A Machine Learning-based Method using the Dynamic Mode Decomposition for Fault Location and Classification.
Abstract not provided.
Wilches-Bernal, Felipe↗
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Results for “Dynamic Mode Decomposition”
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A Machine Learning-based Method using the Dynamic Mode Decomposition for Fault Location and Classification.
Abstract not provided.
Wilches-Bernal, Felipe↗
Learning Compact Physics-Aware Photocurrent Models Using Dynamic Mode Decomposition.
Abstract not provided.
Hanson, Joshua McKinley↗
Dynamic Mode Decomposition for Modeling Geophysical Flows
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Bishnu, Siddhartha↗
Development of Multiresolution Capabilities for the Holistic Energy Resource Optimization Network (HERON) tool A progress update
INL researchers work on technoeconomic analyses for integrated energy systems (IES) using the Framework for Optimization of ResourCes and Economics (FORCE). Within FORCE, researchers use the Holistic Energy Resource Optimization Network (HERON) tool to conduct optimization of grid portfolios under uncertain market conditions. These optimizations determine optimal capacities for all IES components and strategies for resource dispatch which maximize some economic metric (e.g., net present value). Resource dispatch occurs on finer timescales (typically hours) and thus are asked to respond to a given time series (e.g. hourly load demand profiles for a grid, or pre-determined electricity prices). Volatile and complex bidding dynamics as well as poorly forecasted weather events within deregulated markets add uncertainty to the time series; FORCE can address this uncertainty by training a reduced order model on historical time series and generate unique synthetic time series which represent individual scenarios or realizations of the market. The IES configuration can be simulated under these different sampled realizations and a stochastic optimization is conducted which optimizes the expected value of the desired economic metric. The training of a synthetic time series generator is limited by the chosen time resolution; dynamics can occur on different time scales. Seasonal demand trends can dominate faster dynamical events (such as power outages from certain sectors or severe weather events) which might not get captured correctly by the trained model. In this report, we investigate different ways of addressing the training and generation of time series on multiple time scales using three main algorithms: wavelet decomposition, dynamic mode decomposition, and generative adversarial networks for time series. We demonstrate a time series analysis that yields information on not just the frequency space but also temporal space: where a fast Fourier transform can provide what frequencies dominate, the new algorithms can provide when the frequencies dominate as well. These analyses can help improve IES optimization by allowing researchers to couple simulations at different timescales when it is most needed - seasonal, day-ahead, and real time optimization - with greater computational efficiency. Future work will include implementation of a subset of the proposed algorithms into the FORCE toolset and application of these analyses into multiple timescale optimization.
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Data-driven modeling and control of dynamical systems using Koopman and Perron-Frobenius operators
This dissertation studies the data-driven modeling and control problem of nonlinear systems by exploiting the linear operator theoretic framework involving Koopman and Perro-Frobenius operator. A systematic linear-operator based controller design procedure has been established, which can be used to solve a variety of nonlinear control problems, including feedback stabilization using control Lyapunov functions, optimal quadratic regulation using Koopman eigenfunctions and convex optimization formulation of optimal control problem using P-F and Koopman operator approximation. As the core of data-driven modeling, we first propose a new algorithm for the finite-dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time-series data. We argue that the existing approach for the finite-dimensional approximation of these transfer operators such as Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two important properties of these operators, namely positivity and Markov property. The algorithm we propose preserves these two properties. We call the proposed algorithm as naturally structured DMD (NSDMD) since it retains the inherent properties of these operators. Naturally structured DMD algorithm leads to a better approximation of the steady-state dynamics of the system regarding computing Koopman and Perron- Frobenius operator eigenfunctions and eigenvalues. However, preserving positivity property is critical for capturing the real transient dynamics of the system. This positivity property of the transfer operators and it's finite-dimensional approximation play an important role for controller and estimator design of nonlinear systems. To solve the feedback stabilization problem for nonlinear control systems, we tried to take advantage of the Koopman operator framework. The Koopman operator approach provides a linear representation for a nonlinear dynamical system and a bilinear representation for a nonlinear control system. The problem of feedback stabilization of a nonlinear control system is then transformed to the stabilization of a bilinear control system. We