用高斯过程统一动态模式分解，提升非线性系统建模的效率与抗噪能力
Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces
- 将高斯过程与动力模式分解结合，实现核方法在动力系统中的高效建模
- 计算复杂度显著降低，对传感器噪声更具鲁棒性，保持精度
- 适合从事数据驱动动力系统分析的研究者，尤其关注可扩展性与稳定性
Koopman算子作为非线性动力系统的线性表征，近年来在多个科学领域受到关注。本文将Koopman理论与数据科学中流行的再生核希尔伯特空间（Reproducing Kernel Hilbert Space, RKHS）相结合，引入高斯过程方法，有效缓解了基于核的Koopman算法面临的两大难题：一是稀疏性问题——多数核方法难以扩展，需近似以实用；本文证明不仅可降低计算开销，还提升了对传感器噪声的鲁棒性。二是超参数优化与字典学习问题，用于适应具体动力系统。本工作的核心贡献在于统一高斯过程回归与动态模式分解（Dynamic Mode Decomposition, DMD），为数据驱动的动力系统建模提供更高效、稳定的框架。
原文摘要 · Abstract (English)
The Koopman operator, as a linear representation of a nonlinear dynamical system, has been attracting attention in many fields of science. Recently, Koopman operator theory has been combined with another concept that is popular in data science: reproducing kernel Hilbert spaces. We follow this thread into Gaussian process methods, and illustrate how these methods can alleviate two pervasive problems with kernel-based Koopman algorithms. The first being sparsity: most kernel methods do not scale well and require an approximation to become practical. We show that not only can the computational demands be reduced, but also demonstrate improved resilience against sensor noise. The second problem involves hyperparameter optimization and dictionary learning to adapt the model to the dynamical system. In summary, the main contribution of this work is the unification of Gaussian process regression and dynamic mode decomposition.
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