Higher Order 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.

Entropic Regression Dynamic Mode Decomposition (ERDMD) Discovers Informative Sparse and Nonuniformly Time Delayed Models

In this work, we present a method that determines optimal multistep Dynamic Mode Decomposition (DMD) models via Entropic Regression (ER), which is a nonlinear information flow detection algorithm. Motivated by the Higher-Order DMD (HODMD) method of [Le Clainche & Vega, 2017], and the ER technique for network detection and model construction found in [Sun et al., 2015; AlMomani et al., 2020], we develop a method that we call ERDMD, which produces high fidelity time-delay DMD models that allow for nonuniformity in the delays. This optimal choice of delays is discovered by maximizing informativity as measured through ER. These models are shown to be highly efficient and robust. We test our method over several data sets generated by chaotic attractors and show that we are able to build excellent reconstructions using relatively minimal models. We likewise are able to better identify multiscale features via our models which enhances the utility of DMD.

Spatiotemporal Koopman decomposition of second mode instability from a hypersonic schlieren video

Data-driven modal analysis methods provide a powerful way to decompose data into a sum of modes. The spatiotemporal Koopman decomposition (STKD) enables the computation of modes defined by global frequencies and growth rates in various spatial dimensions and time. The method is an extension of the dynamic mode decomposition (DMD) and higher-order dynamic mode decomposition (HODMD) that represents the data as a sum of standing and traveling, possibly growing or decaying, waves. In this paper, the STKD with HODMD is applied to schlieren video highlighting second mode instability waves traveling down the length of a 3-degree half-angle cone and a 7-degree half-angle cone, both at a freestream Mach number of 6. The HODMD is able to compute dominant modes and frequencies that align with those from associated experimental measurements of unsteady pressure fluctuations, and whose mode shapes clearly show the intensifying wavepacket structure of the waves. The STKD algorithm is used to compute streamwise wavenumbers, spatial growth rates, and wave speeds. The spatial growth rates from the STKD and the magnitudes of the HODMD mode shapes are used to compute the N-factor for waves of several frequencies. Overall, the STKD with HODMD is shown to be a useful tool for extracting spatiotemporal disturbance growth from a schlieren video.

Mode decomposition of core dynamics transients using higher-order DMD method

Accurately predicting three-dimensional power distribution within a reactor core during reactivity transients is crucial for optimizing reactor operational control. This paper proposes a higher-order dynamic mode decomposition (DMD) method for analyzing highly nonlinear dynamic systems in nuclear reactors. Through analysis of the 3-D LWR Core Transient Benchmark (3DLWRCT) with rod ejection accident, traditional DMD reconstruction techniques were found inadequate in accurately capturing the system’s dynamic evolution using the given simulation data, primarily due to the nonlinear features and local high-order characteristics of the reactor system. In contrast, the HODMD approach demonstrated its effectiveness in predicting future variations in the core’s state. These findings establish the feasibility of employing the HODMD method for transient analysis of core dynamics, leading to promising prospects for dynamic analysis in nuclear energy systems. Future research directions for improving the reconstructive accuracy of HODMD include capturing nonlinear characteristics, automating parameter selection, enabling multi-scale analysis, enhancing anomaly detection capabilities, and providing a broader context for analysis.

Prediction of spatiotemporal dynamic systems by data-driven reconstruction

Prediction in nonlinear systems is critical and challenging in various fields. Data-driven methods provide the theoretical basis for predicting nonlinear systems. On this basis, a data-driven prediction framework for improved reservoir computing (RC) combined with higher order dynamic mode decomposition (HODMD) is proposed. First, HODMD extracts the eigenmodes of the nonlinear system through higher order Koopman assumptions and retains the principal modes to reconstruct the system. The subsequent process uses the reconstructed system to train a unique weight matrix through the quadratic readout of the reservoir feature vectors; the reusable feature of RC training is employed to accomplish autonomous prediction of the nonlinear system. The practical consideration of the data-driven prediction framework is to enhance the ability of the RC to learn the internal evolutionary laws of the nonlinear system. Numerical results by Kuramoto-Sivashinsky equation demonstrate that the HODMD-RC framework improves the short-term and long-term prediction of nonlinear systems.

