Universal Approximation Theorem Extended to Differentiable Maps on Infinite-Dimensional Manifolds
Researchers have generalized the universal approximation theorem for functional input neural networks (FNNs) to cover differentiable maps on infinite-dimensional weighted manifolds, including the approximation of derivatives. The work introduces a weighted Nachbin theorem and extends classical results beyond compact sets, with applications to non-anticipative functionals and path space functionals via signature methods. This advances the theoretical foundations for using neural networks in settings such as mathematical finance and stochastic analysis, where inputs are functions or paths rather than finite-dimensional vectors.
A new preprint by Philipp Schmocker, submitted to arXiv on June 8, 2026, generalizes the universal approximation theorem (UAT) for functional input neural networks to differentiable maps on possibly infinite-dimensional weighted manifolds. The framework considers FNNs that take inputs from such manifolds, pass them through a nonlinear scalar activation function in a real-valued hidden layer, and return outputs in a Banach space via linear readouts. A central technical contribution is a weighted Nachbin theorem, which enables the approximation results to hold beyond compact sets—a significant extension over classical UAT formulations. The paper further derives approximation results for non-anticipative functionals, including their horizontal and vertical derivatives, which are key objects in functional Itô calculus. Additionally, the authors show that linear functions of the path signature can approximate path space functionals along with their directional derivatives, connecting the work to the growing literature on signature-based machine learning. The paper spans 77 pages with 3 figures and sits at the intersection of functional analysis, probability theory, machine learning, and mathematical finance.
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- arXiv cs.LGCenter
Weighted universal approximation of differentiable maps on infinite-dimensional manifolds
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