发表机构
Catholic University of Eichstätt-Ingolstadt; Chemnitz University of Technology; Technische Universität Ilmenau（天主教艾希施泰特-因戈尔施塔特大学; 开姆尼茨工业大学; 伊尔梅瑙工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于傅里叶字典的扩展动态模式分解变体，在环面上实现闭式解与三角插值等价，导出最优阶误差界，并利用快速傅里叶变换实现无矩阵拟线性计算，在Kuramoto模型上验证至10^7模式。
AI 中文摘要
Koopman算子因其利用扩展动态模式分解（EDMD）等数据驱动方法为非线性系统提供线性视角的能力，在动力系统中引起了广泛关注。在本工作中，我们提出了一种在d维环面上使用傅里叶字典的EDMD变体，其中数据在等距张量网格上采样。在此设置下，EDMD回归问题具有唯一的闭式解，我们证明该解与Koopman像的三角插值一致。这一识别具有两个后果。首先，利用Sobolev空间的Koopman不变性，我们将三角插值的逼近理论结果迁移，以推导Koopman算子逼近的误差界。特别地，所建立的界具有最优阶且带有显式常数。其次，EDMD矩阵无需组装，因为其作用归结为（非等距）快速傅里叶变换，从而EDMD逼近可以以字典规模的拟线性成本进行无矩阵求值。我们以耦合振荡器的Kuramoto模型为例，使用多达10^7个模式的字典来展示结果。
英文摘要
The Koopman operator has gained considerable attention in dynamical systems due to its capability to provide a linear viewpoint for nonlinear systems using data-driven methods such as extended dynamic mode decomposition (EDMD). In this work, we suggest an EDMD-variant with a Fourier dictionary on the $d$-dimensional torus, where the data are sampled on an equispaced tensor grid. In this setting, the EDMD regression problem admits a unique closed-form solution, which we show to coincide with trigonometric interpolation of the Koopman image. This identification has two consequences. First, using Koopman invariance of Sobolev spaces, we transfer approximation-theoretic results for trigonometric interpolation to derive error bounds on approximations of the Koopman operator. In particular, the established bounds are of optimal order with explicit constants. Second, the EDMD matrix never has to be assembled, since its action reduces to (nonequispaced) fast Fourier transforms such that the EDMD-approximation may be evaluated matrix-free with quasi-linear cost in the dictionary size. We illustrate the results for the Kuramoto model of coupled oscillators with dictionaries of up to $10^7$ modes.
