Standard Dynamic Mode Decomposition Research Articles

ON HIGHER ORDER DYNAMIC MODE DECOMPOSITION

This paper introduces an alternative variant of Higher Order Dy namic Mode Decomposition (HODMD), which improves the standard approach from a computational point of view. In the new scheme, time delayed snapshots are used along with the special form of the Koopman operator. An algorithm is derived that allows the calculation of novel decomposition in a stable and efficient way. This method is suitable in cases where standard Dynamic Mode Decomposition (DMD) is not applicable. These are dynamics that show limited spatial complexity, and a very large number of included frequencies. We illustrate and explain the new method using some classical and sample dynamics.

Dynamic mode decomposition for Koopman spectral analysis of elementary cellular automata.

We apply dynamic mode decomposition (DMD) to elementary cellular automata (ECA). Three types of DMD methods are considered, and the reproducibility of the system dynamics and Koopman eigenvalues from observed time series is investigated. While standard DMD fails to reproduce the system dynamics and Koopman eigenvalues associated with a given periodic orbit in some cases, Hankel DMD with delay-embedded time series improves reproducibility. However, Hankel DMD can still fail to reproduce all the Koopman eigenvalues in specific cases. We propose an extended DMD method for ECA that uses nonlinearly transformed time series with discretized Walsh functions and show that it can completely reproduce the dynamics and Koopman eigenvalues. Linear-algebraic backgrounds for the reproducibility of the system dynamics and Koopman eigenvalues are also discussed.

Investigation of unsteady flow mechanisms and modal behavior in a compressor cascade

This paper utilizes experimentally validated Large Eddy Simulation method and the standard Dynamic Mode Decomposition (DMD) analysis technique to investigate the unsteady flow characteristics on a compressor cascade under two Reynolds number (Re) conditions (1.1 × 106 in ground state and 8.0 × 104 in high altitude state). The results demonstrate that at low Re, large-scale vortices are formed on the surface of compressor blades, exhibiting a stable rotating state and dominating the flow. Kelvin-Helmholtz instability serves as the determining factor, inducing chaotic and unstable behavior in the separation shear layer, leading to turbulence. Furthermore, DMD modal decomposition reveals a dominant mode at low Re, gradually amplifying and significantly contributing to the overall flow field, while showing strong correlations with other major modes. Conversely, at high Re, vortices on the blade surface become dispersed and smaller, resulting in intricate interactions between vortices of different scales. Harmonic instability, alongside Kelvin-Helmholtz instability, plays a significant role in accelerating boundary layer transition, heightening flow complexity and instability. Additionally, pronounced interaction and energy transfer between different modes occur at high Re, yielding a relatively balanced energy distribution. These findings have important implications for optimizing blade design and enhancing compressor performance by providing a deeper understanding of vortex dynamics and dynamic modal characteristics in the compressor blade flow field.

Sparsity Promoting Dynamic Mode Decomposition for Data-Driven Modeling of Wind Turbine Wake

High-fidelity numerical simulation is suitable for analyzing the complex unsteady flow field dynamics of wind turbines. For a better understanding of these flow characteristics, the dynamic mode decomposition method can be used to carry out a reduced-order model study on the wakefield of wind turbines based on large-eddy simulations (LES) numerical simulation. In this paper, we abstract material dynamic information from the wakefield of the wind turbine by applying the sparsity-promoting dynamic mode decomposition (SPDMD) method, and the decomposition results are contrasted with the standard dynamic mode decomposition (DMD) method. Indicated that both mode decomposition methods can abstract the dynamic characteristics of wake and reveal the development and variation law of wind turbine wake. However, the frequency and spatial structure of the selected modes are different. For the purpose of demonstrating the extraction impact of the DMD/SPDMD method on the wakefield of wind turbines, DMD/SPDMD reduced-order models are established respectively. The result indicated that the relatively limited number of SPDMD modes is adequate to validly rehabilitate the wakefield of the unabridged wind turbine while standard DMD methods prerequisite more decomposition modes. Therefore, compared with the standard DMD method, the SPDMD method has strong robustness in mode selection, eliminates the feature information that contributes weakly to the flow, and has a smaller performance loss in the reconstruction of the wakefield of the wind turbine. The consumption of computing resources is greatly reduced.

