Modern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non‐autonomous physical systems or non‐autonomous simulation processes are becoming more and more important. A representation of non‐autonomous Markov jump processes is presented as autonomous Markov chains on space‐time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, the so‐called augmented jump chain is derived. The augmented jump chain inherits the sparseness of the infinitesimal generator of the original process and therefore provides a useful tool for studying time‐dependent dynamics even in high dimensions. Furthermore, possible generalizations and applications to the computation of committor functions and coherent sets in the non‐autonomous setting are discussed. After deriving the theoretical foundations, the concepts with a proof‐of‐concept Galerkin discretization of the transfer operator of the augmented jump chain applied to simple examples are illustrated.

The dominant eigenfunctions of the Koopman operator characterize the metastabilities and slow-timescale dynamics of stochastic diffusion processes. In the context of molecular dynamics and Markov state modeling, they allow for a description of the location and frequencies of rare transitions, which are hard to obtain by direct simulation alone. In this article, we reformulate the eigenproblem in terms of the ISOKANN framework, an iterative algorithm that learns the eigenfunctions by alternating between short burst simulations and a mixture of machine learning and classical numerics, which naturally leads to a proof of convergence. We furthermore show how the intermediate iterates can be used to reduce the sampling variance by importance sampling and optimal control (enhanced sampling), as well as to select locations for further training (adaptive sampling). We demonstrate the usage of our proposed method in experiments, increasing the approximation accuracy by several orders of magnitude.

Capturing the macroscopic behavior of molecular dynamics with membership functions was published in Mathematical Optimization for Machine Learning on page 41.

For stochastic diffusion processes the dominant eigenfunctions of the corresponding Koopman operator contain important information about the slow-scale dynamics, that is, about the location and frequency of rare events. In this article, we reformulate the eigenproblem in terms of $\chi$-functions in the ISOKANN framework and discuss how optimal control and importance sampling allows for zero variance sampling of these functions. We provide a new formulation of the ISOKANN algorithm allowing for a proof of convergence and incorporate the optimal control result to obtain an adaptive iterative algorithm alternating between importance sampling and $\chi$-function approximation. We demonstrate the usage of our proposed method in experiments increasing the approximation accuracy by several orders of magnitude.