论文标题
部分可观测时空混沌系统的无模型预测
Sublacunary sequences that are strong sweeping out
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。
An increasing sequence $(a_n)$ of positive integers which satisfies $\frac{a_{n+1}}{a_n}>1+η$ for some positive $η$ is called a lacunary sequence. It has been known for over twenty years that every lacunary sequence is strong sweeping out which means that in every aperiodic dynamical system we can find a set $E$ of arbitrary small measure so that $\limsup_N\frac{1}{N} \sum_{n\le N}\mathbb{1}_E(T^nx)=1$ and $\liminf_N\frac{1}{N} \sum_{n\le N}\mathbb{1}_E(T^nx)=0$ almost everywhere. In this paper we improve this result by showing that if $(a_n)$ satisfies only $\frac{a_{n+1}}{a_n}>1+\frac1{(\log\log n)^{1-η}}$ for some positive $η$ then it is already strong sweeping out.
