Convergence theory for Hermite approximations under adaptive coordinate transformations
For researchers in spectral methods and computational quantum physics, this work offers theoretical justification for using normalizing flows to accelerate Hermite approximations, though the results are incremental as they extend classical theory.
This paper provides the first error estimates for Hermite expansions composed with adaptive coordinate transformations, establishing an equivalence principle that links approximation quality to the regularity of the pullback function. It demonstrates that a nonlinear coordinate transformation can guarantee spectral convergence rates for smooth, decaying functions.
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.