[论文] Weighted universal approximation of differentiable maps on infinite-di...

论文概要

研究领域: ML
作者: Philipp Schmocker, Josef Teichmann
发布时间: 2025-06-06
arXiv: 2506.04839

中文摘要

我们将函数输入神经网络（FNN）的通用近似定理推广到可微映射，包括导数的近似。FNN将输入从可能无限维的加权流形映射到实值隐藏层，应用非线性标量激活函数，然后通过线性读出将输出返回到Banach空间。通过证明加权Nachbin定理，我们建立了可微映射的通用近似定理（UAT），超越了紧集上的通常表述，并包括导数的近似。这导出了非预期泛函的近似结果，包括水平和垂直导数。作为进一步应用，我们证明签名的线性函数能够近似路径空间泛函及其方向导数。

原文摘要

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 tha...

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