Session K09: Nonlinear Dynamics: Model Reduction (8:45am - 9:30am CST)

Data-driven Modeling of Detonation Wave Interactions in Rotating Detonation Engines

Direct observation of a Rotating Detonation Engine (RDE) combustion chamber has enabled the extraction of the kinematics of its detonation waves. The resulting combustion fronts are composed of co- and counter-rotating coherent traveling shock waves whose nonlinear dynamics are rife with mode-locking behavior, bifurcations, and instabilities which are not well understood. Computational fluid dynamics simulations are ubiquitous in the endeavor to characterize the dynamics of RDEs, however, they prove to be prohibitively expensive when considering multiple engine geometries or operating conditions. Reduced order models (ROMs) are preferred to direct calculations because they exploit low-rank structure in the data to minimize computational cost and allow for rapid parameterized studies. ROMs are inherently inhibited by translational invariances such as the traveling waves present in RDEs. In this work, we overcome these roadblocks by using machine learning to discover moving coordinate frames into which the data is shifted. This allows for the application of traditional dimensionality reduction techniques. We explore a suite of data-driven models, ranging from simple representations to neural networks, describing the complex dynamics of the shock waves in the RDE.   

Data-driven nonlinear aeroelastic models of morphing wings for control

Accurate and efficient aeroelastic models are important for enabling the optimization and control of morphing wings, which are characterized by highly-coupled and nonlinear interactions between the aerodynamic and structural dynamics. In this work, we leverage emerging data-driven modeling techniques to develop highly accurate and tractable reduced-order aeroelastic models that are valid over a wide range of operating conditions and are suitable for control. In particular, we develop two extensions to the dynamic mode decomposition with control (DMDc) algorithm to make it suitable for aeroelastic systems: 1) we introduce a formulation to handle algebraic equations, and 2) we develop an interpolation scheme to connect several linear DMDc models developed in different operating regimes. Thus, the innovation lies in accurately modeling the nonlinearities of the coupled aerostructural dynamics over multiple operating regimes. We demonstrate this approach on a high-fidelity model of an airborne wind energy (AWE) system, although the methods are generally applicable to any highly coupled aeroelastic system. Our proposed modeling framework results in real-time prediction and we demonstrate the enhanced model performance for model predictive control. Thus, the proposed architecture may help enable the widespread adoption of next-generation morphing wing technologies.   

Learning dominant physical processes in complex flows with data-driven balance models

Theoretical analysis in fluid mechanics has long relied on judiciously approximating the full flow physics as a balance between a few dominant processes. However, this traditional approach typically only applies in asymptotic regimes where there is a strict separation of scales. We automate and generalize this approach to non-asymptotic regimes by introducing the idea of an equation space, in which different local balances appear as distinct subspace clusters. Unsupervised learning can then automatically identify regions where groups of terms may be neglected. We show that our data-driven balance models successfully delineate dominant balance physics in a broad range of fluid flows, including a laminar bluff body wake, a boundary layer in transition to turbulence, and surface currents in the Gulf of Mexico.   

Robust Principal Component Analysis for Modal Decomposition of Corrupt Fluid Flows

Modal analysis techniques have been used to identify patterns and structures in a variety of fluid applications. However, experimentally acquired flow fields may be corrupted with incorrect and missing entries, which will degrade subsequent modal analyses. Here we explore how robust principal component analysis (RPCA) can be used to leverage global coherent structures to identify and replace spurious data points. RPCA decomposes a data matrix into a sparse component and low-rank matrix of coherent structure. We explore RPCA on a range of fluid simulations and experiments of varying complexity and assess how accurately the low rank structure in the data is recovered.In all cases, we find that RPCA extracts dominant fluid coherent structures and identifies and fills in incorrect or missing measurements. The performance is particularly striking when flow fields are analyzed using dynamic mode decomposition, which is sensitive to noise and outliers.   

