论文标题
部分可观测时空混沌系统的无模型预测
The solution of Liouville's equation (1850, 1853) and its impact
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。
Liouville's 1853 paper, in which he derived in closed form the general local solution of equation $u_{z\bar z}=\exp(u)$, is one of the few papers from the 19th century that 21st century mathematicians routinely quote as motivation for their work. We try and understand the reasons for the enduring importance of this paper. Our conclusions are the following: 1. Mathematics is not cumulative, but lost material may be recovered through examination of ancient works. 2. An apparent paradox is that the degree of dissemination of ideas and concepts among mathematicians is not determined only by geographic or social affiliation (countries, cities, scientific schools, etc.). On the contrary, the existence of common scientific goals makes it possible to transfer and use these concepts. 3. Older papers, because they are not influenced by modern classifications, can help generate ideas that do not arise naturally in a modern framework. Thus, a historical perspective can help bring about new mathematics and indeed, is a necessary dimension of mathematical research.
