论文标题
部分可观测时空混沌系统的无模型预测
Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones
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 $Q$ be a bipartite quiver with vertex set $Q_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $β=(β(x))_{x \in Q_0}$ be a dimension vector of $Q$ with positive integer coordinates. Let $rep(Q, β)$ be the representation space of $β$-dimensional representations of $Q$ and $GL(β)$ the base change group acting on $rep(Q, β)$ be simultaneous conjugation. Let $K^β_{\underlineλ}$ be the multiplicity of the irreducible representation of $GL(β)$ of highest weight $\underlineλ$ in the ring of polynomial functions on $rep(Q, β)$. We show that $K^β_{\underlineλ}$ can be expressed as the number of lattice points of a polytope obtained by gluing together two Knutson-Tao hive polytopes. Furthermore, this polytopal description together with Derksen-Weyman's Saturation Theorem for quiver semi-invariants allows us to use Tardos' algorithm to solve the membership problem for the moment cone associated to $(Q,β)$ in strongly polynomial time.
