This paper presents a new universal approximation theorem for continuous (possibly nonlinear) operators in arbitrary Banach spaces using the Leray-Schauder mapping. Additionally, 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 certain additional assumptions. For the case of $p=2$, we provide sufficient conditions for the approximation results to hold. This article serves as the theoretical framework for a deep learning methodology in operator learning.
Blogger's Review: This paper bridges the gap between operator learning and universal approximation theory, introducing new perspectives and technical means for multivariable functions through orthogonal projection methods. Its contributions hold significant theoretical and practical implications, undoubtedly advancing the field of operator learning.