论文标题
部分可观测时空混沌系统的无模型预测
Koszul duality for simplicial restricted Lie algebras
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。
Let $\mathsf{s}_0\mathsf{Lie}^r$ be the category of $0$-reduced simplicial restricted Lie algebras over a fixed perfect field of positive characteristic $p$. We prove that there is a full subcategory $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_ξ)$ of the homotopy category $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ and an equivalence $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_ξ)\simeq\mathrm{Ho}(\mathsf{s}_1\mathsf{CoAlg}^{tr})$. Here $\mathsf{s}_1\mathsf{CoAlg}^{tr}$ is the category of $1$-reduced simplicial truncated coalgebras; informally, a coaugmented cocommutative coalgebra $C$ is truncated if $x^p=0$ for any $x$ from the augmentation ideal of the dual algebra $C^*$. Moreover, we provide a sufficient and necessary condition in terms of the homotopy groups $π_*(L_\bullet)$ for $L_\bullet \in \mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ to lie in the full subcategory $\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_ξ)$. As an application of the equivalence above, we construct and examine an analog of the unstable Adams spectral sequence of A. K. Bousfield and D. Kan in the category $\mathsf{s}\mathsf{Lie}^r$. We use this spectral sequence to recompute the homotopy groups of a free simplicial restricted Lie algebra.
