论文标题
部分可观测时空混沌系统的无模型预测
Equidistribution of Zeros of Random Polynomials and Random Polynomial mappings on $\mathbb{C}^m$
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 study equidistribution problem of zeros in relation to a sequence of $Z$-asymptotically Chebyshev polynomials on $\mathbb{C}^{m}$. We use certain results obtained in a very recent work of Bayraktar, Bloom and Levenberg and have an equidistribution result in a more general probabilistic setting than what the paper of Bayraktar, Bloom and Levenberg considers even though the basis polynomials they use are more general than $Z$-asymptotically Chebyshev polynomials. Our equidistribution result is based on the expected distribution and the variance estimate of random zero currents corresponding to the zero sets (zero divisors) of polynomials. This equidistribution result of general nature shows that equidistribution result turns out to be true without the random coefficients being i.i.d. (independent and identically distributed), which also means that there is no need to use any specific probability distribution function for these random coefficients. In the last section, unlike from the $1$-codimensional case, we study the basis of polynomials orthogonal with respect to the $L^{2}$-inner product defined by the weighted asymptotically Bernstein-Markov measures on a given locally regular compact set, and with a probability distribution studied well by Bayraktar including the (standard) Gaussian and the Fubini-Study probability distributions as special cases, we have an equidistribution result for codimensions bigger than $1$.