Sequential sampling with heteroscedastic surrogate model to quantify extreme response statistics

We consider a dynamical system with two sources of uncertainties: (1) parameterized input with known probability distribution, and (2) stochastic input-to-response (ItR) map. Our purpose is to efficiently quantify the extreme response statistics when the ItR map is expensive to compute (so a full Monte-Carlo approach is not affordable). This problem setup arises often in physics and engineering, such as weather forecasting and ship motion in irregular wave fields (where the stochasticity in ItR can come from the uncertain subgrid processes and initial conditions in simulation for each input). Our approach in essence leverages sequential sampling in the input parameter space, and accounts for the stochastic ItR map through the heteroscedastic Gaussian process regression (HGPR). A sequential sampling (i.e., next-best sampling) criterion is developed which minimizes the required number of samples to achieve accurate resolution of the extreme response statistics. We demonstrate the effectiveness of the current approach with multiple examples including the ship motion response problem.   

Phase-based analysis of synchronization between laminar cylinder wake and external harmonic actuations

We leverage phase-reduction theory to explore the synchronization properties of the two-dimensional periodic flow over a circular cylinder at Reynolds number of 100. Towards this end, a direct method based on applying weak impulse perturbations at various locations in the near wake and at several times over a period is adopted. The phase response of the wake flow to these impulses reveals how its dynamics can be impacted by external periodic actuations, through enabling the development of a one-dimensional and linear phase-based model with respect to the limit cycle, in place of the full high-dimensional and nonlinear dynamics of the wake. Comparison between the results of the current work and those provided by the Koopman- or adjoint-based analysis of the flow illustrates the commonalities and differences between these methods, while highlighting the characteristic features and capabilities of the phase-reduction analysis. The excellent predictions of the present model with regard to the synchronization properties of the flow under consideration holds promise for its application to complex engineering problems such as structural vibrations and biological swimmers and flyers.   

Network broadcast mode analysis and control with respect to time-varying base flow1

We present a network-based modal analysis that identifies the key dynamical paths along which perturbations amplify in highly unsteady flows. This analysis is built upon the Katz centrality to reveal the flow structures that can effectively spread perturbations over the time-evolving network of vortical elements. Motivated by the resolvent form of the Katz function, we take the singular value decomposition of the resulting communicability matrix, complementing resolvent analysis for fluid flows. The right-singular vector, referred to as the broadcast mode, gives insights into the sensitive regions where introduced perturbations can be effectively spread over the entire fluid-flow network as it evolves over time. We apply the developed formulation to the example of two-dimensional decaying isotropic turbulence. The broadcast mode identifies vortex dipoles as the important structures in amplifying perturbations. By perturbing the flow with the broadcast mode, we demonstrate the effective modification of turbulent flows. The current network-inspired work presents a novel use of network analysis to guide flow control efforts for time-varying base flows.   

Wavelet adaptive POD for large scale flow data

The proper orthogonal decomposition (POD) is a powerful classical tool in fluid mechanics used for instance for model reduction and extraction of coherent flow features. However its applicability to high resolution 3D DNS data is limited due to its computational complexity. Here we propose a wavelet-based adaptive POD, called wPOD, which overcomes this limitation. The size of the analyzed data is reduced by exploiting the compression properties of wavelets with error control, which yields a sparse flow representation. Numerical analysis shows that wavelet compression and POD truncation errors can be balanced and massive 3d high resolution data sets can thus be efficiently handled. A validation of the method will be given for 2D wake flow data. Examples will then be presented for 3D high resolution DNS data of flapping insect flight in turbulence. A comparison with the randomized singular value decomposition illustrates the efficiency and the precision of the wPOD method.   

Projection and Tree Based Reduced Order Modeling for Vortex Particle Simulations.

Vortex particle methods are ubiquitous in modeling vorticity transport phenomena. Example applications include the modeling of a helicopter rotor wake, or wake-body interactions in a school of fish. Unfortunately, these vortex particle methods exhibit poor quadratic $O(N^{2})$ operation-count complexity (OCC), with respect to the number of $N$ particles in the domain. Acceleration techniques, such as the fast-multipole method or other tree-methods, can be used to reduce the OCC. However, these techniques have at best reduced computations to an \textit{N-dependent} linear OCC, i.e. $O(N)$. The presented work addresses the N-dependent OCC bottleneck by introducing a framework that combines hierarchical decomposition and projection-based hyper-reduction to enable \textit{N-independent} OCC. Specifically, the presented framework combines the Barnes-Hut tree method with GNAT hyper-reduction to reduce the pairwise interactions of an N-body problem. The presented method will be showcased on the Biot-Savart kernel to demonstrate fast computations of the induced velocity field for parametric fluid-dynamic example problems.   

