Klus Stefan, Nüske Feliks, Hamzi Boumediene
Department of Mathematics and Computer Science, Freie Universität Berlin, 14195 Berlin, Germany.
Department of Mathematics, Paderborn University, 33098 Paderborn, Germany.
Entropy (Basel). 2020 Jun 30;22(7):722. doi: 10.3390/e22070722.
Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.
许多降维与模型简化技术依赖于从数据中估计相关动力学算子的主导特征函数。重要的例子包括柯普曼算子及其生成器，还有薛定谔算子。我们提出一种基于核的方法来逼近再生核希尔伯特空间中的微分算子，并展示如何通过求解辅助矩阵特征值问题来估计特征函数。所得算法应用于分子动力学和量子化学示例。此外，我们发现，在某些条件下，薛定谔算子可转化为对应于漂移扩散过程的柯尔莫哥洛夫向后算子，反之亦然。这使我们能够将为分析高维随机微分方程而开发的方法应用于量子力学系统。
