Transfer operator methods for Markov processes in high dimensions

Abstract

This dissertation is dedicated to developing new approaches for the discretization and analysis of high-dimensional Markov processes, particularly in molecular dynamics, with a focus on their interactions. The four included articles present methods for addressing associated challenges, such as efficient representation of transfer operators and understanding long-term dynamics. The first two articles focus on representing and understanding the interactions of high- dimensional jump processes through their generators. The first article introduces the Augmented Jump Chain, a method to transform time-dependent Markov processes into time-independent Markov chains. The process description through individual space-time jumps allows for a more efficient numerical treatment of time-dependent processes. The second article develops the Ten- sor Square-Root Approximation, a tensor representation of the generator of diffusion processes that can be explicitly derived from the potential and potentially enables efficient calculations by reducing to low-rank tensors. The final two articles offer new approaches for representing the invariant subspaces of stochas- tic processes using the so-called χ functions, which capture the slowest timescales of the process. These are learned through neural networks with ISOKANN, a method that combines traditional numerical methods and machine learning. The third article focuses on the methodological de- velopment of ISOKANN and the combination of optimal control and adaptive sampling, which allow data to be generated efficiently and iteratively improve the neural network training. The fourth article presents a fundamental interpretation of the χ functions as macro-states, which define a temporal structure for macroscopic transitions, thereby offering a new way to extract representative transition paths. Overall, this dissertation provides new theoretical insights, as well as computation- and data- driven methods, for the analysis of high-dimensional Markov processes, opening perspectives for future applications in molecular dynamics.