A hierarchy structure of local Koopman spectra

Local Koopman spectrum is studied for its role in resolving dynamics of a nonlinear system. For typical linear systems, local Koopman spectra and eigenspace are described by linear theories; for nonlinear systems, the proliferation rule is observed in its recursive application in nonlinear observables. A hierarchy structure of Koopman eigenspace is therefore introduced to depict the nonlinear dynamics in general. With the dynamics being decoupled to base and perturbation components, the perturbation is further separated to linear and nonlinear parts. In the hierarchy, the linear part follows linear spectral theories, and the nonlinear part is defined by linear spectra through recursive proliferation. Local Koopman spectra and eigenfunctions evolve continuously in the whole manifold and are derived from operator perturbation theory. The above analyses were applied first to LTI systems, and then to the nonlinear transition of the flow passing a fixed cylinder and its final periodic state. The triad-chain and the lattice distribution of Koopman spectra are observed numerically to confirm the hierarchy structure of Koopman spectra and the critical role of proliferation rule being involved.   

Least-order model for the transient dynamics of the fluidic pinball

We propose a least-order mean-field model for a flow system undergoing two successive supercritical bifurcations. The fluidic pinball, an incompressible two-dimensional flow crossing three equidistantly spaced cylinders, is numerically investigated using Direct Numerical Simulation. Two generic bifurcations in fluid mechanics are observed: the primary Hopf bifurcation leads to a statistically symmetric vortex shedding and the following pitchfork bifurcation breaks the symmetry at higher Reynolds number. Interestingly, this symmetry-breaking instability works on the steady solution simultaneously, illustrated by the global stability analysis and Floquet analysis. The elementary degrees of freedom are identified under mean-field considerations exploiting the symmetry/asymmetry of the base flow and the fluctuation. An easily interpretable five-dimensional Galerkin model compatible with the quadratic non-linearities of the Navier-Stokes equations is derived, which can reproduce the main features of bifurcating dynamics and the transient behavior to the asymptotic regime. This generalized mean-field Galerkin methodology is considered to be applicable to other transition scenarios and nonlinear model-based control.   

Towards equation-free resolvent analysis

As an equation-based method, resolvent analysis requires knowledge of the exact governing equations of the system so that the resolvent operator can be computed. In this presentation, we explore whether resolvent analysis can be performed purely from data. We base our approach on a nonlinear version of dynamic mode decomposition (DMD) that allows us to approximate the underlying nonlinear operator. Since the system is typically of very high dimension, we leverage techniques from machine learning to project the data onto the principal components of a high (or infinite) dimensional latent space. This approach enables us to disambiguate the linear and nonlinear parts of the operator in the original physical domain. We then establish a connection to the resolvent operator and, finally, validate our approach on a range of high-dimensional problems.   

When deep learning meets ergodic theory

The reliable prediction of the temporal behavior of complex systems is required in numerous fields, including fluid mechanics. This strong interest is inherently connected with modeling issues: often, the governing equations describing the physics of the system under consideration are not accessible or, when known, their solution might require a computational time incompatible with the prediction time constraints. In this view, a data-driven approach is to approximate the system at hand in a generic functional format and inform it from available observations. Numerous successful examples are already available based on deep Neural Networks. Here, we consider Long-Short-Term Memory neural networks and thoroughly investigate the impact of the training set on the long-term prediction. Leveraging metrics and measures from ergodic theory, we analyze the amount of data sufficient for a priori guaranteeing a faithful model of the real system. In particular, we will show how an informed design of the training set, based on symmetries of the system and the topology of underlying attractor, significantly improves the resulting models, opening up avenues of research within the context of active learning. Our developments will be illustrated on the Lorenz’63 model and shear flow models.   

