[论文] Bilevel optimization for data-driven learning of Koopman embeddings us...

论文概要

研究领域: ML 作者: Joel-Pascal Ntwali N'konzi, Feliks Nüske, Stefan Klus 发布时间: 2026-10-08 arXiv: 2610.12370

中文摘要

Koopman 算子理论为分析非线性动力系统提供了线性框架，已成为数据驱动建模的重要工具。核心挑战在于：EDMD 等方法计算的有限维近似需要预先指定字典。近期机器学习路线用人工神经网络自编码器从数据中学习字典，而核方法虽更具可解释性且便于理论分析，在该场景下却鲜受关注。我们提出 EDMD-kDL：一种直接从数据学习有限维 Koopman 嵌入的核方法，结合配点法与双层优化思想，同时学习核字典和对应的 Koopman 近似。在全球海表温度预测、直接从视频数据学习等一系列数值实验中，EDMD-kDL 在所有测试场景下均匹敌或超越 SOTA ANN 方法。与标准核方法不同，该方法的核矩阵大小取决于配点数而非训练集规模，因此在设计上即可扩展到大数据集。

原文摘要

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...

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