Weighted universal approximation of differentiable maps on infinite-dimensional manifolds
无限维流形上可微映射的加权通用逼近
Philipp Schmocker, Josef Teichmann
AI总结 通过加权Nachbin定理,将函数输入神经网络的通用逼近定理推广到可微映射,包括导数逼近,并应用于非预期泛函和路径空间泛函的逼近。
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- 77 pages, 3 figures
我们将函数输入神经网络(FNN)的通用逼近定理推广到可微映射,包括导数的逼近。FNN将输入从可能无限维的加权流形映射到实值隐藏层,在该层上应用非线性标量激活函数,然后通过一些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逼近定理(UAT),该定理超越了紧集上的通常表述,并且还包括导数的逼近。这导致了非预期泛函(包括水平和垂直导数)的逼近结果。作为进一步的应用,我们证明了签名的线性函数能够逼近路径空间泛函,包括它们的方向导数。
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.