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Learning Koopman eigenfunctions of stochastic diffusions with optimal importance sampling and ISOKANN

  • 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.

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Author:Alexander Sikorski, Enric Ribera BorrellORCiD, Marcus Weber
Document Type:Article
Parent Title (English):Journal of Mathematical Physics
First Page:013502
Date of first Publication:2024/01/12
ArXiv Id:http://arxiv.org/abs/2301.00065
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