Projection Methods for Operator Learning and Universal Approximation
This provides a theoretical framework for deep learning in operator learning, addressing a foundational mathematical challenge in machine learning.
The authors tackled the problem of approximating continuous operators on Banach spaces by introducing a method based on orthogonal projections on polynomial bases, achieving a new universal approximation theorem using the Leray-Schauder mapping and deriving sufficient conditions for approximation in specific cases like L^2 spaces.
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.