Application and comparison of dynamic mode decomposition methods in the tip leakage cavitation of a hydrofoil case

The cavitation of the tip leakage vortex (TLV) induced by tip leakage has always been a difficult problem faced by turbomachinery, and its flow structure is complex and diverse. How to accurately extract the main structures that affect the cavitating flow of the TLV from the two-phase flow field is a key problem. In this study, the main mode extraction and low order mode reconstruction accuracy of the cavitation flow field of TLV downstream of National Advisory Committee for Aeronautics (NACA)0009 hydrofoil by two dynamic mode decomposition (DMD) methods are compared. The research shows that the main modes extracted by the standard DMD method contain a large number of noise modes, while the sparsity-promoting DMD eliminates the noise modes, showing obvious advantages in the reconstruction accuracy of the velocity field. The characteristics of cavitation signals are analyzed, and the cavitation signals are divided into four categories, which explains the reason why DMD methods have low reconstruction accuracy in cavitation. This study provides a theoretical basis and strong guarantee for the extraction of mode decomposition characteristics of the two-phase flow field. This is of great significance for accelerating the prediction of multiphase flow fields based on intelligent flow pattern learning in the future. Meanwhile, it also provides a new method and road for the introduction of artificial intelligence technology in future scientific research.

Transonic Shock Buffet Flowfield Assessment Using Various Dynamic Mode Decomposition Techniques

Dynamic mode decomposition (DMD) is a powerful data-driven modal decomposition technique that extracts spatiotemporal coherent structures: a useful process in flow diagnostics and future state estimation of complex nonlinear flow phenomena. Transonic shock buffet is a complicated phenomenon, and modal decomposition techniques such as DMD provide significant insight into its complicated flow physics; but, often, flowfield data are corrupted because of various sources of noise due to the presence of outliers or the absence of critical data components. Therefore, noise corruption renders the modal decomposition inaccurate, and thereby not useful. In this paper, two sources of noise have been considered: simple white noise, and complex salt-and-pepper-type spurious noise. Various DMD techniques including standard DMD, forward–backward DMD, total-least-squares DMD, higher-order DMD, and robust DMD have been implemented. Their effectiveness and limitations in countering noise corruption have been investigated systematically. In the case of white noise corruption, forward–backward DMD, total-least-squares DMD, and higher-order DMD capture the buffet frequency and growth rate with sufficient accuracy, whereas the latter outperforms the other two when the noise variance level is above 5%. In the case of spurious noise, robust DMD handles noise corruption efficiently, with surprisingly high values of pixel corruption of up to 30%.

Analyzing and predicting non-equilibrium many-body dynamics via dynamic mode decomposition

Simulating the dynamics of a nonequilibrium quantum many-body system by computing the two-time Green's function associated with such a system is computationally challenging. However, we are often interested in the time-diagonal of such a Green's function or time-dependent physical observables that are functions of one time. In this paper, we discuss the possibility of using dynamic mode decomposition (DMD), a data-driven model order reduction technique, to characterize one-time observables associated with the nonequilibrium dynamics using snapshots computed within a small time window. The DMD method allows us to efficiently predict long time dynamics from a limited number of trajectory samples. We demonstrate the effectiveness of DMD on a model two-band system. We show that, in the equilibrium limit, the DMD analysis yields results that are consistent with those produced from a linear response analysis. In the nonequilibrium case, the extrapolated dynamics produced by DMD is more accurate than a special Fourier extrapolation scheme presented in this paper. We point out a potential pitfall of the standard DMD method caused by insufficient spatial/momentum resolution of the discretization scheme. We show how this problem can be overcome by using a variant of the DMD method known as higher order DMD.

ONLINE DYNAMIC MODE DECOMPOSITION: AN ALTERNATIVE APPROACH FOR LOW RANK DATASETS

In this study, we provide an alternative approach for computing the dynamic mode decomposition (DMD) in real-time for streaming datasets. It is a low-storage method that updates the DMD approx­imation of a given dynamic as new data becomes available. Unlike the standard online DMD method, which is applicable only to over­constrained and full-rank datasets, the new method is applicable for both overconstrained and underconstrained datasets. The method is equation-free in the sense that it does not require knowledge of the underlying governing equations and is entirely data-driven. Several numerical examples are presented to demonstrate the performance of the method.

