论文标题
部分可观测时空混沌系统的无模型预测
Method for solving inverse spectral problems on quantum star graphs
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。
A new method for solving inverse spectral problems on quantum star graphs is proposed. The method is based on Neumann series of Bessel functions representations for solutions of Sturm-Liouville equations. The representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse spectral problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the representation is sufficient for the recovery of the potential. The method for solving the inverse spectral problem on the graph consists in reducing the problem to a two-spectra inverse Sturm-Liouville problem on each edge. Then a system of linear algebraic equations is derived for computing the first coefficient of the series representation for the solution on each edge and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests.
