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Weighted universal approximation of differentiable maps on infinite-dimensional manifolds

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  • Philipp Schmocker
  • Josef Teichmann

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 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.

Suggested Citation

  • Philipp Schmocker & Josef Teichmann, 2026. "Weighted universal approximation of differentiable maps on infinite-dimensional manifolds," Papers 2606.09820, arXiv.org, revised Jun 2026.
    • Handle: RePEc:arx:papers:2606.09820
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    References listed on IDEAS

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