论文标题
部分可观测时空混沌系统的无模型预测
Convergence of Oja's online principal component flow
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。
Online principal component analysis (PCA) has been an efficient tool in practice to reduce dimension. However, convergence properties of the corresponding ODE are still unknown, including global convergence, stable manifolds, and convergence rate. In this paper, we focus on the stochastic gradient ascent (SGA) method proposed by Oja. By regarding the corresponding ODE as a Landau-Lifshitz-Gilbert (LLG) equation on the Stiefel manifold, we proved global convergence of the ODE. Moreover, we developed a new technique to determine stable manifolds. This technique analyzes the rank of the initial datum. Using this technique, we derived the explicit expression of the stable manifolds. As a consequence, exponential convergence to stable equilibrium points was also proved. The success of this new technique should be attributed to the semi-decoupling property of the SGA method: iteration of previous components does not depend on that of later ones. As far as we know, our result is the first complete one on the convergence of an online PCA flow, providing global convergence, explicit characterization of stable manifolds, and closed formula of exponential convergence depending on the spectrum gap.
