Extended-Kalman-filter-based dynamic mode decomposition for simultaneous system identification and denoising

The dynamic mode decomposition (DMD) method improved by embedded dimension and the proper orthogonal decomposition (POD) method are used to analyze the random wind pressure field of a large-span flat roof. The results show that the fuzzy dynamic features hidden in the wind tunnel test data can be discovered by using the Takens embedding theorem to increase the spatial dimension of the original snapshot matrix. The modal distributions of POD and DMD can reflect the main characteristics of typical vortices in a fluctuating pressure field. However, each mode of POD contains the information of multiple frequencies, which to a certain extent makes the mode of POD become the fluctuating coupling of multiple frequency bands. The mode decomposed by DMD method is a single-frequency mode and the stability of each mode can be obtained. At the same proportion, POD modes contain more energy than DMD modes. But when the same proportion of modes are used to reconstruct the fluctuating wind pressure field, the reconstruction results of DMD method can better describe and fit the local characteristics of the original fluctuating pressure field than those of POD method. This is because DMD method reconstructs the pressure field directly, while POD method mainly reconstructs the energy field. Therefore, the DMD method has more advantages in revealing the spatial-temporal evolution characteristics of a random wind pressure field.

Abstract. We introduce the optimized dynamic mode decomposition (DMD) algorithm for constructing an adaptive and computationally efficient reduced-order model and forecasting tool for global atmospheric chemistry dynamics. By exploiting a low-dimensional set of global spatio-temporal modes, interpretable characterizations of the underlying spatial and temporal scales can be computed. Forecasting is also achieved with a linear model that uses a linear superposition of the dominant spatio-temporal features. The DMD method is demonstrated on 3 months of global chemistry dynamics data, showing its significant performance in terms of computational speed and interpretability. We show that the presented decomposition method successfully extracts and forecasts chemical patterns for leading chemical indicators, including nitric oxide, ozone, nitrogen dioxide, hydroxyl radical, isoprene, and carbon monoxide. Moreover, the DMD algorithm allows for rapid reconstruction of the underlying linear model, which can then easily accommodate non-stationary data and changes in the dynamics.

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.

Understanding sloshing modes in cylindrical tanks with internal components is essential for risk mitigation. Moreover, the complex geometries make theoretical analysis difficult. While dynamic mode decomposition (DMD) has been widely adopted in fluid dynamics, it is meaningful to study its application in sloshing. Unlike conventional flow modes, DMD may face unique challenges due to the discontinuous free surface dynamics. To address this issue, this study selected a container with cylindrical components as the research object. The DMD method was employed to extract sloshing modes from numerical simulations using the Volume of Fluid method. The DMD accuracy is validated through comparison with simulation results. The Froude number (Fr) is used to assess DMD's robustness across different sloshing regimes. Results show that DMD method can effectively capture the sloshing modes. In linear sloshing, the overall modes are governed by the liquid-wall impact. When excited at the first order natural frequency, a nonlinear wave collision pattern is found and generates acceptable errors. Under the second order natural frequency excitation, the errors are sharply increased. Additionally, the relationship between DMD and Fr is independent of the tank filling rate. When Fr < 0.198, the DMD calculation result is reliable. For 0.198 ≤ Fr ≤ 0.368, DMD maintains high reconstruction accuracy, but the prediction error increases by 134.6%. When Fr > 0.368, both prediction and reconstruction errors surge by 128.3% and 127.5%. This work proposes a novel and effective framework in tank's sloshing analysis using DMD. The results provide valuable guidance for tank design, especially with complex geometries and internal components.

Coherent structures decomposition of the flow field in Francis turbine runner under different working conditions

Improvement of mode selection criterion of dynamic mode decomposition in a hydrofoil cavitation multiphase flow case

The modes in the electrocardiogram (ECG) signal can be divided into stable modes and unstable modes. The unstable modes are of great significance for signal analysis and classification. When the traditional dynamic mode decomposition (DMD) method is directly applied to these signals, it is difficult to effectively extract these key modes due to the rank mismatch problem of the data matrix. In order to better capture and analyze unstable modes, this study introduced the Hankel matrix to expand ECG data and used it as the input of DMD to propose the Hankel dynamic mode decomposition (HDMD) method. Although HDMD has been applied in other physiological signal processing, this study is the first to successfully apply it to multi-lead ECG signal analysis. By optimizing the delay parameters of the Hankel matrix and retaining the modes that account for 90% of the singular value decomposition energy, we significantly improve the effectiveness of feature extraction. Each type of abnormal ECG signal data is classified on the PTB Diagnostic ECG database. The experimental results of the proposed model are compared with the direct use of DMD for modal extraction. The highest classification accuracy of multi-modal ECG signal data extracted by HDMD is more than 10% higher than that using DMD. At the same time, the mean squared error (MSE) of the reconstruction using HDMD is 0.282 lower than that of DMD, indicating a significant improvement in reconstruction accuracy. Further illustrating the proposed method, the HDMD method can better capture and analyze the unstable modes in the ECG signal, thereby significantly improving the accuracy and robustness of signal classification. Future work will focus on further optimizing the parameter selection of the HDMD model and exploring its application potential in real-time heart disease monitoring and early warning systems.

