Universal Approximation Theorem Extended to Differentiable Maps on Infinite-Dimensional Manifolds
Researchers have generalized the universal approximation theorem for functional input neural networks to include approximation of derivatives, extending beyond the traditional formulation on compact sets. The work proves a weighted Nachbin theorem and establishes theoretical foundations for approximating differentiable maps in infinite-dimensional spaces. This theoretical advance has implications for neural network design and applications in mathematical finance and functional analysis.
A new mathematical paper extends the classical universal approximation theorem—a foundational result in neural network theory—to a more general setting involving differentiable maps on infinite-dimensional weighted manifolds. The authors prove that functional input neural networks (FNNs) can approximate not only the maps themselves but also their derivatives, going beyond previous results that typically applied only to compact sets. The work introduces a weighted Nachbin theorem as a key technical tool and demonstrates applications to non-anticipative functionals, including horizontal and vertical derivatives, as well as path space functionals. The research spans multiple mathematical disciplines including functional analysis, machine learning, probability theory, and mathematical finance, suggesting broad potential applicability across these fields.
What's missing
The paper does not discuss computational implications or practical algorithms for implementing these theoretical results in real neural networks. Additionally, the specific advantages over existing approximation methods for practitioners are not detailed in the abstract.
What different sources said
- arXiv cs.LGCenter
Weighted universal approximation of differentiable maps on infinite-dimensional manifolds
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