论文标题
部分可观测时空混沌系统的无模型预测
A quantitative formula for the imaginary part of a Weyl coefficient
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 investigate two-dimensional canonical systems $y'=zJHy$ on an interval, with positive semi-definite Hamiltonian $H$, such that limit circle case prevails at the left endpoint and limit point case at the right . Let $q_H$ be the Weyl coefficient of the system. We prove a formula that determines the imaginary part of $q_H$ along the imaginary axis up to multiplicative constants, which are independent of $H$. Using classical Abelian-Tauberian theorems, we deduce characterizations of spectral properties such as integrability of a given comparison function w.r.t. the spectral measure $μ_H$, and boundedness of the distribution function of $μ_H$ relative to a given comparison function. We study in depth Hamiltonians for which $\arg q_H(ir)$ approaches $0$ or $π$ (at least on a subsequence). We show that tangential behavior of $q_H(ir)$ imposes a substantial restriction on the growth of $|q_H(ir)|$. An example is provided where $\arg q_H(ir)$ heavily oscillates. Our results in this context are interesting also from a function theoretic point of view.
