论文标题
部分可观测时空混沌系统的无模型预测
Existence of a Moore graph of degree 57 is still open
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 2020, a paper [arXiv:2010.13443] appeared in the arXiv claiming to prove that a Moore graph of diameter 2 and degree 57 does not exist. (The paper is in Russian; we include a link to a translation of this paper kindly provided to us by Konstantin Selivanov.) The proof technique is reasonable. It employs the fact that such a graph must be distance regular and that there exists a large set of relations which such a graph must satisfy. The argument proceeds by a case analysis that shows that this set of relations cannot be satisfied. We show that this seems not to be correct. The system of equations factors into small diagonal blocks all of which have solutions. As an alternative, we show that there is a family of systems of permutations with the property that the Moore graph exists if and only if there is a member of the family with no solutions.
