Projection-based Reduced-order Modeling Research Articles

Projection-based reduced order modeling of multi-species mixing and combustion

High-fidelity simulations of mixing and combustion processes are computationally demanding and time-consuming, hindering their wide application in industrial design and optimization. This study proposes projection-based reduced order models (ROMs) to predict spatial distributions of physical fields for multi-species mixing and combustion problems in a fast and accurate manner. The developed ROMs explore the suitability of various regression methods, including kriging, multivariate polynomial regression (MPR), k-nearest neighbors (KNN), deep neural network (DNN), and support vector regression (SVR), for the functional mapping between input parameters and reduced model coefficients of mixing and combustion problems. The ROMs are systematically examined using two distinct configurations: steam-diluted hydrogen-enriched oxy-combustion from a triple-coaxial nozzle and fuel-flexible combustion in a practical gas-turbine combustor. The projected low-dimensional manifolds are capable of capturing important combustion physics, and the response surfaces of reduced model coefficients present pronounced nonlinear characteristics of the flowfields with varying input parameters. The ROMs with kriging present a superior performance of establishing the input–output mapping to predict almost all physical fields, such as temperature, velocity magnitude, and combustion products for both test problems. The accuracy of DNN is less encouraging owing to the stringent requirement on the size of training database. KNN performs well in the region near the design points but its effectiveness diminishes when the test points are distant from the sampling points, whereas SVR and MPR exhibit large prediction errors. For the spatial prediction at unseen design points, the ROMs achieve a prediction time of up to eight orders of magnitude faster than conventional numerical simulations, rendering an efficient tool for the fast prediction of mixing and combustion fields and potentially an alternative of a full-order numerical solver.

Projection-based reduced order modeling and data-driven artificial viscosity closures for incompressible fluid flows

Projection-based reduced order models rely on offline–online model decomposition, where the data-based energetic spatial basis is used in the expensive offline stage to obtain equations of reduced states that evolve in time during the inexpensive online stage. The online stage requires a solution method for the dynamic evolution of the coupled system of pressure and velocity states for incompressible fluid flows. The first contribution of this article is to demonstrate the applicability of the incremental pressure correction scheme for the dynamic evolution of pressure and velocity states. The evolution of a large number of these reduced states in the online stage can be expensive. In contrast, the accuracy significantly decreases if only a few reduced states are considered while not accounting for the interactions between unresolved and resolved states. The second contribution of this article is to compare three closure model forms based on global, modal and tensor artificial viscosity approximation to account for these interactions. The unknown model parameters are determined using two calibration techniques: least squares minimization of error in energy approximation and closure term approximation. This article demonstrates that an appropriate selection of solution methods and data-driven artificial viscosity closure models is essential for consistently accurate dynamics forecasting of incompressible fluid flows.

Projection-based reduced order modeling of an iterative scheme for linear thermo-poroelasticity

This paper explores an iterative approach to solve linear thermo-poroelasticity problems, with its application as a high-fidelity discretization utilizing finite elements during the training of projection-based reduced order models. One of the main challenges in addressing coupled multi-physics problems is the complexity and computational expenses involved. In this study, we introduce a decoupled iterative solution approach, integrated with reduced order modeling, aimed at augmenting the efficiency of the computational algorithm. The iterative technique we employ builds upon the established fixed-stress splitting scheme that has been extensively investigated for Biot’s poroelasticity. By leveraging solutions derived from this coupled iterative scheme, the reduced order model employs an additional Galerkin projection onto a reduced basis space formed by a small number of modes obtained through proper orthogonal decomposition. The effectiveness of the proposed algorithm is demonstrated through numerical experiments, showcasing its computational prowess.

