Leveraging the infinite dimensional neural network architecture we proposed
in arXiv:2109.13512v4 and which can process inputs from Fr\'echet spaces, and
using the universal approximation property shown therein, we now largely extend
the scope of this architecture by proving several universal approximation
theorems for a vast class of input and output spaces. More precisely, the input
space $\mathfrak X$ is allowed to be a general topological space satisfying
only a mild condition ("quasi-Polish"), and the output space can be either
another quasi-Polish space $\mathfrak Y$ or a topological vector space $E$.
Similarly to arXiv:2109.13512v4, we show furthermore that our neural network
architectures can be projected down to "finite dimensional" subspaces with any
desirable accuracy, thus obtaining approximating networks that are easy to
implement and allow for fast computation and fitting. The resulting neural
network architecture is therefore applicable for prediction tasks based on
functional data. To the best of our knowledge, this is the first result which
deals with such a wide class of input/output spaces and simultaneously
guarantees the numerical feasibility of the ensuing architectures. Finally, we
prove an obstruction result which indicates that the category of quasi-Polish
spaces is in a certain sense the correct category to work with if one aims at
constructing approximating architectures on infinite-dimensional spaces
$\mathfrak X$ which, at the same time, have sufficient expressive power to
approximate continuous functions on $\mathfrak X$, are specified by a finite
number of parameters only and are "stable" with respect to these parameters.

利用我们在 arXiv:2109.13512v4 提出的无限维神经网络架构，该架构可以处理来自 Frechet 空间的输入，并利用其中证明的通用逼近性质，我们现在通过证明一些普适逼近定理，大大扩展了该架构的适用范围，针对广泛类别的输入和输出空间。

非度量空间中的神经网络

Neural networks in non-metric spaces

Universal approximation theorems are the foundations of classical neural
networks, providing theoretical guarantees that the latter are able to
approximate maps of interest. Recent results have shown that this can also be
achieved in a quantum setting, whereby classical functions can be approximated
by parameterised quantum circuits. We provide here precise error bounds for
specific classes of functions and extend these results to the interesting new
setup of randomised quantum circuits, mimicking classical reservoir neural
networks. Our results show in particular that a quantum neural network with
$\mathcal{O}(\varepsilon^{-2})$ weights and $\mathcal{O} (\lceil
\log_2(\varepsilon^{-1}) \rceil)$ qubits suffices to achieve accuracy
$\varepsilon>0$ when approximating functions with integrable Fourier transform.

本文研究了经典神经网络的普适逼近定理在量子设置下的拓展，通过参数化量子电路近似传统函数，并提供精确的误差界，并将结果推广到模拟经典储备神经网络的随机量子电路。结果表明，一个具有 O (ε^-2) 个权重和 O (⌈log_2 (ε^-1)⌉) 个量子比特的量子神经网络可以在近似具有可积傅里叶变换的函数时达到精度 ε>0。

量子神经网络和量子储备库的通用逼近定理和误差界

Universal Approximation Theorem and error bounds for quantum neural  networks and quantum reservoirs

We propose new symplectic networks (SympNets) for identifying Hamiltonian
systems from data based on a composition of linear, activation and gradient
modules. In particular, we define two classes of SympNets: the LA-SympNets
composed of linear and activation modules, and the G-SympNets composed of
gradient modules. Correspondingly, we prove two new universal approximation
theorems that demonstrate that SympNets can approximate arbitrary symplectic
maps based on appropriate activation functions. We then perform several
experiments including the pendulum, double pendulum and three-body problems to
investigate the expressivity and the generalization ability of SympNets. The
simulation results show that even very small size SympNets can generalize well,
and are able to handle both separable and non-separable Hamiltonian systems
with data points resulting from short or long time steps. In all the test
cases, SympNets outperform the baseline models, and are much faster in training
and prediction. We also develop an extended version of SympNets to learn the
dynamics from irregularly sampled data. This extended version of SympNets can
be thought of as a universal model representing the solution to an arbitrary
Hamiltonian system.

提出了新的辛网络（SympNets）来识别基于线性，激活和梯度模块组合的数据的哈密顿系统。实验结果表明，SymNets 具有很好的泛化能力，并且比基线模型更快。