On the use of dynamic mode decomposition for time-series forecasting of ships operating in waves

In order to guarantee the safety of payload, crew, and structures, ships must exhibit good seakeeping, maneuverability, and structural-response performance, also when they operate in adverse weather conditions. In this context, the availability of forecasting methods to be included within model-predictive control approaches may represent a decisive factor. Here, a data-driven and equation-free modeling approach for forecasting of trajectories, motions, and forces of ships in waves is presented, based on dynamic mode decomposition (DMD). DMD is a data-driven modeling method, which provides a linear finite-dimensional representation of a possibly nonlinear system dynamics by means of a set of modes with associated frequencies. Its use for ship operating in waves has been little discussed and a systematic analysis of its forecasting capabilities is still needed in this context. Here, a statistical analysis of DMD forecasting capabilities is presented for ships in waves, including standard and augmented DMD. The statistical assessment uses multiple time series, studying the effects of the number of input/output waves, time steps, time derivatives, along with the use of time-shifted copies of time series by the Hankel matrix. The assessment of the forecasting capabilities is based on four metrics: normalized root mean square error, Pearson correlation coefficient, average angle measure, and normalized average minimum/maximum absolute error. Two test cases are used for the assessment: the course keeping of a self-propelled 5415M in irregular stern-quartering waves and the turning-circle of a free-running self-propelled KRISO Container Ship in regular waves. Results are overall promising and show how state augmentation (using from four to eight input waves, up to two time derivatives, and four time-shifted copies) improves the DMD forecasting capabilities up to two wave encounter periods in the future. Furthermore, DMD provides a method to identify the most important modes, shedding some light onto the physics of the underlying system dynamics.

Ffowcs Williams – Hawkings analogy for near-field acoustic sources analysis

The paper expands the scope of applying the Ffowcs Williams – Hawkings integration method. We propose using the acoustic field generated from time-dependent data stored on the FW-H control surface as the same common field for computational acoustic beamforming and dynamic mode decomposition methods to analyze the aerodynamic noise sources. We exemplify that it leads to obtaining mutually consistent and complementary information for reliable prediction of acoustic sources characteristics in the process of inverting data produced by a CFD simulation. Moreover, as the results of applying computational acoustic beamforming and dynamic mode decomposition methods depend on many geometric and algorithmic inputs, the proposed approach makes it possible to use various sets of the latter for a comprehensive analysis of obtained inversions and to form the final answer by an averaging procedure. We illustrate this by taking advantage of fast generating the examined acoustic field snapshots in any required region by the FW-H integration method for the recently developed new inverse computational acoustic beamforming algorithm and the standard dynamic mode decomposition method when carrying out a sensitivity study of the predicted acoustic source. The capabilities of the developed approach are demonstrated on the data of CFD scale-resolving simulation of turbulent flow over the 30P30N high-lift configuration.

Experimental Study of Dynamical Airfoil and Aerodynamic Prediction

Dynamic stall is a critical limiting factor for airfoil aerodynamics and a challenging problem for active flow control. In this experimental study, dynamic stall was measured by high-frequency surface pressure tapes and pressure-sensitive paint (PSP). The influence of the oscillation frequency was examined. Dynamic mode decomposition (DMD) with time-delay embedding was proposed to predict the pressure field on the oscillating airfoil based on scattered pressure measurements. DMD with time-delay embedding was able to reconstruct and predict the dynamic stall based on scattered measurements with much higher accuracy than standard DMD. The reconstruction accuracy of this method increased with the number of delay steps, but this also prolonged the computation time. In summary, using the Koopman operator obtained by DMD with time-delay embedding, the future dynamic pressure on an oscillating airfoil can be accurately predicted. This method provides powerful support for active flow control of dynamic stall.

Large Eddy Simulation and Dynamic Mode Decomposition of Turbulent Mixing Layers

Turbulent mixing layers are canonical flow in nature and engineering, and deserve comprehensive studies under various conditions using different methods. In this paper, turbulent mixing layers are investigated using large eddy simulation and dynamic mode decomposition. The accuracy of the computations is verified and validated. Standard dynamic mode decomposition is utilized to flow decomposition, reconstruction and prediction. It was found that the dominant-mode selection criterion based on mode amplitude is more suitable for turbulent mixing layer flow compared with the other three criteria based on singular value, modal energy and integral modal amplitude, respectively. For the mixing layer with random disturbance, the standard dynamic mode decomposition method could accurately reconstruct and predict the region before instability happens, but is not qualified in the regions after that, which implies that improved dynamic mode decomposition methods need to be utilized or developed for the future dynamic mode decomposition of turbulent mixing layers.

