Online Dynamic Mode Decomposition for Time-Varying Systems

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.

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.

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'.

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

In practical engineering applications, the multivariate signal contains more fault feature information than the single-channel signal. How to realize synchronous extraction of fault features from the multivariate signal is of great significance in fault diagnosis of rotary machinery. Dynamic mode decomposition (DMD) has attracted much attention due to its excellent dynamic feature extraction ability. However, DMD lacks mode aliasing property when dealing with the multivariate signal, which may lead to the loss of critical fault feature information. Cater to this problem, this article proposed a multivariate DMD (MDMD) algorithm that is the multivariate extension of DMD. First, snapshot tensors are defined to convert multivariate signals to tensor format. Then, the MDMD algorithm is proposed by introducing tensor operations into the original DMD algorithm, where a tensor low tubal rank component extraction framework is constructed to enable simultaneous extraction of bearing fault features from the multivariate signal, to enhance the robustness and effectiveness of the algorithm. Finally, both numerical simulations and experiments verify that the proposed MDMD has higher fault diagnosis accuracy than other multivariate signal-processing methods.

Addressing climate change through carbon capture and storage (CCS) technologies necessitates advanced computational methodologies for subsurface CO2 storage monitoring. This study focuses on the Illinois Basin Decatur Project (IBDP), a CCS demonstration pilot aimed at CO2 injection into a deep saline reservoir. We introduce a novel framework combining Dynamic Mode Decomposition (DMD), a data-driven model reduction technique, with direct data assimilation to streamline the calibration of CO2 plume evolution models. This approach enhances rapid tracking and overcomes the computational challenges of traditional high-fidelity numerical reservoir simulations known as the full-order model (FOM). Using DMD, we analyzed five distinct FOM simulation cases of the IBDP with varied permeability in the Mt. Simon section to develop reduced-order models (ROMs). These ROMs utilize three state variables: reservoir pressure, CO2 plume saturation, and bottom-hole pressure (BHP) from a CO2 injection well alongside multi-level pressures from a monitoring well derived from the FOM and the actual field data respectively. Initial FOM simulation cases assessed the impact of permeability multipliers on pressure responses. We then transformed these into ROMs using DMD, preserving essential dynamics. Linear interpolation between permeabilities and DMD outputs—modes and eigenvalues—established relationships for rapid BHP prediction under different scenarios. Employing a Kalman filter, we optimized a global permeability multiplier, using the ROMs, to align measured and simulated BHP values, ensuring model calibration. The final calibrated FOM was further decomposed to a DMD-based ROM, enabling quick, accurate predictions, significantly reducing computational time from hours to minutes. Utilizing an ROM derived through DMD, we achieved an order of 160 reduction in computational time (from 8 hours to just 3 minutes) for a 3-year historical CO2 injection period modeled with 547,000 cells of the FOM. The ROM demonstrated remarkable fidelity, with a mean absolute error of 1.46 psi for pressure and 3.7e-05 for CO2 plume saturation, effectively capturing the dynamics of the full-order model. This substantial decrease in computational time illustrates an advantageous trade-off between speed and accuracy, optimizing the potential for long-term forecasting and monitoring of CO2 sequestration. Incorporating the IBDP as a case study, this research contributes a significant advancement to reservoir simulation practices, offering a potent, efficient tool for CCS monitoring. By integrating DMD for ROM construction with precise data assimilation-based calibration, the study provides a comprehensive solution for swift and accurate subsurface CO2 plume tracking, essential for the successful implementation of CCS projects and the broader effort to mitigate climate change impacts.

We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use this result to obtain a representation of the dynamics of systems with continuous spectrum based on the lifting of the coordinates to the space of observables. The numerical application of these methods is demonstrated using well-known dynamical systems and examples from computational fluid dynamics.

Operating convergent–divergent nozzles at low pressure ratios can lead to flow separation. Certain nozzle contours are known to produce a discrete acoustical tone under such conditions that are believed to be generated by a phenomenon known as transonic resonance, which involves a standing pressure wave situated between the separation shock and the nozzle exit plane. This paper reports the findings of a dynamic mode decomposition analysis of a perturbed axisymmetric unsteady Reynolds-averaged Navier–Stokes simulation of a nozzle flow known to exhibit this phenomenon. Two cases of different pressure ratios were studied in depth. The results show that the two cases differ in the shape of the standing pressure wave. The lower pressure ratio produces a standing pressure wave, whereas the higher pressure ratio produces a wave. In both cases, dynamic mode decomposition modes that match the standing pressure wave shape were found to be the least damped and the most energetic modes of the modes produced by the dynamic mode decomposition algorithm. The frequency of the mode for the lower pressure ratio matched the experimentally observed transonic tone frequency extraordinarily well, whereas the higher pressure ratio case was in fair agreement. The dynamic mode decomposition algorithm captured the transonic frequency for five additional pressure ratios.

A note on the use of Dynamic Mode Decomposition in mechanics

We investigate the characteristic of large-scale coherent motions over a large horizontal domain using the Dynamic Mode Decomposition (DMD) spectral analysis algorithm applied on measurements from two long-range pulsed lidars. We show the results and advantages of this methodology on six cases representative of three thermal stratification conditions at two heights relevant for wind energy: near-neutral, unstable and stable stratification at 50m and 200m above ground level. For these cases the DMD algorithm show three types of structures: streaks near the surface for near-neutral for neutral stratification, large-scale convective rolls for the unstable cases and sheet-like rotational patches for stable conditions. The DMD algorithm also shows the stationary effects of the terrain on the flow at 50m above ground level, within the atmospheric surface layer. The possibility of isolating terrain effects from coherent motions makes DMD attractive for studying complex atmospheric flow phenomena as well as to have more realistic input for wind farm flow simulations.

A reduced order method for nonlinear parameterized partial differential equations using dynamic mode decomposition coupled with k-nearest-neighbors regression

Analysis of characteristic wake flow modes on a generic transonic backward-facing step configuration

Dynamic mode decomposition (DMD) is an Arnoldi-like method based on the Koopman operator. It analyzes empirical data, typically generated by nonlinear dynamics, and computes eigenvalues and eigenmodes of an approximate linear model. Without explicit knowledge of the dynamical operator, it extracts frequencies, growth rates, and spatial structures for each mode. We show that expansion in DMD modes is unique under certain conditions. When constructing mode-based reduced-order models of partial differential equations, subtracting a mean from the data set is typically necessary to satisfy boundary conditions. Subtracting the mean of the data exactly reduces DMD to the temporal discrete Fourier transform (DFT); this is restrictive and generally undesirable. On the other hand, subtracting an equilibrium point generally preserves the DMD spectrum and modes. Next, we introduce an “optimized” DMD that computes an arbitrary number of dynamical modes from a data set. Compared to DMD, optimized DMD is superior at calculating physically relevant frequencies, and is less numerically sensitive. We test these decomposition methods on data from a two-dimensional cylinder fluid flow at a Reynolds number of 60. Time-varying modes computed from the DMD variants yield low projection errors.