Projection Methods for Operator Learning and Universal Approximation

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.

17 pages. Comments are welcome! v5: A proof in section 5 has been expanded with details and comments, Remark 5.2 has been corrected (it named Nemytskii operators instead of Urysohn integral operators as it was meant to be)