Prediction of wind energy with the use of tensor‐train based higher order dynamic mode decomposition

AbstractAs the international energy market pays more and more attention to the development of clean energy, wind power is gradually attracting the attention of various countries. Wind power is a sustainable and environmentally friendly resource of energy. However, it is unstable. Therefore, it is important to develop algorithms for its prediction. In this paper, we apply a recently developed algorithm that effectively combines the tensor train decomposition with the higher order dynamic mode decomposition (TT‐HODMD). The dynamic mode decomposition (DMD) is a data‐driven technique that does not need a prior mathematical model. It is based on the measurement data or time slots. As demonstrated, for prediction it is important to use the higher order DMD (HODMD). In turn, HODMD might lead to very large scale arrays that are sparse. The tensor train decomposition provides a highly efficient way to work with such arrays. It is demonstrated that the combined TT‐HODMD algorithm is capable of providing quite accurate prediction of wind power for months ahead.

Dynamic Mode Decomposition of Multiphoton and Stimulated Emission Depletion Microscopy Data for Analysis of Fluorescent Probes in Cellular Membranes.

An analysis of the membrane organization and intracellular trafficking of lipids often relies on multiphoton (MP) and super-resolution microscopy of fluorescent lipid probes. A disadvantage of particularly intrinsically fluorescent lipid probes, such as the cholesterol and ergosterol analogue, dehydroergosterol (DHE), is their low MP absorption cross-section, resulting in a low signal-to-noise ratio (SNR) in live-cell imaging. Stimulated emission depletion (STED) microscopy of membrane probes like Nile Red enables one to resolve membrane features beyond the diffraction limit but exposes the sample to a lot of excitation light and suffers from a low SNR and photobleaching. Here, dynamic mode decomposition (DMD) and its variant, higher-order DMD (HoDMD), are applied to efficiently reconstruct and denoise the MP and STED microscopy data of lipid probes, allowing for an improved visualization of the membranes in cells. HoDMD also allows us to decompose and reconstruct two-photon polarimetry images of TopFluor-cholesterol in model and cellular membranes. Finally, DMD is shown to not only reconstruct and denoise 3D-STED image stacks of Nile Red-labeled cells but also to predict unseen image frames, thereby allowing for interpolation images along the optical axis. This important feature of DMD can be used to reduce the number of image acquisitions, thereby minimizing the light exposure of biological samples without compromising image quality. Thus, DMD as a computational tool enables gentler live-cell imaging of fluorescent probes in cellular membranes by MP and STED microscopy.

Hierarchical higher-order dynamic mode decomposition for clustering and feature selection

This article introduces a novel, fully data-driven method for forming reduced order models (ROMs) in complex flow databases that consist of a large number of variables. The algorithm utilizes higher order dynamic mode decomposition (HODMD), a modal decomposition method, to identify the main frequencies and associated patterns that govern the flow dynamics. By incorporating various normalization techniques into an iterative process, clusters of variables with similar dynamics are identified, allowing the classification of different instabilities and patterns present in the flow. This method, known as hierarchical HODMD (h-HODMD), has been thoroughly tested in the development of ROMs using three different databases obtained from numerical simulations of a nonpremixed coflow methane flame. The effectiveness of h-HODMD has been demonstrated as it consistently outperforms HODMD in terms of modeling and reconstructing flow dynamics using a reduced number of modes. Additionally, the clusters of variables identified by h-HODMD reveal the algorithm's ability to group chemical species whose behavior is consistent from a kinetic perspective. h-HODMD allows for the construction of inexpensive reduced dynamical models that can predict flame liftoff, identify the occurrence of local extinction and blowout conditions, and facilitate control purposes.

Application of noise-filtering techniques to data-driven analysis of electric power systems based on higher-order dynamic mode decomposition

A transition to renewable energy is increasing the long-distance export of power, with reduced spinning inertia and small stability margins. In this work, we apply higher-order variants of a data-driven technique, the dynamic mode decomposition (DMD), for stability analysis and short-term prediction of power systems with renewable sources of energy. In the present paper we study the sampling duration for extraction of physically relevant dynamics suitable for prediction using noisy historical measurements of power system disturbance events. Trends in behaviour of the dominant mode are studied to estimate the singular value threshold and the delay embedding for prediction. In a sampling window around ten times the dominant periodic interval, the higher-order Kalman filtering DMD and extended Kalman filtering DMD (filtering for system identification) are newly applied to predict an historical resonance and a possible near-resonance events of single dominant frequency. In turn, multiple frequencies of a wide-area oscillatory disturbance with low-level non-Gaussian noise are best captured and predicted using the total-least-squares higher-order DMD.

