论文标题
部分可观测时空混沌系统的无模型预测
On the centered co-circular central configurations for the n-body problem
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。
For the power-law potential $n$-body problem, we study a special kind of central configurations where all the masses lie on a circle and the center of mass coincides with the center of the circle. It is also called the centered co-circular central configuration. We get some symmetry results for such central configurations. We show that for positive numbers $α>0$ and integers $n\geq3$ satisfying $\frac{1}{n}\sum_{j=1}^{n-1}\csc^α\frac{jπ}{n}\leq1+\fracα{4}$, the regular $n$-gon with equal masses is the unique centered co-circular central configuration for the $n$-body problem with power-law potential $U_α$. It quickly follows that for the Newtonian $n$-body problem (in the case $α=1$) and $n\leq6$, the regular $n$-gon is the unique centered co-circular central configuration.
