Extension Of Dynamic Mode Decomposition Research Articles

Extension of Dynamic Mode Decomposition for dynamic systems with incomplete information based on t-model of optimal prediction

The Dynamic Mode Decomposition has proved to be a very efficient technique to study dynamic data. This is entirely a data-driven approach that extracts all necessary information from data snapshots which are commonly supposed to be sampled from measurement. The application of this approach becomes problematic if the available data is incomplete because some dimensions of smaller scale either missing or unmeasured. Such setting occurs very often in modeling complex dynamical systems such as power grids, in particular with reduced-order modeling. To take into account the effect of unresolved variables, the optimal prediction approach based on the Mori-Zwanzig formalism can be applied to obtain the most expected prediction under existing uncertainties. This effectively leads to the development of a time-predictive model accounting for the impact of missing data. In the present paper we provide a detailed derivation of the considered method from the Liouville equation and finalize it with the optimization problem that defines the optimal transition operator corresponding to the observed data. In contrast to the existing approach, we consider a first-order approximation of the Mori-Zwanzig decomposition, state the corresponding optimization problem and solve it with the gradient-based optimization method. The gradient of the obtained objective function is computed precisely through the automatic differentiation technique. The numerical experiments illustrate that the considered approach gives practically the same dynamics as the exact Mori-Zwanzig decomposition, but is less computationally intensive.

Higher Order Dynamic Mode Decomposition

This paper deals with an extension of dynamic mode decomposition (DMD), which is appropriate to treat general periodic and quasi-periodic dynamics, and transients decaying to periodic and quasi-periodic attractors, including cases (not accessible to standard DMD) that show limited spatial complexity but a very large number of involved frequencies. The extension, labeled as higher order dynamic mode decomposition, uses time-lagged snapshots and can be seen as superimposed DMD in a sliding window. The new method is illustrated and clarified using some toy model dynamics, the Stuart--Landau equation, and the Lorenz system. In addition, the new method is applied to (and its robustness is tested in) some permanent and transient dynamics resulting from the complex Ginzburg--Landau equation (a paradigm of pattern forming systems), for which standard DMD is seen to only uncover trivial dynamics, and the thermal convection in a rotating spherical shell subject to a radial gravity field.

A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a "black box" integrator. We will show that this approach is, in effect, an extension of Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the "stochastic Koopman operator" [1]. Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data, and two that show potential applications of the Koopman eigenfunctions.