论文标题
部分可观测时空混沌系统的无模型预测
Closed $k$-Schur Katalan functions as $K$-homology Schubert representatives of the affine Grassmannian
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。
Recently, Blasiak-Morse-Seelinger introduced symmetric functions called Katalan functions, and proved that the $K$-theoretic $k$-Schur functions due to Lam-Schilling-Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called the closed $k$-Schur Katalan functions are identified with the Schubert structure sheaves in the $K$-homology of the affine Grassmannian. The main result is a proof of the conjecture. We also study a $K$-theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a non-geometric manner, based on the unipotent solution of the relativistic Toda lattice of Ruijsenaars. We prove that the map sends a Schubert class of the quantum $K$-theory ring of the flag variety to a closed $K$-$k$-Schur Katalan function up to an explicit factor related to a translation element with respect to an anti-dominant coroot. In fact, we prove the above map coincides with a map whose existence was conjectured by Lam, Li, Mihalcea, Shimozono, and proved by Kato, and more recently by Chow and Leung.
