Approximation Rates in Fr\'echet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliers

TL;DR

This paper investigates the approximation of linear differential operators in Fourier space using Fréchet metrics, providing conditions for error bounds and demonstrating approximation in Barron spaces, relevant for operator learning in PDEs.

Contribution

It introduces a new framework for approximating PDE operators via Fourier symbols with error control in Fréchet metrics, extending existing Barron space results.

Findings

Established sufficient conditions for operator approximation errors

Extended approximation results to broader semi-norm sequences

Provided concrete examples of approximable symbols in Barron spaces

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 H\"ormander-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 Fr\'echet 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.