

Abstract

Recent work has shown that parameterizing and optimizing coordinate transformations using normalizing flows, i.e., invertible neural networks, can significantly accelerate the convergence of spectral approximations. We present the first error estimates for approximating functions using Hermite expansions composed with adaptive coordinate transformations. Our analysis establishes an equivalence principle: approximating a function $f$ in the span of the transformed basis is equivalent to approximating the pullback of $f$ in the span of Hermite functions. This allows us to leverage the classical approximation theory of Hermite expansions to derive error estimates in transformed coordinates in terms of the regularity of the pullback. We present an example demonstrating how a nonlinear coordinate transformation can enhance the convergence of Hermite expansions. Focusing on smooth functions decaying along the real axis, we construct a monotone transport map that aligns the decay of the target function with the Hermite basis. This guarantees spectral convergence rates for the corresponding Hermite expansion. Our analysis provides theoretical insight into the convergence behavior of adaptive Hermite approximations based on normalizing flows, as recently explored in the computational quantum physics literature.