Neural networks that take functions as inputs—common in operator learning—are typically shown to be universal through existence arguments. According to the arXiv abstract, these proofs either reduce the problem to finite-dimensional networks or rely on Stone–Weierstrass-type density results. Such arguments establish that a universal approximator exists, but they do not necessarily show how to construct one.
The new paper, titled "Constructive Universal Approximation by Infinite-Dimensional Neural Networks via Gaussian-Weighted Ridgelet Transform," proposes a different approach. It uses a Gaussian-weighted ridgelet transform to give a constructive universal approximation theorem for infinite-dimensional neural networks. This suggests an explicit method for building approximating networks rather than only proving their existence.
The abstract is brief and does not include full technical details, so the exact construction and its conditions are not yet available from this source. Still, the shift from non-constructive density arguments to a constructive transform-based proof could be useful for designing operator-learning architectures with guarantees.