propose a control Lyapunov function (CLF)-based approach for the design of stabilizing feedback controllers for the bilinear system. The search for finding a CLF for the bilinear control system is formulated as a convex optimization problem. This leads to a schematic procedure for designing CLF-based stabilizing feedback controllers for the bilinear system and hence the original nonlinear system. Another advantage of the proposed controller design approach outlined in this dissertation is that it does not require explicit knowledge of system dynamics. In particular, the bilinear representation of a nonlinear control system in the Koopman eigenfunction space can be obtained from time-series data. Next, we study the optimal quadratic regulation problem for nonlinear systems. The linear operator theoretic framework involving the Koopman operator is used to lift the dynamics of nonlinear control system to an infinite-dimensional bilinear system. The optimal quadratic regulation problem for nonlinear system is formulated in terms of the finite-dimensional approximation of the bilinear system. A convex optimization-based approach is proposed for solving the quadratic regulator problem for bilinear system. We applied a variety of examples and compared the simulation results between our framework and conventional LQR control using linearized model. For more general optimal control problems, we provide a density-function based convex formulation for the optimal control problem of the nonlinear system. The convex formulation relies on the duality result in the stability theory of a dynamical system involving density function and Perron-Frobenius operator. The optimal control problem is formulated as an infinite-dimensional convex optimization program. The finite-dimensional approximation of the optimization problem relies on the recent advances made in the data-driven computation of the Koopman operator, which is dual to the Perron-Frobenius operator. Simulation results are presented to demonstrate the application of the developed framework.
Huang, Bowen↗
Characterization of Acoustic Emissions From Analogue Rocks Using Sparse Regression‐DMDc
Abstract Moisture loss in rock is known to generate acoustic emissions (AE). Phenomena that result in AE during drying are related to the movement of fluids through the pores and induced‐cracks that arise from differential mineral shrinkage, especially in clay‐bearing rock. AE from the movement of fluids occurs from the reconfiguration of fluid interfaces during drying, while AE from mineral shrinkage involves the debonding within or between minerals. Here, analogue rock samples were used to examine the differences in the AE signatures when one or both AE source‐types are present. An unsupervised sparse regression model, Dynamic Mode Decomposition with control, that extends Dynamic Mode Decomposition is used to characterize the AE signals recorded during the drying of porous analogue rock samples fabricated with ordinary Portland cement, with and without clay. This method can effectively and accurately reconstruct acoustic signals emitted from samples that only experience moisture loss without cracking. However, the method struggles to reconstruct signals from samples with intricate crack networks that formed during drying because AE generating mechanisms can emit contemporaneously, and the resulting waves propagate through drying‐induced cracks that can lead to multiple internal reflections. Thus, the differential reconstruction accuracy of time series generated by different underlying physical processes provides a robust filter for reducing large data catalogs. In general, both dynamics and sparse initiating events are learned directly from data and this method exposes a data hierarchy based on the complexity of the intrinsic dynamics.
58 GEOSCIENCES↗
Efficient Measurement-Driven Eigenenergy Estimation with Classical Shadows
Quantum algorithms exploiting real-time evolution under a target Hamiltonian have demonstrated remarkable efficiency in extracting key spectral information. However, the broader potential of these methods, particularly beyond ground-state calculations, is underexplored. In this work, we introduce the framework of multiobservable dynamic mode decomposition (MODMD), which combines the observable dynamic mode decomposition (DMD), a measurement-driven eigensolver tailored for near-term implementation, with classical shadow tomography. MODMD leverages random scrambling in the classical shadow technique to construct, with exponentially reduced resource requirements, a signal subspace that encodes rich spectral information. Notably, we replace typical Hadamard-test circuits with a protocol designed to predict low-rank observables, thereby broadening the use of classical shadow tomography for predicting many low-rank observables. We establish theoretical guarantees on the spectral approximation from MODMD, taking into account distinct sources of error. In the ideal case, we prove that the spectral error scales as exp (−Δ𝐸𝑡 max ), where Δ𝐸 is the Hamiltonian spectral gap and 𝑡 max is the maximal simulation time. This analysis provides a rigorous justification of the rapid convergence observed across simulations. To demonstrate the utility of our framework, we consider its application to fundamental tasks, such as determining the low-lying, i.e., ground or excited, energies of representative many-body systems. Our work paves the path for efficient designs of measurement-driven algorithms on near-term and early fault-tolerant quantum devices.
