Tensor-based computation of the Koopman generator via operator logarithm
This work addresses the challenge of system identification for nonlinear dynamical systems, particularly in high-dimensional settings, offering a scalable solution that is incremental over existing operator-logarithm approaches.
The paper tackled the curse of dimensionality in operator-logarithm methods for identifying governing equations of nonlinear dynamical systems by proposing a data-driven method that computes the Koopman generator in a low-rank tensor train format, achieving accurate recovery of vector field coefficients and scalability to higher-dimensional systems as demonstrated on 4D and 10D examples.
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.