Towards an Adaptive Dynamic Mode Decomposition

Dynamic Mode Decomposition (DMD) is a tool that creates an approximate model from spatio-temporal data. We have developed an architecture of this tool that will adapt to the data from a given problem by leveraging time delay coordinates, projections, and robust principal component analysis. Our scheme which we call Adaptive Dynamic Mode Decomposition (ADMD) can be used in its exact form or the user may even utilize parts of the scheme for generating a DMD model that is more accurate and reliable compared to the one given by standard DMD. ADMD is demonstrated on several datasets of varying complexities and its performance appears to be promising.

Flow prediction using dynamic mode decomposition with time-delay embedding based on local measurement

We develop a method for the prediction of flow fields based on local particle image velocimetry (PIV) measurement. High spatial resolution can be achieved by focusing PIV on local flow regions; however, it is difficult for standard dynamic mode decomposition (DMD) to predict the temporally resolved flow field based on limited information in sub-domain. In this regard, the local flow field is embedded using time-delay to augment the spatial dynamics. As such, both high temporally and spatially resolved flow fields can be faithfully obtained from local PIV measurement using the DMD method. Using fabricated patterns, we demonstrate that DMD with time-delay embedding can faithfully predict dynamic patterns over a long time interval, whereas the standard DMD can only match the ground truth briefly following initiation. Using PIV measurement of a wake flow and a highly dynamic sweeping jet flow, the DMD with time-delay embedding can increase the temporal resolution up to 100 times with a prediction error rate of approximately 8%. Compared with wake flow, where unsteady flow patterns are relatively weak, a sweeping jet flow demonstrates that the prediction performance is improved even more significantly using time-delay embedding compared with standard DMD when the flow is highly dynamic. For sweeping jet flow, the prediction error rate can drop from 56% using standard DMD to 8.3% by embedding a time-delay smaller than five steps, for a small cost of calculation time. In addition, the DMD with time-delay embedding shows robustness to small noise. For data with high noise whose signal-to-noise ratio is 15, the method has an error rate of less than 5%.

Dynamic wind turbine wake reconstruction: A Koopman-linear flow estimator

A challenging topic arising in dynamic wind turbine wake is modeling, especially the low-order approximation. The central problem is the fact that it has high-dimensional and nonlinear wake characteristics. In this paper, a Koopman-linear flow estimator is designed according to the Koopman operator theory. Different from the conventional flow reconstruction with the linear stochastic estimation method, a dynamic state-space model with physical states is constructed. The wake dynamics are approximated using a limited number of measurable physical parameters by the dynamic part; then, the full wake flow is reconstructed from the low-order states by the estimation part. The flow estimator is designed into three different forms following Extended Dynamic Mode Decomposition (EDMD) method. Each form has its unique advantages. Precisely, probe sensors are placed in the studied space and provide direct information of the wake, and a few in-directly physical parameters are also included. Nonlinear integer programming is further adopted using a heuristic optimization algorithm, by which the sensor configurations are optimized. Comparisons with the standard Dynamic Mode Decomposition (DMD)-based wake model are adopted in time domain and frequency domain to verify the effectiveness of the proposed flow estimators. The results show acceptable accuracy in typical modeling cases and maintain good estimation accuracy when the measurement noises are involved. Finally, the proposed Koopman-linear flow estimator is compared with related stochastic estimation methods, in which the connections of the proposed estimator with stochastic ones are also discussed.

Extended dynamic mode decomposition for inhomogeneous problems

Dynamic mode decomposition (DMD) is a powerful data-driven technique for construction of reduced-order models of complex dynamical systems. Multiple numerical tests have demonstrated the accuracy and efficiency of DMD, but mostly for systems described by homogeneous partial differential equations (PDEs) with homogeneous boundary conditions. We propose an extended dynamic mode decomposition (xDMD) approach to cope with the potential unknown sources/sinks in PDEs. Motivated by similar ideas in deep neural networks, we equip our xDMD with two new features. First, it has a bias term, which accounts for inhomogeneity of PDEs and/or boundary conditions. Second, instead of learning a flow map, xDMD learns the residual increment by subtracting the identity operator. Our theoretical error analysis demonstrates the improved accuracy of xDMD relative to standard DMD. Several numerical examples are presented to illustrate this result.