quantum algorithms & computation↗
Flow field characteristics of a 127-pin rod bundle with hexagonal spacer grids
A determination of nominal flow phenomena in liquid metal fast reactor (LMFR) fuel assemblies is critical toward generation-IV reactor development. Axially positioned spacer grids are used to maintain the geometry of hexagonal rod bundles and simultaneously introduce perturbations in the flow. Three-dimensional (3D) printed asymmetric honeycomb spacer grids were installed in a prototypical 127-pin LMFR fuel assembly model to study complex fluid dynamics interactions induced by the spacer grid and rods. To characterize flow dynamics in this intricate geometry, time-resolved particle image velocimetry (TR-PIV) using the matched-index-of-refraction method was employed to obtain non-intrusive velocity measurements for three axial planes (one near-wall and two interior planes) at a Reynolds number of 6000. The statistical TR-PIV results compared sub-channel-dependent normalized time-averaged velocity, velocity fluctuations, Reynolds stress, vorticity, and turbulence kinetic energy distributions. TR-PIV line profiles characterized downstream spacer grid flow dynamics. Two-point spatial and spatial–temporal cross-correlation fields revealed local coherent structures and quantified convection velocities of traveling vortices. Spatial–temporal decomposition using dynamic mode decomposition (DMD) applied to the near-wall vorticity fields extracted turbulent structures and flow instabilities in the wake region of the spacer grid, along with their decay and frequency rates. Reduced-order velocity fields from DMD reconstructions identified the most energy-containing coherent structures persistent in the near-wall region. This research provides experimental data sets and analyses of flow behavior in rod bundles with hexagonal spacer grids. In conclusion, the results are critical toward LMFR design and geometry optimization, crucial for the validation of computational fluid dynamics and reduced-order flow models.
42 ENGINEERING↗
Reduced order models for thermal radiative transfer problems based on moment equations and data-driven approximations of the Eddington tensor
Here a new group of structure and asymptotic preserving reduced-order models (ROMs) for multidimensional nonlinear thermal radiative transfer (TRT) problems is presented. They are formulated by means of the nonlinear projective approach and data compression techniques. The nonlinear projection is applied to the Boltzmann transport equation (BTE) to derive a hierarchy of low-order moment equations. Approximation of the Eddington tensor that provides exact closure for the system of moment equations is found with projection-based data-driven methodologies. These include the (i) proper orthogonal decomposition (POD), (ii) dynamic mode decomposition (DMD) and (iii) a variant of the DMD. A parameterization is derived for this ROM for the temperature of radiation incoming to the problem domain (the radiation drive temperature). This parameterization is informed from results of a dimensionless study of the TRT problem. Analysis of the ROMs is performed on the classical Fleck-Cummings TRT multigroup test problem in 2D geometry with a radiation-driven Marshak wave. Numerical results are presented to demonstrate the performance of these ROMs for the simulation of evolving radiation and heat waves. Results show these models to be sufficiently accurate for practical computations with rather low-rank representations of the Eddington tensor. As the rank of the approximation is increased, the errors of solutions generated by the ROMs gradually decreases.
