论文标题
部分可观测时空混沌系统的无模型预测
Absolutely continuous and BV-curves in 1-Wasserstein spaces
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 extend the result of Lisini (Calc Var Partial Differ Equ 28:85-120, 2007) on the superposition principle for absolutely continuous curves in $p$-Wasserstein spaces to the special case of $p=1$. In contrast to the case of $p>1$, it is not always possible to have lifts on absolutely continuous curves. Therefore, one needs to relax the notion of a lift by considering curves of bounded variation, or shortly BV-curves, and replace the metric speed by the total variation measure. We prove that any BV-curve in a 1-Wasserstein space can be represented by a probability measure on the space of BV-curves which encodes the total variation measure of the Wasserstein curve. In particular, when the curve is absolutely continuous, the result gives a lift concentrated on BV-curves which also characterizes the metric speed. The main theorem is then applied for the characterization of geodesics and the study of the continuity equation in a discrete setting.
