Data-Driven Approximation of Transfer Operators: Naturally Structured Dynamic Mode Decomposition

Abstract

In this paper, we provide a new algorithm for the finite dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time series data. We argue that existing approach for the finite dimensional approximation of these transfer operators such as Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two important properties of these operators, namely positivity and Markov property. The algorithm we propose in this paper preserve these two properties. We call the proposed algorithm as naturally structured DMD since it retains the inherent properties of these operators. Naturally structured DMD algorithm leads to a better approximation of the steady-state dynamics of the system regarding computing Koopman and Perron-Frobenius operator eigenfunctions and eigenvalues. However preserving positivity properties is critical for capturing the real transient dynamics of the system. This positivity of the transfer operators and it's finite dimensional approximation also has an important implication on the application of the transfer operator methods for controller and estimator design for nonlinear systems from time series data.

Data-Driven Approximations of Dynamical Systems Operators for Control

“…Other data-driven methods to approximate the Perron-Frobenius operator include blind source separation, the variational approach for conformation dynamics (VAC) [129] and DMD-based variants exploiting the duality between Koopman and Perron-Frobenius operators, e.g. naturally structured DMD (NSDMD) [73] and constrained Ulam dynamic mode decomposition [66]. Data-driven control based on the Perron-Frobenius operator appears more challenging as it relies on good statistical estimates requiring large amounts of data.…”

On the application of the dynamic mode decomposition in problems of computational fluid dynamics

“…Эта линеаризация основана на том, что оператор Купмана (линейный) является сопряженным оператором Перрона-Фробениуса [15][16][17][18][19].…”

“…Оператор Купмана тесно связан (сопряжен) с оператором Перрона-Фробениуса (пропагатором обобщенного уравнения Лиувилля), описывающим линейную динамику плотности вероятности [14][15][16][17][18][19].…”

On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems

“…Towards this goal various data-driven methods are proposed for the finite dimensional approximation of these operators [2], [15]- [18] with Dynamic Mode Decomposition (DMD) and extended DMD being the popular ones. By exploiting the duality between Koopman and P-F operators the work in [19] provides novel naturally structured DMD algorithm for data-driven approximation of both Koopman and P-F operator that preserves positivity and Markov properties of these operators. Recent work has focused on the data-driven approximation of Koopman operator for random dynamical systems (RDS) [20], [21].…”