42 ENGINEERING↗
Self-Aligned Focusing Schlieren and OH Planar Laser-Induced Fluorescence Flow Visualization in a Dual-Mode Scramjet
Ahigh-speed self-aligned focusing schlieren (SAFS) system was used to visualize density gradients in and around the cavity flameholder of the combustor section of the University of Virginia Supersonic Combustion Facility (UVASCF). Images with this system were acquired at a framing rate of 110 kHz with no fuel injection, with fuel injection but no flame, and for fuel injection with combustion corresponding to a global equivalence ratio of 𝜙 = 0.18. Images with an air throttle in operation to modify the shock train location with fuel injection and with flame were also acquired. Simultaneous OH planar laser-induced fluorescence (PLIF) images were also acquired at a framing rate of 20 Hz. Results obtained with both visualization techniques are compared to one another to highlight how SAFS can complement more advanced flow visualization techniques and resolve dynamic behavior that may not otherwise be captured. Both proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) analysis techniques are applied to the SAFS image sequences to identify coherent periodic structures for the runs with fuel injection and combustion.
Brett F Bathel↗
Heterogeneous Mixtures of Dictionary Functions to Approximate Subspace Invariance in Koopman Operators: Why Deep Koopman Operators Work
Abstract Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary . In a widely used algorithm, extended dynamic mode decomposition (EDMD), the dictionary functions are drawn from a fixed class of functions. Deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper, we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Typically, a Koopman dictionary’s nonlinear functions are homogeneous. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm Yeung et al. ( In: 2019 American Control Conference (ACC), 2019). We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves similar accuracy and dimensional scaling to deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.
Johnson, Charles A.↗
Data-scarce surrogate modeling of shock-induced pore collapse process
Understanding the mechanisms of shock-induced pore collapse is of great interest in various disciplines in sciences and engineering, including materials science, biological sciences, and geophysics. However, numerical modeling of the complex pore collapse processes can be costly. To this end, a strong need exists to develop surrogate models for generating economic predictions of pore collapse processes. Here, in this work, we study the use of a data-driven reduced-order model, namely dynamic mode decomposition, and a deep generative model, namely conditional generative adversarial networks, to resemble the numerical simulations of the pore collapse process at representative training shock pressures. Since the simulations are expensive, the training data are scarce, which makes training an accurate surrogate model challenging. To overcome the difficulties posed by the complex physics phenomena, we make several crucial treatments to the plain original form of the methods to increase the capability of approximating and predicting the dynamics. In particular, physics information is used as indicators or conditional inputs to guide the prediction. In realizing these methods, the training of each dynamic mode composition model takes only around 30 s on CPU. In contrast, training a generative adversarial network model takes 8 h on GPU. Moreover, using dynamic mode decomposition, the final-time relative error is around 0.3% in the reproductive cases. We also demonstrate the predictive power of the methods at unseen testing shock pressures, where the error ranges from 1.3 to 5% in the interpolatory cases and 8 to 9% in extrapolatory cases.
97 MATHEMATICS AND COMPUTING↗
Reduced-order modeling on a near-term quantum computer
Quantum computing is an advancing area of research in which computer hardware and algorithms are developed to take advantage of quantum mechanical phenomena. In recent studies, quantum algorithms have shown promise in solving linear systems of equations as well as systems of linear ordinary differential equations (ODEs) and partial differential equations (PDEs). Reducedorder modeling (ROM) algorithms for studying fluid dynamics have shown success in identifying linear operators that can describe flowfields, where dynamic mode decomposition (DMD) is a particularly useful method in which a linear operator is identified from data. In this work, DMD is reformulated as an optimization problem to propagate the state of the linearized dynamical system on a quantum computer. This reformulation was chosen as a means of facilitating implementation on a near-term quantum computer. Quadratic unconstrained binary optimization (QUBO), a technique for optimizing quadratic polynomials in binary variables, allows for quantum annealing algorithms to be applied. A quantum circuit model (quantum approximation optimization algorithm, QAOA) is utilized to obtain predictions of the state trajectories. Results are shown for the quantum-ROM predictions for flow over a 2D cylinder at Re = 220 and flow over a NACA0009 airfoil at Re = 500 and α = 15°. The quantum-ROM predictions are found to depend on the number of bits utilized for a fixed point representation and the truncation level of the DMD model. Comparisons with DMD predictions from a classical computer algorithm are made, as well as an analysis of the computational complexity and prospects for future, more fault-tolerant quantum computers.
