First Universal Approximation Theorems for Differentiable Operators in Banach Spaces

A new study on arXiv introduces the first Universal Approximation Theorems (UATs) for nonlinear k-times differentiable operators in Banach spaces, along with their derivatives. This work builds on Hornik's 1991 research, pushing the boundaries into infinite-dimensional spaces and addressing important gaps in nonlinear functional analysis and operator learning. The theorems apply consistently across compact sets and in weighted Sobolev norms for different finite input measures through operator learning methods. The authors also discuss the potential for achieving high-order accuracy in operator learning and speeding up constrained optimization within Banach spaces.