The ISOKANN (Invariant Subspaces of Koopman Operators Learned by Artificial Neural Networks) framework provides a data-driven route to extract collective variables (CVs) and effective dynamics from complex molecular systems. In this work, we integrate the theoretical foundation of Koopman operators with Krylov-like subspace algorithms, and reduced dynamical modeling to build a coherent picture of how to describe metastable transitions in high-dimensional systems based on CVs. Starting from the identification of CVs based on dominant invariant subspaces, we derive the corresponding effective dynamics on the latent space and connect these to transition rates and times, committor functions, and transition pathways. The combination of Koopman-based learning and reduced-dimensional effective dynamics yields a principled framework for computing transition rates and pathways from simulation data. Numerical experiments on one-, two-, and three-dimensional benchmark potentials illustrate the ability of ISOKANN to reconstruct the coarse-grained kinetics and reproduce transition times across enthalpic and entropic barriers.

翻译：ISOKANN（通过人工神经网络学习的Koopman算子不变子空间）框架提供了一种从复杂分子系统中提取集体变量（CVs）和有效动力学的数据驱动途径。本研究将Koopman算子的理论基础与Krylov-like子空间算法及降维动力学建模相结合，构建了一个基于CVs描述高维系统中亚稳态转变的连贯图像。从基于主导不变子空间的CVs识别出发，我们推导了潜空间上的相应有效动力学，并将其与转变速率和时间、漫射函数以及转变路径相关联。基于Koopman的学习与降维有效动力学的结合，为从模拟数据计算转变速率和路径提供了原则性框架。在一维、二维和三维基准势能上的数值实验展示了ISOKANN重建粗粒化动力学并再现跨焓势垒和熵势垒转变时间的能力。