97 MATHEMATICS AND COMPUTING↗
Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive
We investigate the sensitivity of reduced order models (ROMs) to training data spatial resolution as well as sampling rate. In particular, we consider proper orthogonal decomposition (POD), coupled with Galerkin projection (POD-GP), as an intrusive reduced order modeling technique. For non-intrusive ROMs, we consider two frameworks. The first is using dynamic mode decomposition (DMD), and the second is based on artificial neural networks (ANNs). For ANN, we utilized a residual deep neural network, and for DMD we have studied two versions of DMD approaches; one with hard thresholding and the other with sorted bases selection. Also, we highlight the differences between mean-subtracting the data (centering) and using the data without mean-subtraction. We tested these ROMs using a system of 2D shallow water equations for four different numerical experiments, adopting combinations of sampling rates and spatial resolutions. For these cases, we found that the DMD basis obtained with hard threshodling is sensitive to sampling rate. The sorted DMD algorithm helps to mitigate this problem and yields more stabilized and converging solution. Furthermore, we demonstrate that both DMD approaches without mean subtraction provide significantly more accurate results than DMD with mean-subtracting the data. On the other hand, POD is relatively insensitive to sampling rate and yields better representation of the flow field. Meanwhile, spatial resolution has little effect on both POD and DMD performances. Here, the numerical results reveal that an ANN on POD subspace (POD-ANN) performs remarkably better than POD-GP and DMD in capturing system dynamics, even with a small number of modes.
42 ENGINEERING↗
Data-Centric Approach to Capture Non-Polynomial Nonlinear Dynamics
We propose an analytical construction of observable functions in the extended dynamic mode decomposition (EDMD) algorithm. EDMD is a numerical method for approximating the spectral properties of the Koopman operator. The choice of observable functions is fundamental for applying EDMD to nonlinear problems arising in systems and control. Existing methods either start from a set of dictionary functions and look for the subset that best fits the underlying nonlinear dynamics or rely on machine learning algorithms to “learn” observable functions. Conversely, in this paper, we start from the dynamical system model and lift it through the Lie derivatives, rendering it into a polynomial form. This proposed transformation into a polynomial form is exact and provides an adequate set of observable functions. The strength of the proposed approach is its applicability to a broader class of nonlinear dynamical systems, particularly those with nonpolynomial functions and compositions thereof. Moreover, it retains the physical interpretability of the underlying dynamical system and can be readily integrated into existing numerical libraries. We demonstrate the proposed approach with an application to electric power systems. The modeled system consists of a single generator connected to an infinite bus, where nonlinear terms include sine and cosine functions. The results demonstrate the effectiveness of the proposed procedure in off-attractor nonlinear dynamics for estimation and prediction; the observable functions obtained from the proposed construction outperform methods that use dictionary functions comprising monomials or radial basis functions.
extended dynamic mode decomposition↗
Data-Driven Adaptive Damping Controller for Wind Power Plants with Doubly-Fed Induction Generators
This paper presents an adaptive damping controller for wind power plants in which the turbines are equipped with doubly-fed induction generators. The controller is designed to respond to an input control signal that is triggered according to the system operating conditions. A processing unit continuously estimates the electromechanical modes of oscillation based on real-time streaming data acquired from a phasor measurement unit that is strategically positioned on the grid. The decision to trigger (or not trigger) the control signal is automatic, based on the relative damping of the dominant mode. The modes are estimated using the dynamic mode decomposition algorithm with time-delay embedding. Numerical simulations performed on the two-area system demonstrate that the proposed controller enhances the rotor angle stability for both small-signal and large disturbances, and is adaptive to changing grid conditions.
damping controller↗