论文标题
随机多重操作员积分
Random Multiple Operator Integrals
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 introduction of Schur multipliers into the context of Double Operator Integrals (DOIs) was proposed by V. V. Peller in 1985. This work extends theorem on Schur multipliers from measurable functions to their closure space and generalizes the definition of DOIs to Multiple Operator Integrals (MOIs) for integrand functions as Schur multipliersconstructible by taking the limit of projective tensor product and by taking the limit of integral projective tensor product. According to such closure space construction for integrand functions, we demonstrate that any function defined on a compact set of a Euclidean space can be expressed by taking the limit of the projective tensor product of linear functions. We also generalize previous works about random DOIs with respect to finite dimensional operators, tensors, to MOIs with respect to random operators, which are defined from spectral decomposition perspectives. Based on random MOIs definitions and their properties, we derive several tail bounds for norms of higher random operator derivatives, higher random operator difference and Taylor remainder of random operator-valued functions.
