证明了神经网络可逼近无限维流形上的可微映射及其导数。
Weighted universal approximation of differentiable maps on infinite-dimensional manifolds
- 基于加权Nachbin定理，扩展了函数输入神经网络的通用逼近能力。
- 首次实现对路径空间泛函及其方向导数的线性签名近似。
- 适用于金融、随机分析等领域中的非前瞻泛函建模。
我们将函数输入神经网络（FNN）的通用逼近定理推广至可微映射，包含对导数的逼近。FNN将输入从可能无限维的加权流形映射到实值隐层，应用非线性标量激活函数后，通过线性读出返回到巴拿赫空间。通过证明加权Nachbin定理，建立了可微映射的通用逼近定理，突破了传统紧集限制，并涵盖导数逼近。由此得到对非前瞻泛函（包括水平与垂直导数）的逼近结果。进一步表明，签名的线性函数可逼近路径空间泛函及其方向导数。
原文摘要 · Abstract (English)
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.
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