论文标题
卢辛（Lusin）空间作为当地紧凑的波兰空间的图像
Lusin spaces as images of locally compact Polish spaces
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
A Lusin space is a Hausdorff space being the image of a Polish space under a continuous bijection. Such spaces have multiple applications, in particular, as state spaces of various stochastic systems. In this work, we consider the spaces obtained as the images of a noncompact and locally compact Polish space $(X, \mathcal{T})$, which we call $c$-Lusin. The main result is the statement that a $c$-Lusin space $Y=f(X)$, can be written as $Z\cup Y_1$, where $Z$ is a locally compact Polish space whereas $Y_1$ is $c$-Lusin. At the same time, $Y_1$ is the set of the discontinuity points of $f^{-1}$ which is a closed subset of $Y$. Moreover, $Y_1$ is nowhere dense if (and only if) $Y$ is a Baire space. By the same arguments, $Y_1$ can also be decomposed as $Z_1 \cup Y_2$ with the properties as above. In the case where $f$ can be extended to a continuous map $f:X\cup \{\infty\} \to Y$, and thus $Y_1$ is a singleton, we construct a metric on $X$ such that the corresponding metric space is compact and homeomorphic to the $c$-Lusin space $(f(X), \mathcal{T}')$.
