论文标题
部分可观测时空混沌系统的无模型预测
Spinning switches on a wreath product
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。
We classify an algebraic phenomenon on certain families of wreath products that can be seen as coming from a family of puzzles about switches on the corners of a spinning table. Such puzzles have been written about and generalized since they were first popularized by Martin Gardner in 1979. In this paper, we provide perhaps the fullest generalization yet, modeling both the switches and the spinning table as arbitrary finite groups combined via a wreath product. We classify large families of wreath products depending on whether or not they correspond to a solvable puzzle, completely classifying the puzzle in the case when the switches behave like abelian groups, constructing winning strategies for all wreath product that are $p$-groups, and providing novel examples for other puzzles where the switches behave like nonabelian groups, including the puzzle consisting of two interchangeable copies of the monster group $M$. Lastly, we provide a number of open questions and conjectures, and provide other suggestions of how to generalize some of these ideas further.