Higher order dynamic mode decomposition beyond aerospace engineering

It is a well known fact that fluid dynamics play a crucial rule in countless fields in scientific and industrial applications, including nature and medicine (ocean currents, fluid motion around jellyfish, blood circulation...), in energy production (wind turbines, tidal energy, combustion...) and of course in transportation and aerospace (trains, automobiles, aerospace shape optimization, cross-flow instability...). Understanding all these complex fluid dynamics problems is not an easy task and it requires a routinely description and quantification. Data-science and its diverse technique have arisen as a powerful tool to tackle these fluid dynamics problems, techniques such as machine learning and artificial intelligence, data processing and decision making, modeling and simulations are but a few of the promising tools described so far. In this work, we focus on data-driven techniques. In particular, we are introducing the fully data-driven technique, the higher order dynamic mode decomposition (HODMD). This technique was originally developed for the fluid mechanics field, but rapidly grew and covered several other fields and applications. A short review comparing two different applications will be shown in this contribution. We will first present our proposed technique, explain its solid mathematical foundations and highlight its capabilities (robustness, denoising properties, efficiency regardless of the amounts of data...). Next, we show the applicability of the HODMD technique to two, very different and quite complex problems: (i) a compressible, turbulent jet and (ii) pattern identification in medical dataset, consisting of echocardiography images.

Time series prediction of ship course keeping in waves using higher order dynamic mode decomposition

A novel reduced-order model (ROM) based on higher order dynamic mode decomposition (HODMD) is proposed for the time series prediction of ship course-keeping motion in waves. The proposed ROM is validated by using the data of course-keeping tests of an ONR tumblehome ship model. First, modes are decomposed from the model test data by standard DMD and HODMD, and the dominant modes are selected according to the energy index. Then, the decomposed dominant modes are used to reconstruct and predict the dynamics of ship motion. The dynamic characteristics in the dynamical systems are revealed according to the energy index, growth rates, and frequencies of the decomposed modes. In addition, the effects of the tunable parameter in HODMD on prediction accuracy and computational times are analyzed by a parametric study. The prediction results by HODMD show better agreement with the model test data than those by standard DMD.

Extraction and analysis of flow features in planar synthetic jets using different machine learning techniques

Synthetic jets are useful fluid devices with several industrial applications. In this study, we use the flow fields generated by two synchronously operating synthetic jets and simulated using direct numerical simulations. These flow fields are characterized by a jet Reynolds number, Re=100, 150, and 200, and a Strouhal number, St=0.03. We benchmark four different dimensionality reduction techniques: (1) higher-order dynamic mode decomposition (HODMD), (2) proper orthogonal decomposition, (3) vector quantization via principal component analysis (VQPCA), and (4) linear autoencoders. These techniques are often used in generating reduced-order models (ROMs). The performances of these techniques are compared (i) in terms of their ability to accurately reconstruct the high-dimensional flow fields from their low-dimensional manifolds and (ii) in terms of their ability to extract meaningful low-dimensional patterns/features/structures that best describe the main dynamics of the synthetic jets. The similarity between the extracted features is also quantitatively assessed with the help of Procrustes analysis, showing how manifolds from different techniques become more similar when a larger number of modes are retained. Accurate reconstruction and model complexity (or interpretability) are often two counter-balancing objectives. In this comparative study, we found that among the four techniques, VQPCA has clear advantages for developing accurate ROMs, while HODMD is useful for understanding the dynamics of synthetic jets, providing additional information that is not readily available with other methods.

Experimental characterization of shock-separation interaction over wavy-shaped geometries through feature analysis

A canonical wavy surface exposed to a Mach 2 flow is investigated through high-frequency Pressure Sensitive Paint, Kulite measurements, and shadowgraph imaging. The wavy surface features a compression and expansion region, two shock-boundary layer interactions, and two shock-separation regions. The unsteady characteristics of the wall pressure and shock angles are presented, demonstrating an increase in the amplitude of the instabilities when traveling through the shock systems. The pressure sensitive paint measurements confirm a two-dimensional flow pattern with small transversal unsteadiness. Higher-Order Dynamic Mode Decomposition and Spectral Proper Orthogonal Decomposition are implemented to dissect the different flow features, revealing several dominant low-frequency and medium-frequency phenomena. The separation region appears at frequencies with Strouhal numbers between 0.01 and 0.2, confirmed by the frequency content in the local pressure measurement using Kulites and the pressure sensitive paint.

