论文标题
部分可观测时空混沌系统的无模型预测
Finite range decompositions of Gaussian fields with applications to level-set percolation
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。
In a recent work (arXiv:2206.10724), Muirhead has studied level-set percolation of (discrete or continuous) Gaussian fields, and has shown sharpness of the associated phase transition under the assumption that the field has a certain multiscale white noise decomposition, a variant of a finite-range decomposition. We show that a large class of Gaussian fields have such a white noise decomposition with optimal decay parameter. Examples include the discrete Gaussian free field, the discrete membrane model, and the mollified continuous Gaussian free field. This answers various questions from Muirhead's paper. Our construction of the white-noise decomposition is a refinement of Bauerschmidt's construction of finite-range decomposition. In the continuous setting our construction is very similar to Bauerschmidt's, while in the discrete setting several new ideas are needed, including the use of a result by Pólya and Szegő on polynomials that take positive values on the positive real line.
