AI 中文总结
该研究利用arXiv:2109.13512v4的神经架构，证明了无限维拓扑空间上$L^p(\mu)$的全局通用逼近定理，且向量情形也获类似结果。
AI 中文摘要
利用我们在arXiv:2109.13512v4中引入的神经架构，我们在$L^p(\mu)$拓扑下证明了一个全局通用逼近定理，其中$1\le p<\infty$，$\mu$是合适的无限维拓扑空间$\mathfrak X$上的Radon概率测度。即，$L^p(\mu)$中的任意函数$f:\mathfrak X\to \mathbb R$都可通过合适的无限维架构以任意精度逼近，而这些架构又可仅由有限个参数指定的近乎经典神经网络逼近。本文还考虑了向量情形（其中$f=f(x)\in E$，$E$为Banach空间），并得到了类似结果。
英文摘要
Leveraging the neural architectures which we introduced in arXiv:2109.13512v4, we show a global universal approximation theorem in the topology of $L^p(μ)$, where $1\le p<\infty$ and $μ$ is a Radon probability measure on a suitable infinite dimensional topological space $\mathfrak X$. Namely, any function $f:\mathfrak X\to \mathbb R$ in $L^p(μ)$ can be approximated to any degree of accuracy by suitable infinite dimensional architectures. These architectures can be in turn approximated by almost classical neural networks which are specified by a finite number of parameters only. The vectorial case (where $f=f(x)\in E$ and $E$ is a Banach space) is also considered and analogous results are obtained.
