Universal Approximation of Nonlinear Operators and Their Derivatives
We show that Universal Approximation (UA) of nonlinear operators and their derivatives via Operator Learning (OL) architectures fails in ${C^k_F}$ (Fréchet) compact-open topologies and in Fréchet--Sobolev norms (i.e. under operator norms). We solve this obstruction by restoring UA in natural weaker topologies: $C^k_B$ (Bastiani) compact-open topologies and (novel) weighted Bastiani--Sobolev spaces for general finite input measures. In full Banach-space generality, these are the first complete generalizations of the corresponding influential classical results in [Hornik, 1991] to infinite-dimensional spaces and OL. Based on our UATs, we formulate Bastiani--Sobolev training in DIOL. These r...
| Authors | Filippo de Feo |
|---|---|
| Primary category | cs.LG |
| Published | 14 May 2026 |
| Last updated at source | 1 October 2026 |
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