论文标题
部分可观测时空混沌系统的无模型预测
Optimal periodic $L_2$-discrepancy and diaphony bounds for higher order digital sequences
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。
We present an explicit construction of infinite sequences of points $(\boldsymbol{x}_0,\boldsymbol{x}_1, \boldsymbol{x}_2, \ldots)$ in the $d$-dimensional unit-cube whose periodic $L_2$-discrepancy satisfies $$L_{2,N}^{\rm per}(\{\boldsymbol{x}_0,\boldsymbol{x}_1,\ldots, \boldsymbol{x}_{N-1}\}) \le C_d N^{-1} (\log N)^{d/2} \quad \mbox{for all } N \ge 2,$$ where the factor $C_d > 0$ depends only on the dimension $d$. The construction is based on higher order digital sequences as introduced by J. Dick in the year 2008. The result is best possible in the order of magnitude in $N$ according to a Roth-type lower bound shown first by P.D. Proinov. Since the periodic $L_2$-discrepancy is equivalent to P. Zinterhof's diaphony the result also applies to this alternative quantitative measure for the irregularity of distribution.