DRIPS: A framework for dimension reduction and interpolation in parameter space

Reduced-order models are often used to describe the behavior of complex systems, whose simulation with a full model is too expensive, or to extract salient features from the full model's output. We introduce a new model-reduction framework DRIPS (dimension reduction and interpolation in parameter space) that combines the offline local model reduction with the online parameter interpolation of reduced-order bases (ROBs). The offline step of this framework relies on dynamic mode decomposition (DMD) to build a low-rank linear surrogate model, equipped with a local ROB, for quantities of interest derived from the training data generated by repeatedly solving the (nonlinear) high-fidelity model for multiple parameter points. The online step consists of the construction of a parametric reduced-order model for each target/test point in the parameter space, with the interpolation of ROBs done on a Grassman manifold and the interpolation of reduced-order operators done on a matrix manifold. The DMD component enables DRIPS to model (typically low-dimensional) quantities of interest directly, without having to access the (typically high-dimensional and possibly nonlinear) operators in a high-fidelity model that governs the dynamics of the underlying high-dimensional state variables, as required in projection-based reduced-order modeling. A series of numerical experiments suggests that DRIPS yields a model reduction, which is computationally more efficient than the commonly used projection-based proper orthogonal decomposition; it does so without requiring a prior knowledge of the governing equation for quantities of interest. Moreover, for the nonlinear systems considered, DRIPS is more accurate than Gaussian-process interpolation (Kriging).

Real-Time Response Estimation of Structural Vibration with Inverse Force Identification

Virtual sensing is an effective method to identify the inaccessible state of the structural systems by compensating the limitations of the conventional physical sensing techniques. Recently, it becomes popular in structural vibration field and their relevant physical domains such as civil, mechanical, aerospace engineering, thermal dynamics, and acoustics. This study aimed to develop a virtual sensing algorithm of structural vibration for the real-time identification of unmeasured information. First, certain local point vibration responses (such as displacement and acceleration) are measured using physical sensors, and the data sets are extended using a numerical model to cover the unmeasured quantities through the entire spatial domain in the real-time computation process. A modified time integrator is then proposed to synchronize the physical sensors and the numerical model using inverse dynamics. In particular, an efficient inverse force identification method is derived using implicit time integration. The second-order ordinary differential formulation and its projection-based reduced-order modeling are used to avoid two times larger degrees of freedom within the state-space form. Then, the Tikhonov regularization noise-filtering algorithm is employed instead of Kalman filtering. The performance of the proposed method is investigated on both numerical and experimental testbeds under sinusoidal and random excitation loading conditions. In the numerical test, the system could identify the status of the motor housing structure in the speed of 16,181 S a m p l e s / s . The FDE and RMSE values are bounded under 0.1816 and 1503.2. In the case of the experimental test, the algorithm is implemented to the beam structure using a single-board computer, including inverse force identification and unmeasured response prediction. Even in the limited computational environment, the system could identify the applied forces in real time in the speed of 2,173 S a m p l e s / s . Through all experimental cases, FDE and RMSE values are bounded under 0.2925 and 0.1660. The results show that the virtual sensing algorithm can accurately identify unmeasured information, forces, and displacements throughout the vibration model in real time in a very limited computing environment.

A Bayesian Nonlinear Reduced Order Modeling Using Variational AutoEncoders

This paper presents a new nonlinear projection based model reduction using convolutional Variational AutoEncoders (VAEs). This framework is applied on transient incompressible flows. The accuracy is obtained thanks to the expression of the velocity and pressure fields in a nonlinear manifold maximising the likelihood on pre-computed data in the offline stage. A confidence interval is obtained for each time instant thanks to the definition of the reduced dynamic coefficients as independent random variables for which the posterior probability given the offline data is known. The parameters of the nonlinear manifold are optimized as the ones of the decoder layers of an autoencoder. The parameters of the conditional posterior probability of the reduced coefficients are the ones of the encoder layers of the same autoencoder. The optimization of both sets of the encoder and the decoder parameters is obtained thanks to the application of a variational Bayesian method, leading to variational autoencoders. This Reduced Order Model (ROM) is not a regression model over the offline pre-computed data. The numerical resolution of the ROM is based on the Chorin projection method. We apply this new nonlinear projection-based Reduced Order Modeling (ROM) for a 2D Karman Vortex street flow and a 3D incompressible and unsteady flow in an aeronautical injection system.

Projection-tree reduced-order modeling for fast N-body computations

This work presents a data-driven reduced-order modeling framework to accelerate the computations of nonlocal and N-body methods that model dynamical systems. The proposed framework differs from traditional acceleration methods, like the Barnes–Hut method, which requires online tree building of the state space, or the fast-multipole method, which requires rigorous a priori analysis of governing kernels and online tree building. Our approach combines Barnes–Hut hierarchical decomposition, projection-based reduced-order modeling via the least-squares Petrov–Galerkin (LSPG) projection, and hyper-reduction by way of the Gauss–Newton with approximated tensor (GNAT) approach. The resulting projection-tree reduced-order model (PTROM) enables a drastic reduction in operational count complexity by constructing sparse hyper-reduced pairwise interactions of the non-compact N-body dynamical system. As a result, the presented framework is capable of achieving an operational count complexity that is independent of N, the number of bodies in the numerical domain. Capabilities of the PTROM method are demonstrated on the two-dimensional fluid-dynamic Biot–Savart kernel within a parametric and reproductive setting. Results show the PTROM is capable of achieving over 2000× wall-time speed-up with respect to the full-order model, where the speed-up increases with N. The resulting solution delivers quantities of interest with errors that are less than 0.1% with respect to full-order model.

