发表机构
Maxwell Institute for Mathematical Sciences, The University of Edinburgh; Heriot–Watt University; Max Planck Institute for Dynamics of Complex Technical Systems; LAAS-CNRS; Université de Toulouse; School of Mathematical & Computer Sciences, Heriot–Watt University（爱丁堡大学麦克斯韦数学研究所; 赫瑞瓦特大学; 马克斯·普朗克复杂技术系统动力学研究所; 法国国家科学研究中心自动控制与系统分析实验室; 图卢兹大学; 赫瑞瓦特大学数学与计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出EDMD-kDL方法，结合配置法与双层优化，通过核字典学习从数据中学习Koopman嵌入，在海温预测、视频学习等实验中性能优于或相当ANN方法，且可扩展至大数据集。
AI 中文摘要
Koopman算子理论为非线性动力系统分析提供了线性框架，已成为数据驱动建模的核心工具。然而，扩展动态模态分解（EDMD）等方法计算得到的有限维近似需要预先指定字典，这是一个核心挑战。近期机器学习方法通过从数据中学习字典来解决这一局限，主要采用人工神经网络（ANN）自编码器架构。尽管核方法具有更强的可解释性和理论分析的易处理性，但在该场景下却鲜受关注。我们提出了带核字典学习的扩展动态模态分解（EDMD-kDL），这是一种直接从数据中学习有限维Koopman嵌入的核方法。该方法结合了配置法和双层优化的思想，可同时学习核字典和对应的Koopman近似。我们在一系列数值实验中，包括全球海表温度预测和直接从视频数据学习，将EDMD-kDL与最先进的基于ANN的方法进行了评估。在所有测试场景中，EDMD-kDL的性能与基于ANN的方法相当或更优。此外，与标准核方法不同，所提方法设计上可扩展至大型数据集，因为所需核矩阵的大小取决于配置点的数量，而非训练数据集的规模。
英文摘要
Koopman operator theory provides a linear framework for analyzing nonlinear dynamical systems and has become a major tool for data-driven modeling. A central challenge, however, is that finite-dimensional approximations computed by methods such as extended dynamic mode decomposition (EDMD) require the dictionary to be specified a priori. Recent machine-learning approaches address this limitation by learning the dictionary from data, predominantly using artificial neural network (ANN) autoencoder architectures. Although kernel methods offer an alternative with greater interpretability and tractability for theoretical analysis, they have received little attention in this setting. We introduce extended dynamic mode decomposition with kernel-based dictionary learning (EDMD-kDL), a kernel-based method for learning finite-dimensional Koopman embeddings directly from data. The method combines ideas from collocation methods and bilevel optimization to simultaneously learn a kernel dictionary and the corresponding Koopman approximation. We evaluate EDMD-kDL against state-of-the-art ANN-based approaches on a range of numerical experiments, including global sea-surface-temperature forecasting and learning directly from video data. Across all tested settings, EDMD-kDL achieves performance comparable to or better than the ANN-based methods. Moreover, in contrast to standard kernel methods, the proposed approach is scalable to large datasets by design since the size of the required kernel matrices depends on the number of collocation points rather than the size of the training dataset.
