论文标题
部分可观测时空混沌系统的无模型预测
Approaching the Soundness Barrier: A Near Optimal Analysis of the Cube versus Cube Test
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 Cube versus Cube test is a variant of the well-known Plane versus Plane test of Raz and Safra, in which to each $3$-dimensional affine subspace $C$ of $\mathbb{F}_q^n$, a polynomial of degree at most $d$, $T(C)$, is assigned in a somewhat locally consistent manner: taking two cubes $C_1, C_2$ that intersect in a plane uniformly at random, the probability that $T(C_1)$ and $T(C_2)$ agree on $C_1\cap C_2$ is at least some $ε$. An element of interest is the soundness threshold of this test, i.e. the smallest value of $ε$, such that this amount of local consistency implies a global structure; namely, that there is a global degree $d$ function $g$ such that $g|_{C} \equiv T(C)$ for at least $Ω(ε)$ fraction of the cubes. We show that the cube versus cube low degree test has soundness ${\sf poly}(d)/q$. This result achieves the optimal dependence on $q$ for soundness in low degree testing and improves upon previous soundness results of ${\sf poly}(d)/q^{1/2}$ due to Bhangale, Dinur and Navon.
