论文标题
部分可观测时空混沌系统的无模型预测
Une introduction aux périodes
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。
This survey article is the outgrowth of two talks given at the Journées X-UPS "Périodes et transcendance" at École polytechnique. Periods are complex numbers whose real and imaginary parts can be written as integrals of rational functions over domains defined by polynomial inequalities, everything with rational coefficients. According to a conjecture by Kontsevich-Zagier, every algebraic relation among these numbers should follow from the obvious rules of calculus: additivity, change of variables, Stokes's formula. I first explain the definition of periods and some ensuing elementary properties, by illustrating them with a host of examples. Then I gently move to the interpretation of these numbers as entries of the integration pairing between algebraic de Rham cohomology and singular homology of algebraic varieties defined over $\mathbf{Q}$, the point of view which is at the origin of all recent breakthroughs in their study.
