论文标题
基于内核的Koopman Generator和Schrödinger操作员
Kernel-based approximation of the Koopman generator and Schrödinger operator
论文作者
论文摘要
许多维度和模型还原技术都依赖于从数据中估算相关动力运算符的主要本征函数。重要示例包括Koopman操作员及其发电机，以及Schrödinger操作员。我们提出了一种基于内核的方法，用于在复制内核Hilbert空间中近似差异操作员，并通过解决辅助矩阵特征值问题来估算特征功能。所得算法应用于分子动力学和量子化学实例。此外，我们认为，在某些条件下，可以将Schrödinger操作员转变为对应于漂移扩散过程的Kolmogorov向后的操作员，反之亦然。这使我们能够将开发的方法应用于量子机械系统的高维随机微分方程分析。
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.
