论文标题
部分可观测时空混沌系统的无模型预测
Spectral Theory of Self-adjoint Finitely Cyclic Operators and Introduction to Matrix Measure $L^2$-spaces
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 study finitely cyclic self-adjoint operators in a Hilbert space, i.e. self-adjoint operators that posses such a finite subset in the domain that the orbits of all its elements with respect to the operator are linearly dense in the space. One of the main goals here is to obtain the representation theorem for such operators in a form analogous to the one well-known in the cyclic self-adjoint operators case. To do this, we present here a detailed introduction to matrix measures, to the matrix measure $L^2$ spaces, and to the multiplication by scalar functions operators in such spaces. This allows us to formulate and prove in all the details the less known representation result, saying that the finitely cyclic self-adjoint operator is unitary equivalent to the multiplication by the identity function on $\mathbb{R}$ in the appropriate matrix measure $L^2$ space. We study also some detailed spectral problems for finitely cyclic self-adjoint operators, like the absolute continuity.