Cluster-based network model

We propose an automatable data-driven methodology for robust nonlinear reduced-order modelling from time-resolved snapshot data. In the kinematical coarse-graining, the snap-shots are clustered into few centroids representable for the whole ensemble. The dynamics is conceptualized as a directed network, where the centroids represent nodes and the directed edges denote possible finite-time transitions. The transition probabilities and times are inferred from the snapshot data. The resulting cluster-based network model constitutes a deterministic-stochastic grey-box model resolving the coherent-structure evolution. This model is motivated by limit-cycle dynamics, illustrated for the chaotic Lorenz attractor and successfully demonstrated for the laminar two-dimensional mixing layer featuring Kelvin-Helmholtz vortices and vortex pairing, and for an actuated turbulent boundary layer with complex dynamics. Cluster-based network modelling opens a promising new avenue with unique advantages over other model-order reductions based on clustering or proper orthogonal decomposition.   

Data-driven sensor selection method using ADMM for a large-scale problem

The present study proposes a sensor selection method based on the sparsity-promoting framework with the A-optimality criterion and the alternating direction method of multipliers algorithm. The performance of the proposed method was evaluated with a random sensor problem and compared with the previously proposed methods such as the greedy method and the convex relaxation. The performance of the proposed method was better than the existing method in terms of the trace of the inverse of the Fisher information matrix. The computational complexity of the proposed method is the first order of the size of the problem ($n)$ and the square order of the number of latent state variables ($r)$. In the case of the convex approximation, the computational complexity is cubic orders of the size of the problem. The Considered problem in the present study is $r$~\textless \textless ~$n$, and thus the computational cost of the proposed method is quite smaller than that of the convex relaxation. The proposed method was applied to the data-driven sparse-sensor-selection problem. A data set adopted is the NOAA OISST V2 mean sea surface temperature set. At the number of sensors larger than that of the latent state variables, the proposed method showed better performance compared to previously proposed methods in terms of the trace of the inverse of the Fisher information matrix and reconstruction error.   

Effect of Objective Function on Sparse Sensor Placement using Greedy Method

In the present study, the objective functions based on D-optimal, A-optimal, and E-optimal criteria of optimal design are adopted to the data-driven sparse sensor selection using the greedy method. The D-, A-, and E-optimal design maximizes the determinant, minimizes the trace of inverse, and maximizes the minimum eigenvalue of the Fisher information matrix, respectively. The greedy methods based on D-, A-, and E-optimality are applied to a random sensor problem, and computational results are compared. In terms of the determinant and trace of the inverse, the sensors selected by the D-optimality objective function works better than those by A- and E-optimality. On the other hand, in terms of the minimum eigenvalue, those by A-optimality works the best and those by E-optimality works better than those by D-optimality. Furthermore, the climate datasets of the National Oceanic and Atmospheric Administration (NOAA) are reconstructed using the sensors selected by the D-, A-, and E-optimality-based greedy methods. The results indicate that the greedy method based on D-optimality is the most suitable for high accurate reconstruction with low computational cost in the case of training datasets are the same as test datasets.   

Assessment of hybrid data-driven models to predict unsteady flows

This work systematically assesses two hybrid data-driven reduced-order models for predicting unsteady wake dynamics for single and side-by-side cylinders. These models rely on recurrent neural networks (RNNs) to evolve low-dimensional unsteady flow states. The first model, termed POD-RNN, projects the high-fidelity Navier-Stokes data to a low-dimensional subspace via proper orthogonal decomposition (POD). The time-dependent coefficients in the POD subspace are propagated by recurrent net (closed-loop/encoder-decoder updates) and mapped to a high-dimensional state via the mean flow field and POD basis vectors. The second model, referred as convolution recurrent autoencoder network (CRAN), employs convolutional neural networks (CNN) (instead of POD), as layers of linear kernels with nonlinear activations, to extract low-dimensional features from flow snapshots. The flattened features are advanced using a recurrent (closed-loop manner) net and up-sampled gradually to high-dimensional snapshots. For flow past a cylinder, the predictive performance of both models is remarkable, with CRAN being a bit overkill. However, CRAN outperforms POD-RNN for longer time predictions of bi-stable flow past side-by-side cylinders.   

