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Register a new userChallenges in Dynamic Mode Decomposition
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Dynamic Mode Decomposition (DMD) is a powerful tool for extracting spatial and temporal patterns from multi-dimensional time series, and it has been used successfully in a wide range of fields, including fluid mechanics, robotics, and neuroscience. Two of the main challenges remaining in DMD research are noise sensitivity and issues related to Krylov space closure when modeling nonlinear systems. Here, we investigate the combination of noise and nonlinearity in a controlled setting, by studying a class of systems with linear latent dynamics which are observed via multinomial observables. Our numerical models include system and measurement noise. We explore the influences of dataset metrics, the spectrum of the latent dynamics, the normality of the system matrix, and the geometry of the dynamics. Our results show that even for these very mildly nonlinear conditions, DMD methods often fail to recover the spectrum and can have poor predictive ability. Our work is motivated by our experience modeling multilegged robot data, where we have encountered great difficulty in reconstructing time series for oscillatory systems with slow transients, which decay only slightly faster than a period.
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Koopman mode analysis has provided a framework for analysis of nonlinear phenomena across a plethora of fields. Its numerical implementation via Dynamic Mode Decomposition (DMD) has been extensively deployed and improved upon over the last decade. We address the problems of mean subtraction and DMD mode selection in the context of finite dimensional Koopman invariant subspaces. Preprocessing of data by subtraction of the temporal mean of a time series has been a point of contention in companion matrix-based DMD. This stems from the potential of said preprocessing to render DMD equivalent to temporal DFT. We prove that this equivalence is impossible when the order of the DMD-based representation of the dynamics exceeds the dimension of the system. Moreover, this parity of DMD and DFT is mostly indicative of an inadequacy of data, in the sense that the number of snapshots taken is not enough to represent the true dynamics of the system. We then vindicate the practice of pruning DMD eigenvalues based on the norm of the respective modes. Once a minimum number of time delays has been taken, DMD eigenvalues corresponding to DMD modes with low norm are shown to be spurious, and hence must be discarded. When dealing with mean-subtracted data, the above criterion for detecting synthetic eigenvalues can be applied after additional pre-processing. This takes the form of an eigenvalue constraint on Companion DMD, or yet another time delay.
Extended dynamic mode decomposition (EDMD) provides a class of algorithms to identify patterns and effective degrees of freedom in complex dynamical systems. We show that the modes identified by EDMD correspond to those of compact Perron-Frobenius and Koopman operators defined on suitable Hardy-Hilbert spaces when the method is applied to classes of analytic maps. Our findings elucidate the interpretation of the spectra obtained by EDMD for complex dynamical systems. We illustrate our results by numerical simulations for analytic maps.
The Dynamic-Mode Decomposition (DMD) is a well established data-driven method of finding temporally evolving linear-mode decompositions of nonlinear time series. Traditionally, this method presumes that all relevant dimensions are sampled through measurement. To address dynamical systems in which the data may be incomplete or represent only partial observation of a more complex system, we extend the DMD algorithm by including a Mori-Zwanzig Decomposition to derive memory kernels that capture the averaged dynamics of the unresolved variables as projected onto the resolved dimensions. From this, we then derive what we call the Memory-Dependent Dynamic Mode Decomposition (MDDMD). Through numerical examples, the MDDMD method is shown to produce reasonable approximations of the ensemble-averaged dynamics of the full system given a single time series measurement of the resolved variables.
A data-driven and equation-free approach is proposed and discussed to model ships maneuvers in waves, based on the dynamic mode decomposition (DMD). DMD is a dimensionality-reduction/reduced-order modeling method, which provides a linear finite-dimensional representation of a possibly nonlinear system dynamics by means of a set of modes with associated oscillation frequencies and decay/growth rates. DMD also allows for short-term future estimates of the systems state, which can be used for real-time prediction and control. Here, the objective of the DMD is the analysis and forecast of the trajectories/motions/forces of ships operating in waves, offering a complementary efficient method to equation-based system identification approaches. Results are presented for the course keeping of a free-running naval destroyer (5415M) in irregular stern-quartering waves and for the free-running KRISO Container Ship (KCS) performing a turning circle in regular waves. Results are overall promising and show how DMD is able to identify the most important modes and forecast the systems state with reasonable accuracy up to two wave encounter periods.
Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables that spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky PDE as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.
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