Generalized Reproducing Kernel Banach Spaces: A Functional Analytic Framework for Abstract Neural Networks

Abstract

In this paper, we introduce a generalization of Reproducing Kernel Banach Spaces (RKBS), which we term \emph{Generalized Reproducing Kernel Banach Spaces} (GRKBS). The motivation stems from recent results showing that classical fully connected neural networks can be understood as finite-dimensional subspaces of RKBS. Our generalization extends this perspective to settings with Banach-valued codomains, allowing the construction of \emph{abstract neural networks} (AbsNN) as compositions of GRKBS. This framework provides a natural pathway to model neural architectures that go beyond classical machine learning paradigms, including physically-informed structures governed by differential equations. We establish a unified definition of GRKBS, prove structural uniqueness results, and analyze the existence of sparse minimizers for the corresponding abstract training problem. This contributes to bridging functional analytic theory and the design of new neural architectures with applications in both approximation theory and mathematical modeling.