Large-eddy simulation and challenges for projection-based reduced-order modeling of a gas turbine model combustor

Computationally efficient modeling of gas turbine combustion is challenging due to the chaotic multi-scale physics and the complex non-linear interactions between acoustic, hydrodynamic, and chemical processes. A large-eddy simulation, referred to as the full order model (FOM), is performed for a gas turbine model combustor with turbulent combustion effects modeled using a flamelet-based method. Modal analysis reveals a high degree of correlation with averaged and instantaneous high-frequency particle image velocimetry fields. The dynamics of the precessing vortex core is quantitatively characterized using dynamic mode decomposition. The governing equations of the FOM are projected onto a low-dimensional linear manifold to construct a reduced-order model (ROM). A discretely-consistent least squares projection is used to guarantee global stability. The ROM provides an accurate reconstruction of the combustion dynamics within the training region, but faces a significant challenge in future state predictions. This limitation is mainly due to the increased projection error, which in turn is a direct consequence of the highly chaotic nature of the flow field, involving a wide range of dispersed coherent structures. This shortcoming is overcome using an adaptive basis method which yields accurate predictions of dynamics beyond the training region consistent with the FOM. Formal projection-based ROMs have not been applied to a problem of this scale and complexity, and achieving accurate and efficient ROMs is a grand challenge problem. A production-ready ROM method will significantly decrease the computational cost of the flame dynamics as well as the portability of this prediction to smaller-scale computers.

Surrogate-assisted inversion for large-scale history matching: Comparative study between projection-based reduced-order modeling and deep neural network

History matching can play a key role in improving geological characterization and reducing the uncertainty of reservoir model predictions. Application of reservoir history matching is restricted by the huge computational cost by amongst others the many runs of the full model. Surrogate models with a reduced complexity are therefore used to reduce the computational demands. This paper presents an efficient surrogate-assisted deterministic inversion framework to primarily explore the possibility of applying deep neural network (DNN) surrogate to approximate the gradient of large-scale history matching by using auto-differentiation (AD). In combination with the deep neural network model, the AD enables us to evaluate the gradients efficiently in a parallel manner. Furthermore, the benefits of using stochastic gradient optimizers in the deep learning practice, instead of full gradient optimizers in conventional deterministic inversions, is investigated as well. Numerical experiments are conducted on a 3D benchmark reservoir model in the context of a water-flooding production scenario. The quantity of interest, e.g., dynamic saturation for an ensemble of test models, can be accurately predicted. The proposed surrogate-assisted inversion with stochastic gradient optimizer obtains a very quick convergence rate against the model and data noise for the high-dimensional history matching problem with a large number of data and parameters. In addition, we also conduct several comparisons and evaluations with our previously proposed projection-based subdomain POD-TPWL approach in terms of computational efficiency and accuracy. The subdomain POD-TPWL constructs a local surrogate model, which is repeatedly reconstructed a number of times for maintaining a satisfactory accuracy, while DNN constructs a global surrogate model based on the entire training data and generally does not require additional reconstructions. The subdomain POD-TPWL is very sensitive to how the domain is decomposed, increasing the training samples does not infinitely improve the history matching results by a fixed decomposition. Overall, these two kinds of surrogate models have demonstrated great potential in solving large-scale history matching problem. The DNN surrogate is particularly useful to generate multiple posteriors for model uncertainty quantification. • Deep neural network is used to approximate the gradient of seismic history matching by using auto-differentiation (AD). • Deep-learning inversion with stochastic gradient optimizer demonstrates good convergence performance against both model and data noise. • This method demonstrates a significant reduction in computational cost while reasonable accuracy remains for seismic history matching.

