论文标题
部分可观测时空混沌系统的无模型预测
A Novel Regularity Criterion For The three-dimensional Navier-Stokes Equations Based On Finitely many observations
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 this paper we present two results: (1) A data assimilation algorithm for the 3D Navier-Stokes equation (3D NSE) using nodal data, and, as a consequence (2) a novel regularity criterion for the 3D NSE based on finitely many observations of the velocity. The data assimilation algorithm we employ utilizes nudging, a method based on a Newtonian relaxation scheme motivated by feedback-control. The observations, which may be either modal, nodal or volume elements, are drawn from a weak solution of the 3D NSE and are collected almost everywhere in time over a finite grid and our results, including the regularity criterion, hold for data of any of the aforementioned forms. The regularity criterion we propose follows from our data assimilation algorithm and is hence intimately connected to the notion of determining functionals (modes, nodes and volume elements). To the best of our knowledge, all existing regularity criteria require knowing the solution of the 3D NSE almost everywhere in space. Our regularity criterion is fundamentally different from any preexisting regularity criterion as it is based on finitely many observations (modes, nodes and volume elements). We further prove that the regularity criterion we propose is both a necessary and sufficient condition for regularity. Thus our result can be viewed as a natural generalization of the notion of determining modes, nodes and volume elements as well as the asymptotic tracking property of the nudging algorithm for the 2D NSE to the 3D setting.