Forecasting through deep learning and modal decomposition in two-phase concentric jets

This work aims to improve fuel chamber injectors’ performance in turbofan engines, thus implying improved performance and reduction of pollutants. This requires the development of models that allow real-time prediction and improvement of the fuel/air mixture. However, the work carried out to date involves using experimental data (complicated to measure) or the numerical resolution of the complete problem (computationally prohibitive). The latter involves the resolution of a system of partial differential equations (PDE). These problems make difficult to develop a real-time prediction tool. Therefore, in this work, we propose using machine learning in conjunction with (complementarily cheaper) single-phase flow numerical simulations in the presence of tangential discontinuities to estimate the mixing process in two-phase flows. In this meaning we study the application of two proposed neural network (NN) models11The code of one model and two data sets are available at https://github.com/RAbadiaH/forecasting-through-deep-learning-and-modal-decomposition-in-multi-phase-concentric-jets.git. as PDE surrogate models. Where the future dynamics is predicted by the NN, given some preliminary information. We show the low computational cost required by these models, both in their training and inference phases. We also show how NN training can be improved by reducing data complexity through a modal decomposition technique called higher order dynamic mode decomposition (HODMD), which identifies the main structures inside flow dynamics and reconstructs the original flow using only these main structures. This reconstruction has the same number of samples and spatial dimension as the original flow, but with a less complex dynamics and preserving its main features. The core idea of this work is to test the limits of applicability of deep learning models to data forecasting in complex fluid dynamics problems. Generalization capabilities of the models are demonstrated by using the same NN architectures to forecast the future dynamics of four different two-phase flows.

Tensor Train-Based Higher-Order Dynamic Mode Decomposition for Dynamical Systems

Higher-order dynamic mode decomposition (HODMD) has proved to be an efficient tool for the analysis and prediction of complex dynamical systems described by data-driven models. In the present paper, we propose a realization of HODMD that is based on the low-rank tensor decomposition of potentially high-dimensional datasets. It is used to compute the HODMD modes and eigenvalues to effectively reduce the computational complexity of the problem. The proposed extension also provides a more efficient realization of the ordinary dynamic mode decomposition with the use of the tensor-train decomposition. The high efficiency of the tensor-train-based HODMD (TT-HODMD) is illustrated by a few examples, including forecasting the load of a power system, which provides comparisons between TT-HODMD and HODMD with respect to the computing time and accuracy. The developed algorithm can be effectively used for the prediction of high-dimensional dynamical systems.

Data-driven modal decomposition methods as feature detection techniques for flow problems: A critical assessment

Modal decomposition techniques are showing a fast growth in popularity for their wide range of applications and their various properties, especially as data-driven tools. There are many modal decomposition techniques, yet Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) are the most widespread methods, especially in the field of fluid dynamics. Following their highly competent performance on various applications in several fields, numerous extensions of these techniques have been developed. In this work, we present an ambitious review comparing eight different modal decomposition techniques, including most established methods, i.e., POD, DMD, and Fast Fourier Transform; extensions of these classical methods: based either on time embedding systems, Spectral POD and Higher Order DMD, or based on scales separation, multi-scale POD (mPOD) and multi-resolution DMD (mrDMD); and also a method based on the properties of the resolvent operator, the data-driven Resolvent Analysis. The performance of all these techniques will be evaluated on four different test cases: the laminar wake around cylinder, a turbulent jet flow, the three-dimensional wake around a cylinder in transient regime, and a transient and turbulent wake around a cylinder. All these mentioned datasets are publicly available. First, we show a comparison between the performance of the eight modal decomposition techniques when the datasets are shortened. Next, all the results obtained will be explained in detail, showing both the conveniences and inconveniences of all the methods under investigation depending on the type of application and the final goal (reconstruction or identification of the flow physics). In this contribution, we aim at giving a—as fair as possible—comparison of all the techniques investigated. To the authors' knowledge, this is the first time a review paper gathering all these techniques have been produced, clarifying to the community what is the best technique to use for each application.

