[Paper] Weighted universal approximation of differentiable maps on infinite-dimensional manifolds

Source: arXiv - 2606.09820v1

Overview

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.

Key Contributions

This paper presents research in the following areas:

• math.FA
• cs.LG
• math.PR
• q-fin.MF
• stat.ML

Methodology

Please refer to the full paper for detailed methodology.

Practical Implications

This research contributes to the advancement of math.FA.

Authors

• Philipp Schmocker
• Josef Teichmann

Paper Information

• arXiv ID: 2606.09820v1
• Categories: math.FA, cs.LG, math.PR, q-fin.MF, stat.ML
• Published: June 8, 2026
• PDF: Download PDF