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

Abstract

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.

References in corpus (4)

• On the numerical approximation of the Perron-Frobenius and Koopman operator
• Importance sampling in path space for diffusion processes with slow-fast variables
• A Koopman framework for rare event simulation in stochastic differential equations
• Convexity of a stochastic control functional related to importance sampling of Itô diffusions

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