Higher order dynamic mode decomposition to model reacting flows

This work presents a new application of higher order dynamic mode decomposition (HODMD) for the analysis of reactive flows. Due to the high complexity of the data analysed, consisting of more than 80 variables (i.e., temperature and chemically reacting species) a new extension of HODMD has been developed combining the multi-dimensional HODMD algorithm with classical preprocessing techniques generally used in machine learning analyses, such as principal component analysis (PCA). This new methodology has proved to be suitable to identify the main patterns driving the main dynamics of the flow, as well as to develop reduced order models grounded in physical principles. The new algorithm was tested by means of a database obtained from a Computational Fluid Dynamics simulation of an axy-symmetric time varying non-premixed co-flow nitrogen-diluted methane flame, carried out by means of a detailed kinetic mechanism. Different dynamics were identified, and they were associated to the different variables of the reactive flow analysed. Also, this algorithm has been coupled with an additional feature selection step carried out via PCA and varimax rotation. The results that were obtained by coupling PCA with varimax rotation show the outstanding capabilities of this novel methodology to compress original databases as function of the different preprocessing techniques that are used.

Mode interpretation and force prediction surrogate model of flow past twin cylinders via machine learning integrated with high-order dynamic mode decomposition

Understanding and modeling the flow field and force development over time for flow past twin tandem cylinders can promote insight into underlying physical laws and efficient engineering design. In this study, a new surrogate model, based on a convolutional neural network and higher-order dynamic mode decomposition (CNN-HODMD), is proposed to predict the unsteady fluid force time history specifically for twin tandem cylinders. Sampling data are selected from a two-dimensional direct numerical simulation flow solution over twin tandem cylinders at different aspect ratios (AR = 0.3–4), gap spacing (L* = 1–8), and Re = 150. To promote insight into underlying physical mechanisms and better understand the surrogate model, the HODMD analysis is further employed to decompose the flow field at selected typical flow regimes. Results indicate that CNN-HODMD performs well in discovering a suitable low-dimensional linear representation for nonlinear dynamic systems via dimensionality augment and reduction technique. Therefore, the CNN-HODMD surrogate model can efficiently predict the time history of lift force at various AR and L* within 5% error. Moreover, fluid forces, vorticity field, and power spectrum density of twin cylinders are investigated to explore the physical properties. It was found three flow regimes (i.e., overshoot, reattachment, and coshedding) and two wake vortex patterns (i.e., 2S and P). It was found the lift force of the upstream cylinder for AR < 1 is more sensitive to the gap increment, while the result is reversed for the downstream cylinder. It was found that the fluctuating component of the wake of cylinders decreases with increasing AR at L* = 1. Moreover, flow transition was observed at L* = 4.

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.

Data-driven assessment of arch vortices in simplified urban flows

Understanding flow structures in urban areas is widely recognized as a challenging concern due to its effect on urban development, air quality, and pollutant dispersion. In this study, state-of-the-art data-driven methods for modal analysis of simplified urban flows are used to study the dominant flow processes in these environments. Higher order dynamic mode decomposition (HODMD), a highly-efficient method to analyze turbulent flows, is used together with traditional techniques such as proper-orthogonal decomposition (POD) to analyze high-fidelity simulation data of a simplified urban environment. Furthermore, the spatio-temporal Koopman decomposition (STKD) will be applied to the temporal modes obtained with HODMD to perform spatial analysis. The flow interaction within the canopy influences the flow structures, particularly the arch vortex. The latter is a vortical structure generally found downstream of wall-mounted obstacles, which is generated as a consequence of flow separation. Therefore, the main objective of the present study is to characterize the mechanisms that promote these phenomena in urban areas with different geometries. Remarkably, among all the vortical structures identified by the HODMD algorithm, low- and high-frequency modes are classified according to their relation with the arch vortex. They are referred to as vortex-generator and vortex-breaker modes, respectively. This classification implies that one of the processes driving the formation and destruction of major vortical structures in between the buildings is the interaction between low- and high-frequency structures. The high energy revealed by the POD for the vortex-breaker modes points to this destruction process as the mechanism driving the flow dynamics. Furthermore, the results obtained with the STKD method show how the generating- and breaking-mechanisms originated along with the streamwise and spanwise directions.