论文标题
部分可观测时空混沌系统的无模型预测
Distribution of similar configurations in subsets of $\mathbb{F}_q^d$
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。
Let $\mathbb{F}_q$ be a finite field of order $q$ and $E$ be a set in $\mathbb{F}_q^d$. The distance set of $E$ is defined by $Δ(E):=\{\lVert x-y \rVert :x,y\in E\}$, where $\lVert α\rVert=α_1^2+\dots+α_d^2$. Iosevich, Koh and Parshall (2018) proved that if $d\geq 2$ is even and $|E|\geq 9q^{d/2}$, then $$\mathbb{F}_q= \frac{Δ(E)}{Δ(E)}=\left\{\frac{a}{b}: a\in Δ(E),\ b\in Δ(E)\setminus\{0\} \right\}.$$ In other words, for each $r\in \mathbb{F}_q^*$ there exist $(x,y)\in E^2$ and $(x',y')\in E^2$ such that $\lVert x-y\rVert\neq0$ and $\lVert x'-y' \rVert=r\lVert x-y\rVert$. Geometrically, this means that if the size of $E$ is large, then for any given $r \in \mathbb{F}_q^*$ we can find a pair of edges in the complete graph $K_{|E|}$ with vertex set $E$ such that one of them is dilated by $r\in \mathbb{F}_q^*$ with respect to the other. A natural question arises whether it is possible to generalize this result to arbitrary subgraphs of $K_{|E|}$ with vertex set $E$ and this is the goal of this paper. In this paper, we solve this problem for $k$-paths $(k\geq 2)$, simplexes and 4-cycles. We are using a mix of tools from different areas such as enumerative combinatorics, group actions and Turán type theorems.
