Tensor-based computation of the Koopman generator via operator logarithm

TL;DR

This paper introduces a tensor-based method to compute the Koopman generator from data using operator logarithms, improving scalability and accuracy for high-dimensional nonlinear systems.

Contribution

It proposes a novel low-rank tensor train approach that computes the Koopman generator via operator logarithms, addressing dimensionality challenges.

Findings

Accurately recovers vector field coefficients in 4D and 10D systems.

Preserves tensor train format while computing the Koopman generator.

Demonstrates scalability to higher-dimensional systems.

Abstract

Identifying governing equations of nonlinear dynamical systems from data is challenging. While sparse identification of nonlinear dynamics (SINDy) and its extensions are widely used for system identification, operator-logarithm approaches use the logarithm to avoid time differentiation, enabling larger sampling intervals. However, they still suffer from the curse of dimensionality. Then, we propose a data-driven method to compute the Koopman generator in a low-rank tensor train (TT) format by taking logarithms of Koopman eigenvalues while preserving the TT format. Experiments on 4-dimensional Lotka-Volterra and 10-dimensional Lorenz-96 systems show accurate recovery of vector field coefficients and scalability to higher-dimensional systems.