论文标题
Gardiner-Masur紧凑型边界点的最佳测量学
Optimal geodesics for boundary points of the Gardiner-Masur compactification
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。
The Gardiner-Masur compactification of Teichmüller space is homeomorphic to the horofunction compactification of the Teichmüller metric. Let $ξ$ and $η$ be a pair of boundary points in the Gardiner-Masur compactification that fill up the surface. We show that there is a unique Teichmüller geodesic which is optimal for the horofunctions corresponding to $ξ$ and $η$. In particular, when $ξ$ and $η$ are Busemann points that fill up the surface, the geodesic converges to $ξ$ in forward direction and to $η$ in backward direction. As an application, we show that if $\mathbf{G}_n$ is a sequence of Teichmüller geodesics passing through $X_n$ and $Y_n$ such that $X_n \to ξ$ and $Y_n \to η$, then $\mathbf{G}_n$ converges to a unique Teichmüller geodesic.
