Session ZC29: Modeling Methods IV: Data-driven and Machine-Learning Techniques

Shape-morphing modes for reduced-order modeling of advection-dominated flows with shallow neural networks

Reduced-order modeling of fluid flows that are dominated by advection is notoriously difficult. We introduce a new method which tackles this issue by incorporating time-dependent shifts in the modes to which the flow is reduced. The evolution of the shift parameter is determined automatically using the method of reduced-order nonlinear solutions (RONS). We show that these shifts are equivalent to a rotation of the reduced linear subspace and can be interpreted as a shallow neural network with time-dependent biases. In addition, any number of conserved quantities of the flow can be readily enforced in the reduced model. We demonstrate the application of our method on a number of examples, including vortex dynamics and surface waves.   

A phase-based proper orthogonal decomposition that accounts for intrinsic large scale motion

A number of important fluid flows are driven by an intrinsic large scale within the flow, whose dynamics modulate the behavior of other flow scales via nonlinear mechanisms. Traditional modal analysis approaches, such as snapshot proper orthogonal decomposition (POD),  represent this behavior inefficiently, using a cascade of modes even when representing a small number of physical scales interacting. To address this issue, we present a new space-phase POD method that extracts modes informed by this nonlinear interplay, within a formal POD framework. The modes utilize a transformation between time and phase of the large scale motion to create modes that coherently evolve along the large scale’s dynamics. We demonstrate the method’s utility using two examples: a low-Reynolds-number bluff-body flow and a more complex turbulent shock problem, using data from Duvvuri et al (“Large- and small-amplitude shock-wave oscillations over axisymmetric bodies in high-speed flow”, Journal of Fluid Mechanics, 913 (2021)). The proposed approach distills the large-scale behavior in the former example, and in the latter example yields modes that represent both the shock motion that drives large scale dynamics as well as the smaller scale turbulence occurring about the shock. In both cases, the space-phase POD technique is able to capture additional information about the dynamic structures within the flow as compared to classical data-driven techniques.   

Optimal linear model reduction using SPOD modes

Spectral proper orthogonal decomposition (SPOD) modes provide an optimal linear representation of the long-term evolution of stationary flows as measured by a space-time norm. In other words, the truncated SPOD representation of a trajectory becomes more accurate, on average, than the representation in any other space-time basis as the time interval becomes long. We present a method to solve for the exact SPOD coefficients that represent a trajectory in forced linear systems, thereby obtaining this optimal representation given the initial condition and forcing. The method works by projecting the unreduced equations in the frequency domain onto the SPOD modes at each frequency, and may be formulated as a frequency domain Petrov-Galerkin method. The method requires an ensemble of realizations of the system from which to calculate the SPOD modes and the entire time series of the forcing for the realization to be calculated. With these inputs, the method is observed to be substantially more accurate than standard methods, such as POD-Galerkin and balanced truncation, which is expected given the SPOD optimality property. It also scales favorably to large systems, and is significantly faster than standard linear model reduction methods.   

Dynamics-preserving compression for modal flow analysis

Numerical simulations of complex, multi-physics flow problems frequently produce large datasets, with the analysis of these datasets often constrained by the memory required to manipulate the data using standard decomposition algorithms. Many existing compression algorithms succeed at reducing spatial complexity but usually distort the underlying dynamics. We propose a dynamics-preserving compression technique, based on locality-sensitive hashing, that results in considerable dimensional reduction while controlling the distorting of the dynamics within a user-specified threshold. We apply this technique to a model turbomachinery flow and extract coherent modal structures covering proper orthogonal (POD) and dynamic modes (DMD). Compression rates of up to two orders of magnitudes can be achieved with, for example, only a one-percent distortion of the dynamics. This technique can be viewed as an alternative to Hankelized systems that encode the snapshot dynamics at the expense of increased spatial dimensionality. Extensions and further developments of the method will be discussed as well.   

Interpolatory input and output projections for flow control

Eigenvectors of the observability and controllability Gramians represent responsive and receptive flow structures that enjoy a well-established connection to resolvent forcing and response modes. However, whereas resolvent modes have demonstrated great potential to guide sensor and actuator placement, observability and controllability modes have been exclusively leveraged in the context of model reduction via input and output projections. In this work, we introduce interpolatory, rather than orthogonal, input and output projections, that can be leveraged for sensor and actuator placement and open-loop control design. An interpolatory projector is an oblique projector with the property of preserving certain entries in the vector being projected. We review the connection between the resolvent operator and the Gramians and present several numerical examples where we perform both orthogonal and interpolatory input and output projections onto the dominant forcing and response subspaces. Input projections are used to identify dynamically relevant disturbances, place sensors to measure disturbances, and place actuators for feedforward control in the linearized Ginzburg--Landau equation. Output projections are used to identify coherent structures and place sensors aiming at state reconstruction in the turbulent flow in a minimal channel at Re_τ=185. The framework does not require data snapshots and relies only on knowledge of the steady or mean flow.   

