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

TL;DR

This paper introduces a novel method combining importance sampling and neural networks to accurately compute Koopman eigenfunctions of stochastic diffusions, revealing slow-scale dynamics and rare events.

Contribution

It reformulates the eigenproblem using $hi$-functions within the ISOKANN framework, enabling convergence proofs and adaptive importance sampling for improved accuracy.

Findings

Significantly increased approximation accuracy

Effective zero-variance importance sampling for eigenfunctions

Demonstrated convergence and robustness of the method

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.