Learning fluid flow physics from noisy, incomplete, experimental data

Purely data-driven methods have shown a lot of promise in identifying models of simple, low-dimensional systems from data which have a low level of noise and provide a complete description of the system state. However, they fall apart for data that is high-dimensional, noisy, or incomplete, which is common in fluid dynamics. We show that this challenge can be addressed by augmenting the data-driven approach with a few general physical constraints and using a weak formulation of the model. To illustrate this, we construct a quantitative two-dimensional model of a weakly turbulent flow in a thin layer of electrolyte driven by Lorentz force from PIV data on a coarse spatiotemporal grid. Our hybrid approach also allows reconstruction of the latent variables that cannot be measured directly, e.g., pressure and forcing field.   

Adaptive-Robust Dynamic Mode Decomposition (DMD) for Reduced-Order Modeling and Control

We develop a data-driven reduced-order modeling framework that adapts to changing flow regimes. Such a framework ensures the reduced-order model (ROM) corrects itself to account for changing parameters. Generally, ROMs developed from the data-assimilation are built on specific flow parameters for which the data is available. However, it is ubiquitous that the flow regime is often changing in systems such as HVAC applications, wind farms, etc. Therefore, ROMs need to account for parameter variations to be useful in an “off-design” setting. In this work, we develop a data-driven ROM at a nominal parameter using Dynamic Mode Decomposition (DMD), following which the ROM is appended with a correction/closure term that adapts itself in an online manner to correct for varying parameters. First, we show that the correction term robustly stabilizes the system by guaranteeing stability. Secondly, we use an extremum seeking optimization approach to update the correction term parameters to adapt to the aforementioned variations. Lastly, we discuss the results of using our proposed framework on the viscous Burgers equation. We show that our framework can adapt and account for parametric uncertainties by self-correcting even though the ROM is developed in a flow regime ``very far'' from the gr   

Probabilistic cluster-based feedback control of fluid flow dynamics

We develop a cluster-based feedback control strategy to modify the evolution of flow states to a desired dynamics using limited temporal measurements capturing the flow’s attractor. The flow state is encoded into a low-dimensional feature space, which is partitioned into a small number of clusters and where each flow state is represented by a cluster probability vector. The evolution of the cluster probability vector is given based on the transition probability matrix, which encodes the transitions among the clusters. We formulate an LQR problem to control the cluster probability vector to a desired distribution. The control gain matrix is translated to an optimal feedback control law in the physical space using an estimated scaling function. We first demonstrate this probabilistic control approach on two canonical oscillators: a Lorenz-63 system with bistability and a coupled Fitz-Hugh Nagumo oscillator system that exhibit extreme events. Using DNS, we also employ this control strategy to reduce the pressure fluctuations over a 2D cavity using limited sensor measurements on the cavity wall. The present approach is developed for real-time characterization and control of complex flow dynamics.   

Neural Network approach to reduced order modeling of multiphase flows

We explore the use of Neural Networks (NN) to learn black as well as grey box models of the Partial Differential Equations (PDE) that govern multiphase flows in a 2-Dimensional (2-D) vertical channel. The data is generated using Direct Numerical Simulations (DNS). The covariance method is used to perform Proper Orthogonal Decomposition (POD) on the velocity and void fraction to filter the data so we can learn an effective PDE. The selected POD modes are further reduced through an autoencoder. The selected minimum number of non-linear projections have a one-to-one correspondence with the first few POD modes while reducing the loss function. The POD modes are used to evolve the solution in time using NN. We also use a NN to learn the functional form of the PDE and use the learned PDE to predict the dynamics. The closure terms in the averaged multiphase flow equations are predicted using NN and the predicted PDE is used to evolve in time the velocity and the void fraction, in another method. The developed models are used to predict the dynamics of flows with different initial and boundary conditions.