We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parameterization of a low-dimensional transition manifold in a certain function space. In this article, we enhance this approach by embedding and learning this transition manifold in a reproducing kernel Hilbert space, exploiting the favorable properties of kernel embeddings. Under mild assumptions on the kernel, the manifold structure is shown to be preserved under the embedding, and distortion bounds can be derived. This leads to a more robust and more efficient algorithm compared to the previous parameterization approaches.

我们提出了一种新颖的基于核的机器学习算法，用于识别高维多尺度随机系统有效动力学的低维几何。最近，作者开发了一种数学框架，用于计算此类系统的最佳反应坐标，该框架基于学习特定功能空间中低维过渡歧管的参数化。在本文中，我们利用内核嵌入的有利特性，通过在可复制的内核希尔伯特空间中嵌入和学习此转换流形来增强这种方法。在对内核的温和假设下，流形结构显示为在嵌入下得以保留，并且可以导出变形边界。