论文标题
伪空间的完整性，闭合度和度量反射
Completeness, Closedness and Metric Reflections of Pseudometric 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。
It is well-known that a metric space $(X, d)$ is complete iff the set $X$ is closed in every metric superspace of $(X, d)$. For a given pseudometric space $(Y, ρ)$, we describe the maximal class $\mathbf{CEC}(Y, ρ)$ of superspaces of $(Y, ρ)$ such that $(Y, ρ)$ is complete if and only if $Y$ is closed in every $(Z, Δ) \in \mathbf{CEC}(Y, ρ)$. We also introduce the concept of pseudoisometric spaces and prove that spaces are pseudoisometric iff their metric reflections are isometric. The last result implies that a pseudometric space is complete if and only if this space is pseudoisometric to a complete pseudometric space.
