Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alexander Sikorski, Enric Ribera Borrell, Marcus Weber

TL;DR

This article reformulates 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.

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 $χ$-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 $χ$-function approximation. We demonstrate the usage of our proposed method in experiments increasing the approximation accuracy by several orders of magnitude.

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

TL;DR

This article reformulates 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.

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 -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 -function approximation. We demonstrate the usage of our proposed method in experiments increasing the approximation accuracy by several orders of magnitude.
Paper Structure (16 sections, 4 theorems, 52 equations, 6 figures, 1 algorithm)

This paper contains 16 sections, 4 theorems, 52 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. ISOKANN Theory
  3. The Koopman operator
  4. ISOKANN -- Computing the dominant eigenspace
  5. Illustrative example
  6. Choice of the transformation S
  7. 1D-ISOKANN with explicit PCCA+
  8. Restoring the eigenfunctions
  9. Computational procedure
  10. Optimal sampling of Koopman eigenfunctions and chi-functions
  11. Importance Sampling for random variables
  12. Optimal Importance Sampling for Diffusion Processes
  13. Optimal sampling of eigenfunctions of the Koopman operator
  14. Application to ISOKANN
  15. Example: Controlled 1D-ISOKANN for the double well
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\chi,\ \bar{S}$ and $S$ be chosen as above in eqn:chibareqn:shiftscaleeqn:fullS. For generic $\bar{\chi}_0:\mathbf{X} \rightarrow\mathbb{R}$, i.e. containing components of $v_1$ and $v_2$, the 1D-ISOKANN iteration converges to the $\chi$-function for some $\alpha,\beta \in \mathbb{R}$, which in turn solves the ISOKANN problem

Figures (6)

  • Figure 1: Potential function $U$ of the double well with two metastable regions separated by a potential barrier.
  • Figure 2: The two dominant eigenfunctions $v_1, v_2$ of the Koopman operator for the double well potential.
  • Figure 3: Both components of the two-dimensional $\chi$-function, which are linear combinations of the eigenfunctions (\ref{['fig:eigenfun']}). In this case the application of $K$, which in general is linear in $\chi$, corresponds to a shift-scale (see \ref{['sec:isokannI']})
  • Figure 4: Scatter plot depicting $v_1$ against $v_2$ and $\chi_1$ against $\chi_2$ on a uniform grid over $\mathbf{X}$. PCCA+ constructs the linear map $S$ mapping $v$ onto the unit-simplex $\chi$.
  • Figure 5: Training performance over $50$ power iterations / batches with 500 ADAM steps each. The blue line shows the training loss and the red line shows the standard deviation of the Monte-Carlo samples in the training data.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2
  • Corollary 2.1
  • proof