A parameterized LSTM deep neural network framework to model unsteady flow problems

We propose a deep learning framework that can predict the space-time evolution of complex flow problems across a range of parametric regimes. Our approach is based on a Proper Orthogonal Decomposition (POD) for dimensionality reduction combined with a Long-Short Term Memory (LSTM) deep learning neural network for the temporal modeling of dynamical systems. The specific contributions of our work are focused on the LSTM architecture, where the problem parameters, such as those governing flow, body shape, or body kinematics, are considered independent inputs to the LSTM deep learning neural network. This enables the LSTM network to predict different flow states within a wide problem space as defined by the parametrization, and/or switch dynamically between them. We demonstrate the benefits of this approach on the 2D modeling of flow past a flapping ellipse and show that our approach is capable of real-time modeling flow patterns across a set of kinematic parameters.   

Data-driven closure of the harmonic-balanced Navier-Stokes equations in the frequency domain

The Fourier-Galerkin method is employed to calculate the multifrequency and multiscale asymptotic nonlinear flow response in the frequency domain, by expanding the solution as Fourier series. The resulting equations are known as the harmonic-balanced Navier-Stokes (HBNS) equations. Although near the threshold of transition a small number of harmonics suffice to achieve convergence, further away the computational cost becomes intractable because energy is transferred to higher harmonics, which can no longer be neglected. In this study, we propose a data-driven framework to model the residual (nonlocal closure) terms for the frequency-truncated HBNS equations. By splitting the sought solution into low-frequency (resolved) and high-frequency (unresolved) harmonics, we systematically express the low-frequency residual as a function of the resolved frequency harmonics only. A consistent deep learning architecture, which parameterizes the residual function, is designed and trained using high-fidelity results near the thresholds of transition for two-dimensional (2D) cylinder flow. We show that our proposed framework achieves low generalization error by predicting accurately the coarse-grained residual for unseen Reynolds numbers, and significantly reduces the computational cost by solving accurately for the coarse-grained dynamics.   

A Shift Procedure for Identifying Low Rank Behavior from Non-Stationary Dynamical System Data

Many model order reduction techniques for dynamical systems leverage pe-

riodicity or statistical stationarity properties in collected data to identify the

underlying physics and to build reduced order models. However, many sys-

tems of interest, such as scramjet unstart, do not exhibit these properties and

traditional approaches to applying data-driven decomposition techniques, such

as Proper Orthogonal Decomposition (POD), yield results that are difficult to

interpret, or use for reduced order models. However, shifting the data into a

suitably defined reference frame that travels with an identified feature of interest

can recover many of the attractive properties of typical decomposition methods,

at least to first-order. We propose a robust, data-driven way to execute this shift

by computing an instantaneously varying translational velocity vector from the

Empirical Mode Decomposition (EMD) of time series data which tracks the lo-

cation of targeted space-time features. Application of decomposition techniques

in the shifting reference frame recovers many of the low rank dynamics that

are not easily apparent in the original data. The approach is demonstrated by

application to four example problems ranging in complexity from a convecting

Gaussian pulse to unstarting shocks in a scramjet isolator.   

A co-kurtosis PCA based dimensionality reduction with neural network reconstruction for chemical kinetics in reacting flows

Identifying low-dimensional manifolds (LDMs) to represent the thermo-chemical state in reacting flows is crucial for significantly reducing the computational cost. The widely used principal component analysis (PCA) achieves this by obtaining an eigenvector basis for the LDM through an eigenvalue decomposition of the data covariance matrix. However, recent studies have revealed that PCA is not very sensitive to extreme-valued samples representing stiff chemical dynamics in spatiotemporally localized reaction zones. An alternative technique that focuses on higher-order joint statistical moments, co-kurtosis PCA (CoK-PCA), has demonstrated remarkable accuracy in capturing stiff chemical dynamics. However, the effectiveness of the CoK-PCA method has been comparatively assessed with PCA only in an a priori setting with a linear reconstruction method. In this work, we employ a nonlinear artificial neural network (ANN) based technique for reconstructing the original thermo-chemical state and evaluate the quality of the CoK-PCA LDM compared to PCA. Results from the a priori analyses of different datasets, which include a two-stage auto-ignition of dimethyl ether-air mixture, demonstrate the robustness of the CoK-PCA-ANN approach in accurately capturing the overall thermo-chemistry.   

Toward Real-Time Simulation of Cardiovascular Flows by Introducing a Stabilized Frequency Finite Element Methods

The finite element methods for the solution of the Navier-Stokes equation have found common use for simulating cardiovascular flows. These simulations typically use periodic boundary conditions for physiological relevance. This results in a solution that is unsteady and often periodic. To capture this behavior, a conventional finite element method uses time-stepping to resolve the unsteady behavior of the flow to obtain cycle-to-cycle convergence. As a result, the overall cost of these simulations is significant. In fact, for most cardiovascular CFD simulations, more than 90% of the computational cost is spent on numerical convergence due to its unsteady and periodic nature. 

In this talk we propose an alternative way of simulating these flows in the frequency rather than the time domain, thereby obviating the need to simulate many cycles for convergence. In addition, since the periodic flows can be represented with a few modes in the frequency domain, we no longer need to rely on thousands of time steps to resolve the unsteady nature of these flows. We show that our formulation reduces the overall cost of a cardiovascular simulation by several orders of magnitude from hours or days to seconds or minutes. In this talk, we describe the underlying numerical method and compare it against the conventional time formulation when simulating hemodynamics in Norwood operation at low Reynolds number.