论文标题
部分可观测时空混沌系统的无模型预测
Approximate Carathéodory bounds via Discrepancy Theory
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 approximate Carathéodory problem in general form is as follows: Given two symmetric convex bodies $P,Q \subseteq \mathbb{R}^m$, a parameter $k \in \mathbb{N}$ and $\mathbf{z} \in \textrm{conv}(X)$ with $X \subseteq P$, find $\mathbf{v}_1,\ldots,\mathbf{v}_k \in X$ so that $\|\mathbf{z} - \frac{1}{k}\sum_{i=1}^k \mathbf{v}_i\|_Q$ is minimized. Maurey showed that if both $P$ and $Q$ coincide with the $\| \cdot \|_p$-ball, then an error of $O(\sqrt{p/k})$ is possible. We prove a reduction to the vector balancing constant from discrepancy theory which for most cases can provide tight bounds for general $P$ and $Q$. For the case where $P$ and $Q$ are both $\| \cdot \|_p$-balls we prove an upper bound of $\sqrt{ \frac{\min\{ p, \log (\frac{2m}{k}) \}}{k}}$. Interestingly, this bound cannot be obtained taking independent random samples; instead we use the Lovett-Meka random walk. We also prove an extension to the more general case where $P$ and $Q$ are $\|\cdot \|_p$ and $\| \cdot \|_q$-balls with $2 \leq p \leq q \leq \infty$.
