论文标题
部分可观测时空混沌系统的无模型预测
A bijection between the sets of $(a,b,b^2)$-Generalized Motzkin paths avoiding $\mathbf{uvv}$-patterns and $\mathbf{uvu}$-patterns
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。
A generalized Motzkin path, called G-Motzkin path for short, of length $n$ is a lattice path from $(0, 0)$ to $(n, 0)$ in the first quadrant of the XOY-plane that consists of up steps $\mathbf{u}=(1, 1)$, down steps $\mathbf{d}=(1, -1)$, horizontal steps $\mathbf{h}=(1, 0)$ and vertical steps $\mathbf{v}=(0, -1)$. An $(a,b,c)$-G-Motzkin path is a weighted G-Motzkin path such that the $\mathbf{u}$-steps, $\mathbf{h}$-steps, $\mathbf{v}$-steps and $\mathbf{d}$-steps are weighted respectively by $1, a, b$ and $c$. Let $τ$ be a word on $\{\mathbf{u}, \mathbf{d}, \mathbf{v}, \mathbf{d}\}$, denoted by $\mathcal{G}_n^τ(a,b,c)$ the set of $τ$-avoiding $(a,b,c)$-G-Motzkin paths of length $n$ for a pattern $τ$. In this paper, we consider the $\mathbf{uvv}$-avoiding $(a,b,c)$-G-Motzkin paths and provide a direct bijection $σ$ between $\mathcal{G}_n^{\mathbf{uvv}}(a,b,b^2)$ and $\mathcal{G}_n^{\mathbf{uvu}}(a,b,b^2)$. Finally, the set of fixed points of $σ$ is also described and counted.