Trading strategy based on dynamic mode decomposition: Tested in Chinese stock market

The identification of coherent structures using proper orthogonal decomposition and dynamic mode decomposition

The objective of this study is to develop a reduced order model (ROM) that accurately predicts the key features of the wake generated by a wind turbine, based on the results of a computational fluid dynamics (CFD) simulation with the actuator line model (ALM). The simulation of a three-bladed up-wind wind turbine was performed using Reynolds-averaged Navier-Stokes (RANS) equations and the finite volume method. The two-equation turbulence model K-ωSST was used to approximate the Reynolds stresses and the Actuator Line Method (ALM) was used to simulate the effect of the wind turbine rotor on the flow field, representing the rotor blades as a distribution of forces acting on the fluid. Data from the RANS simulation was collected to be used by the Dynamic Mode Decomposition (DMD) algorithm. Velocity field was defined as the object of study for the construction of the reduced order model (ROM). From the implementation of the DMD algorithm, the eigenvalues and energy content were obtained for the field of interest. From these results, the main modes and their spectrum were calculated. The velocity field reconstruction was performed using the DMD algorithm, and the RMSE was computed relative to the simulation and experimental data, yielding a good approximation. Due to the turbulence model used, very stable results were obtained in the reduced order model, showing great capability to predict the fields of interest in time.

Analysis of aggregated load consumption forecasting in short, medium and long term horizons using Dynamic Mode Decomposition

Dynamic mode decomposition (DMD) provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of temporal, or spatio-temporal, data. A variety of regression techniques have been developed for producing the linear model approximation whose solutions are exponentials in time. For spatio-temporal data, DMD provides low-rank and interpretable models in the form of dominant modal structures along with their exponential/oscillatory behaviour in time. The majority of DMD algorithms, however, are prone to bias errors from noisy measurements of the dynamics, leading to poor model fits and unstable forecasting capabilities. The optimized DMD algorithm minimizes the model bias with a variable projection optimization, thus leading to stabilized forecasting capabilities. Here, the optimized DMD algorithm is improved by using statistical bagging methods whereby a single set of snapshots is used to produce an ensemble of optimized DMD models. The outputs of these models are averaged to produce a bagging, optimized dynamic mode decomposition (BOP-DMD). BOP-DMD improves performance by stabilizing and cross-validating the DMD model by ensembling; it also robustifies the model and provides both spatial and temporal uncertainty quantification (UQ). Thus, unlike currently available DMD algorithms, BOP-DMD provides a stable and robust model for probabilistic, or Bayesian, forecasting with comprehensive UQ metrics. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

The objective of this paper is to identify the dominant coherent structures within cavitating flow around a Clark-Y hydrofoil using two data-driven modal decomposition methods, proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). A snapshot data sequence is obtained using a large eddy simulation and the interaction between cavitation and the vortex during cloud cavity shedding evolution is investigated. Modal decomposition via POD and DMD indicates that the dominant coherent structures include the large-scale cavity–vortex, re-entrant jet, shear layer, and small-scale vortex in the wake. In addition, the flow field can be reconstructed from the most energetic POD or DMD modes. The errors in the flow reconstructions produced using the first four POD modes, first eight POD modes, and first eight DMD modes are 3.884%, 3.240%, and 3.889%, respectively. Furthermore, transient cavitating flow can be predicted via the DMD method with an error of 8.081%. The largest errors in the reconstructed and predicted results occur mostly in the shear layer, trailing edge, and near wake. POD and DMD provide accurate and practically beneficial techniques for understanding cavitating flow, although substantial challenges remain with regard to predicting this intense nonlinear system.

Dynamic mode decomposition (DMD) is a popular technique for modal decomposition, flow analysis, and reduced-order modeling. In situations where a system is time varying, one would like to update the system's description online as time evolves. This work provides an efficient method for computing DMD in real time, updating the approximation of a system's dynamics as new data becomes available. The algorithm does not require storage of past data and computes the exact DMD matrix using rank-1 updates. A weighting factor that places less weight on older data can be incorporated in a straightforward manner, making the method particularly well suited to time-varying systems. A variant of the method may also be applied to online computation of “windowed DMD,” in which only the most recent data are used. The efficiency of the method is compared against several existing DMD algorithms: for problems in which the state dimension is less than about 200, the proposed algorithm is the most efficient for real-time computation, and it can be orders of magnitude more efficient than the standard DMD algorithm. The method is demonstrated on several examples, including a time-varying linear system and a more complex example using data from a wind tunnel experiment. In particular, we show that the method is effective at capturing the dynamics of surface pressure measurements in the flow over a flat plate with an unsteady separation bubble.

Lagrangian dynamic mode decomposition for construction of reduced-order models of advection-dominated phenomena