Projection Methods for Operator Learning and Universal Approximation

TL;DR

This paper establishes a new universal approximation theorem for nonlinear operators on Banach spaces and introduces a projection-based method for operator learning in function spaces, providing a theoretical foundation for deep learning approaches.

Contribution

It presents a novel universal approximation theorem for operators using Leray-Schauder mappings and develops a projection-based operator learning method in Banach spaces, especially $L^p$ spaces.

Findings

Universal approximation theorem for operators on Banach spaces.

A projection-based operator learning method in $L^p$ spaces.

Conditions under which approximation results hold for $p=2$.

Abstract

We obtain a new universal approximation theorem for continuous (possibly nonlinear) operators on arbitrary Banach spaces using the Leray-Schauder mapping. Moreover, we introduce and study a method for operator learning in Banach spaces $L^p$ of functions with multiple variables, based on orthogonal projections on polynomial bases. We derive a universal approximation result for operators where we learn a linear projection and a finite dimensional mapping under some additional assumptions. For the case of $p=2$, we give some sufficient conditions for the approximation results to hold. This article serves as the theoretical framework for a deep learning methodology in operator learning.