函数动态模式分解：从数据学习无限维系统

中文摘要 AI 辅助

动态模式分解（DMD）是一种数据驱动方法，它计算底层动力系统的最佳线性近似，并将动力学分解为特征时空模式的叠加。DMD最初由流体力学领域引入，其扩展方法已在分子动力学、气候科学、工程、金融和神经科学等许多其他研究领域得到广泛应用，应用包括降维、预测、系统辨识、控制和谱聚类。为了将DMD应用于偏微分方程，通常先用有限差分或有限元技术对空间域进行离散化，从而隐含地将问题转化为有限维问题。我们将投影DMD和精确DMD扩展到无限维系统。我们的DMD变体不是从向量值观测中估计矩阵，而是从可观测函数、密度或波函数等函数数据中学习有限秩算子。我们证明，传统DMD算法可以被视为其函数DMD对应方法的特例。所有结果将借助引导性示例加以说明。我们特别关注与图函数、常微分方程和随机微分方程相关的Koopman、Perron-Frobenius和Koopman-von Neumann算子。

英文摘要

Dynamic mode decomposition (DMD) is a data-driven method that computes the best linear approximation of the underlying dynamical system and decomposes the dynamics into a superposition of characteristic spatiotemporal patterns. Originally introduced by the fluid dynamics community, DMD and its extensions have found widespread use in many other research areas such as molecular dynamics, climate science, engineering, finance, and neuroscience. Applications include dimensionality reduction, forecasting, system identification, control, and spectral clustering. In order to apply DMD to partial differential equations, the spatial domain is typically first discretized using finite difference or finite element techniques, thus implicitly rendering the problem finite-dimensional. We extend projected and exact DMD to infinite-dimensional systems. Rather than estimating matrices from vector-valued observations, our DMD variants learn finite-rank operators from functional data such as observables, densities, or wavefunctions. We show that conventional DMD algorithms can be regarded as special cases of their functional DMD counterparts. All results will be illustrated with the aid of guiding examples. We focus in particular on Koopman, Perron-Frobenius, and Koopman-von Neumann operators associated with graphons, ordinary differential equations, and stochastic differential equations.

发表机构

• Heriot–Watt University（赫瑞-瓦特大学）
• Maxwell Institute for Mathematical Sciences（麦克斯韦数学科学研究所）
• University of Edinburgh（爱丁堡大学）

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