Weighted universal approximation of differentiable maps on infinite-dimensional manifolds

2026-06-08

AI Summary

This work addresses the universal approximation problem for differentiable mappings and their derivatives on infinite-dimensional weighted manifolds. By introducing a weighted Nachbin theorem, it overcomes the classical restriction to compact sets and extends the approximation capability of function-input neural networks from function values to their derivatives. The approach integrates Banach space linear readouts, horizontal and vertical derivatives, and path signature techniques to establish the first universal approximation theorem that simultaneously approximates both a functional and its derivative on non-anticipative functionals and path-space functionals. Theoretically, it is shown that linear functionals of path signatures can effectively approximate such functionals along with their directional derivatives, substantially enhancing the expressive power and applicability of infinite-dimensional neural networks.

Abstract

We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives. A FNN maps the input from a possibly infinite-dimensional weighted manifold to the real-valued hidden layer, on which a non-linear scalar activation function is applied, and then returns the output into a Banach space via some linear readouts. By proving a weighted Nachbin theorem, we establish a universal approximation theorem (UAT) for differentiable maps, which goes beyond the usual formulation on compact sets and also includes the approximation of the derivatives. This leads us to approximation results for non-anticipative functionals including the horizontal and vertical derivatives. As a further application, we show that linear functions of the signature are able to approximate path space functionals including their directional derivatives.

Problem

• universal approximation
• differentiable maps
• infinite-dimensional manifolds
• weighted approximation
• non-anticipative functionals

Innovation

• universal approximation theorem
• differentiable maps
• infinite-dimensional manifolds
• functional neural networks
• signature method