论文标题
部分可观测时空混沌系统的无模型预测
A New Feasibility Condition for the AT4 Family
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 $Γ$ be an antipodal distance-regular graph with diameter $4$ and eigenvalues $θ_0>θ_1>θ_2>θ_3>θ_4$. Then $Γ$ is tight in the sense of Jurišić, Koolen, and Terwilliger [12] whenever $Γ$ is locally strongly regular with nontrivial eigenvalues $p:=θ_2$ and $-q:=θ_3$. Assume that $Γ$ is tight. Then the intersection numbers of $Γ$ are expressed in terms of $p$, $q$, and $r$, where $r$ is the size of the antipodal classes of $Γ$. We denote $Γ$ by $\mathrm{AT4}(p,q,r)$ and call this an antipodal tight graph of diameter $4$ with parameters $p,q,r$. In this paper, we give a new feasibility condition for the $\mathrm{AT4}(p,q,r)$ family. We determine a necessary and sufficient condition for the second subconstituent of $\mathrm{AT4}(p,q,2)$ to be an antipodal tight graph. Using this condition, we prove that there does not exist $\mathrm{AT4}(q^3-2q,q,2)$ for $q\equiv3$ $(\mathrm{mod}~4)$. We discuss the $\mathrm{AT4}(p,q,r)$ graphs with $r=(p+q^3)(p+q)^{-1}$.
