Approximation Rates in Fréchet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliers
Ahmed Abdeljawad, Thomas Dittrich
TL;DR
The paper addresses the problem of approximating linear differential operators by manipulating their Fourier symbols within a Fréchet-space framework, enabling operator learning with error measured by a Fréchet metric $d_{\mathcal{V}}$. It develops two main theorems: one under monotone growth of a separable semi-norm sequence and another under bounded growth, and then instantiates these results for exponential spectral Barron spaces and Gelfand-Shilov/Pseudodifferential-symbol settings. By embedding exponential spectral Barron spaces into smooth symbol classes and leveraging existing two-layer cosine-activated network results, the authors derive explicit rates (e.g., $r_\ell(N)=C_\ell e^{-c_\ell N^{\beta/d}}$) and identify regimes where Fréchet-approximation is feasible or blocked by analytic/derivative-boundedness constraints. The work thus furnishes a principled, Fréchet-metric foundation for neural-operator approximation, clarifying when and how dimension-robust rates can be achieved and guiding the selection of target function classes for operator-learning tasks.
Abstract
Operator learning is a recent development in the simulation of Partial Differential Equations (PDEs) by means of neural networks. The idea behind this approach is to learn the behavior of an operator, such that the resulting neural network is an (approximate) mapping in infinite-dimensional spaces that is capable of (approximately) simulating the solution operator governed by the PDE. In our work, we study some general approximation capabilities for linear differential operators by approximating the corresponding symbol in the Fourier domain. Analogous to the structure of the class of Hörmander-Symbols, we consider the approximation with respect to a topology that is induced by a sequence of semi-norms. In that sense, we measure the approximation error in terms of a Fréchet metric, and our main result identifies sufficient conditions for achieving a predefined approximation error. Secondly, we then focus on a natural extension of our main theorem, in which we manage to reduce the assumptions on the sequence of semi-norms. Based on existing approximation results for the exponential spectral Barron space, we then present a concrete example of symbols that can be approximated well.