arXiv:2406.12264v3 Announce Type: replace-cross
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.
OptoLoop: An optogenetic tool to probe the functional role of genome organization
The genome folds inside the cell nucleus into hierarchical architectural features, such as chromatin loops and domains. If and how this genome organization influences the