A Data-Driven Framework for Koopman Semigroup Estimation in Stochastic Dynamical Systems

TL;DR

This paper introduces SDMD, a data-driven method for approximating the Koopman semigroup in stochastic systems, incorporating sampling time for stability, using neural networks for basis selection, and providing theoretical guarantees.

Contribution

The paper presents SDMD, a novel framework that efficiently approximates the stochastic Koopman semigroup directly, integrating sampling time and neural networks, with proven convergence guarantees.

Findings

SDMD accurately approximates eigenvalues and eigenfunctions.

The method ensures numerical stability and efficiency.

Theoretical guarantees confirm convergence in large data limits.

Abstract

We present Stochastic Dynamic Mode Decomposition (SDMD), a novel data-driven framework for approximating the Koopman semigroup in stochastic dynamical systems. Unlike existing methods, SDMD explicitly incorporates sampling time into its approximation, ensuring numerical stability and precision. By directly approximating the Koopman semigroup instead of the generator, SDMD avoids computationally expensive matrix exponential computations, which offers a more efficient and practical pathway for analyzing stochastic dynamics. The framework further integrates neural networks to automate basis selection, which reduces the reliance on manual intervention while maintaining computational efficiency. Rigorous theoretical guarantees, including convergence in the large data limit, zero-limit of sampling time, and large dictionary size, establish the method's reliability. Numerical experiments on canonical stochastic systems validate SDMD's effectiveness in approximating eigenvalues and eigenfunctions of the stochastic Koopman operator.