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This research extends the universal approximation theorem to differentiable maps on infinite-dimensional manifolds, rigorously proving that functional input neural networks can approximate derivatives alongside function values. By establishing a weighted Nachbin theorem, the work provides a theoretical foundation for handling non-anticipative functionals and path space approximations beyond standard compact set formulations. These results offer significant theoretical value for functional analysis and advanced machine learning architectures dealing with complex, high-dimensional data structures.
Weighted universal approximation of differentiable maps on infinite-dimensional manifolds
arXiv:2606.09820v1 Announce Type: cross 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.
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.
Functional Analysis (math.FA); Machine Learning (cs.LG); Probability (math.PR); Mathematical Finance (q-fin.MF); Machine Learning (stat.ML)
arXiv:2606.09820 [math.FA]
(or arXiv:2606.09820v1 [math.FA] for this version)
https://doi.org/10.48550/arXiv.2606.09820 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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