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The study of universal approximation of arbitrary functions $f: \mathcal{X}\to \mathcal{Y}$ by neural networks has a rich and thorough history dating backto Kolmogorov (1957). In the case of learning finite dimensional maps, manyauthors have shown various forms of the universality of both fixed depth andfixed width neural networks. However, in many cases, these classical resultsfail to extend to the recent use of approximations of neural networks withinfinitely many units for functional data analysis, dynamical systemsidentification, and other applications where either $\mathcal{X}$ or$\mathcal{Y}$ become infinite dimensional. Two questions naturally arise: whichinfinite dimensional analogues of neural networks are sufficient to approximateany map $f: \mathcal{X} \to \mathcal{Y}$, and when do the finite approximationsto these analogues used in practice approximate $f$ uniformly over its infinitedimensional domain $\mathcal{X}$?  In this paper, we answer the open question of universal approximation ofnonlinear operators when $\mathcal{X}$ and $\mathcal{Y}$ are both infinitedimensional. We show that for a large class of different infinite analogues ofneural networks, any continuous map can be approximated arbitrarily closelywith some mild topological conditions on $\mathcal{X}$. Additionally, weprovide the first lower-bound on the minimal number of input and output unitsrequired by a finite approximation to an infinite neural network to guaranteethat it can uniformly approximate any nonlinear operator using samples from itsinputs and outputs.

On Universal Approximation by Neural Networks with Uniform Guarantees on Approximation of Infinite Dimensional Maps