论文标题
部分可观测时空混沌系统的无模型预测
Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup
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。
In this paper, we obtain a full classification of reductive dual pairs in a, real or complex, Lie superalgebra $\mathfrak{spo}(E)$ and Lie supergroup $\textbf{SpO}(E)$. Moreover, by looking at the natural action of the orthosymplectic Lie supergroup $\textbf{SpO}(E)$ on the Weyl-Clifford algebra $\textbf{WC}(E)$, we prove that for a reductive dual pair $(\mathscr{G}\,, \mathscr{G}') = ((G\,, \mathfrak{g})\,, (G'\,, \mathfrak{g}'))$ in $\textbf{SpO}(E)$, the superalgebra $\textbf{WC}(E)^{\mathscr{G}}$ consisting of $\mathscr{G}$-invariant elements in $\textbf{WC}(E)$ is generated by the Lie superalgebra $\mathfrak{g}'$. We obtain a full classification of reductive dual pairs in the (real or complex) Lie superalgebra $\mathfrak{spo}(\mathrm E)$ and the Lie supergroup $\textbf{SpO}(\mathrm E)$. Using this classification we prove that for a reductive dual pair $(\mathscr{G}\,, \mathscr{G}') = ((\mathrm G\,, \mathfrak{g})\,, (\mathrm G'\,, \mathfrak{g}'))$ in $\textbf{SpO}(\mathrm E)$, the superalgebra $\textbf{WC}(\mathrm E)^{\mathscr{G}}$ consisting of $\mathscr{G}$-invariant elements in the Weyl-Clifford algebra $\textbf{WC}(\mathrm E)$, equipped with the natural action of the orthosymplectic Lie supergroup $\textbf{SpO}(\mathrm E)$, is generated by the Lie superalgebra $\mathfrak{g}'$. As an application, we prove that Howe duality holds for the dual pairs $({\textbf{SpO}}(2n|1)\,, {\textbf{OSp}}(2k|2l)) \subseteq {\textbf{SpO}}(\mathbb{C}^{2k|2l} \otimes \mathbb{C}^{2n|1})$.
