论文标题
部分可观测时空混沌系统的无模型预测
On the Wasserstein median of probability measures
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。
The primary choice to summarize a finite collection of random objects is by using measures of central tendency, such as mean and median. In the field of optimal transport, the Wasserstein barycenter corresponds to the Fréchet or geometric mean of a set of probability measures, which is defined as a minimizer of the sum of squared distances to each element in a given set with respect to the Wasserstein distance of order 2. We introduce the Wasserstein median as a robust alternative to the Wasserstein barycenter. The Wasserstein median corresponds to the Fréchet median under the 2-Wasserstein metric. The existence and consistency of the Wasserstein median are first established, along with its robustness property. In addition, we present a general computational pipeline that employs any recognized algorithms for the Wasserstein barycenter in an iterative fashion and demonstrate its convergence. The utility of the Wasserstein median as a robust measure of central tendency is demonstrated using real and simulated data.
