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.

翻译：库普曼算子作为非线性动力系统的线性表示，已在多个科学领域引起广泛关注。近期，库普曼算子理论与数据科学中另一流行概念——再生核希尔伯特空间——相结合。我们沿着这一思路深入高斯过程方法，并阐明这些方法如何缓解基于核的库普曼算法的两个普遍性问题。首先是稀疏性问题：大多数核方法难以扩展规模，需要近似处理才能实际应用。我们证明不仅能够降低计算需求，还能展示其对传感器噪声的增强鲁棒性。第二个问题涉及超参数优化与字典学习，以使模型适应动力系统。总之，本工作的主要贡献在于将高斯过程回归与动态模态分解进行了统一。