Bayesian Dynamic Mode Decomposition with Variational Matrix Factorization

Dynamic mode decomposition (DMD) and its extensions are data-driven methods that have substantially contributed to our understanding of dynamical systems. However, because DMD and most of its extensions are deterministic, it is difficult to treat probabilistic representations of parameters and predictions. In this work, we propose a novel formulation of a Bayesian DMD model. Our Bayesian DMD model is consistent with the procedure of standard DMD, which is to first determine the subspace of observations, and then compute the modes on that subspace. Variational matrix factorization makes it possible to realize a fully-Bayesian scheme of DMD. Moreover, we derive a Bayesian DMD model for incomplete data, which demonstrates the advantage of probabilistic modeling. Finally, both of nonlinear simulated and real-world datasets are used to illustrate the potential of the proposed method.

Dynamic mode decomposition analysis of rotating detonation waves

A rotating detonation combustor (RDC) is a novel approach to achieving pressure gain combustion. Due to the steady propagation of the detonation wave around the perimeter of the annular combustion chamber, the RDC dynamic behavior is well suited to analysis with reduced-order techniques. For flow fields with such coherent aspects, the dynamic mode decomposition (DMD) has been shown to capture well the dominant oscillatory features corresponding to stable limit-cycle or quasi-periodic behavior within its dynamic modes. Details regarding the application of the technique to RDC—such as the number of frames, the effect of subtracting the temporal mean from the processed dataset, the resulting dynamic mode shapes, and the reconstruction of the dynamics from a reduced set of dynamic modes—are analyzed and interpreted in this study. The DMD analysis is applied to two commonly observed operating conditions of rotating detonation combustion, viz., (1) a single spinning wave with weak counter-rotating waves and (2) a clapping operating mode with two counter-propagating waves at equal speed and strength. We show that care must be taken when applying DMD to RDC datasets due to the presence of standing waves (expressed as either counter-propagating azimuthal waves or longitudinal pulsations). Without accounting for these effects, the reduced-order reconstruction fails using the standard DMD approach. However, successful application of the DMD allows for the reconstruction and separation of specific wave modes, from which models of the stabilization and propagation of the primary and counter-rotating waves can be derived.

Numerical investigation of centrifugal compressor stall with compressed dynamic mode decomposition

Heightened interest in visualizing the flow structure under critical conditions for aero-engine led to further understanding of the underlying physical mechanism as well as to a better knowledge of the issues faced with their design and flow control. In this work, an unsteady full annular simulation of a centrifugal compressor coupled with experimental validations was presented. The diffuser stall was determined with various criterion principles and the wave-maker region under asymmetric flow field was investigated. Then, a dynamic mode decomposition (DMD) procedure concerning the whole domain and individual planes was applied to the numerical data so as to capture and visualize the dominating flow structure. With the purpose of improving calculation efficiency, a compressed DMD was adopted and validated against the standard DMD analysis results. Finally, the nonlinear development and spatial propagation between components of the coherent perturbations in diffuser and volute were reconstructed. The results indicate that the representative organized structure of flow instability can be captured with standard DMD approach. And by applying the compressed DMD approach, the flow mode can also be obtained with calculating time saving of 88.4% when the compression ratio is 1%. Considering the flow construction, four stall cells oscillating with little circumferential phase change are recovered during 6 rotor-cycles, indicating that the origin of the diffuser stall lies at the hub wall with fixed circumferential position near the diffuser outlet. The spatial propagation of stall mode except for the one near the volute tongue extends to the volute flow with a dumping effect of high-frequency perturbations.

Dynamic mode decomposition via dictionary learning for foreground modeling in videos

Accurate extraction of foregrounds in videos is one of the challenging problems in computer vision. In this study, we propose dynamic mode decomposition via dictionary learning (dl-DMD), which is applied to extract moving objects by separating the sequence of video frames into foreground and background information with a dictionary learned using block patches on the video frames. Dynamic mode decomposition (DMD) decomposes spatiotemporal data into spatial modes, each of whose temporal behavior is characterized by a single frequency and growth/decay rate and is applicable to split a video into foregrounds and the background when applying it to a video. And, in dl-DMD, DMD is applied on coefficient matrices estimated over a learned dictionary, which enables accurate estimation of dynamical information in videos. Due to this scheme, dl-DMD can analyze the dynamics of respective regions in a video based on estimated amplitudes and temporal evolution over patches. The results on synthetic data exhibit that dl-DMD outperforms the standard DMD and compressed DMD (cDMD) based methods. Also, the results of an empirical performance evaluation in the case of foreground extraction from videos using publicly available dataset demonstrates the effectiveness of the proposed dl-DMD algorithm and achieves a performance that is comparable to that of the state-of-the-art techniques in foreground extraction tasks.