Adjoint and Acceleration Methods for Projection-based Reduced Order Modeling

Adjoint and Acceleration Methods for Projection-based Reduced Order Modeling

Model fusion with physics-guided machine learning: Projection-based reduced-order modeling

The unprecedented amount of data generated from experiments, field observations, and large-scale numerical simulations at a wide range of spatiotemporal scales has enabled the rapid advancement of data-driven and especially deep learning models in the field of fluid mechanics. Although these methods are proven successful for many applications, there is a grand challenge of improving their generalizability. This is particularly essential when data-driven models are employed within outer-loop applications like optimization. In this work, we put forth a physics-guided machine learning (PGML) framework that leverages the interpretable physics-based model with a deep learning model. Leveraging a concatenated neural network design from multi-modal data sources, the PGML framework is capable of enhancing the generalizability of data-driven models and effectively protects against or inform about the inaccurate predictions resulting from extrapolation. We apply the PGML framework as a novel model fusion approach combining the physics-based Galerkin projection model and long- to short-term memory (LSTM) network for parametric model order reduction of fluid flows. We demonstrate the improved generalizability of the PGML framework against a purely data-driven approach through the injection of physics features into intermediate LSTM layers. Our quantitative analysis shows that the overall model uncertainty can be reduced through the PGML approach, especially for test data coming from a distribution different than the training data. Moreover, we demonstrate that our approach can be used as an inverse diagnostic tool providing a confidence score associated with models and observations. The proposed framework also allows for multi-fidelity computing by making use of low-fidelity models in the online deployment of quantified data-driven models.

‘On-the-fly’ snapshots selection for Proper Orthogonal Decomposition with application to nonlinear dynamics

Over the last few decades, Reduced Order Modeling (ROM) has slowly but surely inched towards widespread acceptance in computational mechanics, as well as other simulation-based fields. Projection-based Reduced Order Modeling (PROM) relies on the construction of an appropriate Reduced Basis (RB), which is typically a low-rank representation of a set of “observations” made using full-field simulations, usually obtained through truncated Singular Value Decomposition (SVD). However, SVD encounters limitations when dealing with a large number of high-dimensional observations, requiring the development of alternatives such as the incremental SVD. The key advantages of this approach are reduced computational complexity and memory requirement compared to a regular “single pass” spectral decomposition. These are achieved by only using relevant observations to enrich the low-rank representation as and when available, to avoid having to store them. In addition, the RB may be truncated ’on-the-fly’ so as to reduce the size of the matrices involved as much as possible and, by doing so, avoid the quadratic scale-up in computational effort with the number of observations. In this paper, we present a new error estimator for the incremental SVD, which is shown to be an upper bound for the approximation error, and propose an algorithm to perform the incremental SVD truncation and observation selection ’on-the-fly’, instead of using a prohibitively large number of frequently “hard to set” parameters. The performance of the approach is discussed on the reduced-order Finite Element (FE) model simulation of impact on a Taylor beam.

The Adjoint Petrov–Galerkin method for non-linear model reduction

We formulate a new projection-based reduced-order modeling technique for non-linear dynamical systems. The proposed technique, which we refer to as the Adjoint Petrov–Galerkin (APG) method, is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. A Markovian finite memory assumption within the Mori–Zwanzig formalism is then used to develop a reduced-order representation of the coarse scales. This procedure leads to a closed reduced-order model that displays commonalities with the adjoint stabilization method used in finite elements. The formulation is shown to be equivalent to a Petrov–Galerkin method with a non-linear, time-varying test basis, thus sharing some similarities with the Least-Squares Petrov–Galerkin method. Theoretical analysis examining a priori error bounds and computational cost is presented. Numerical experiments on the compressible Navier–Stokes equations demonstrate that the proposed method can lead to improvements in numerical accuracy, robustness, and computational efficiency over the Galerkin method on problems of practical interest. Improvements in numerical accuracy and computational efficiency over the Least-Squares Petrov–Galerkin method are observed in most cases.

Koopman-Based Approach to Nonintrusive Reduced Order Modeling: Application to Aerodynamic Shape Optimization and Uncertainty Propagation

A methodology for nonintrusive projection-based nonlinear model reduction originally presented by Renganathan et al. (“Koopman-Based Approach to Nonintrusive Projection-Based Reduced-Order Modeling with Black-Box High-Fidelity Models,” AIAA Journal, Vol. 56, No. 10, 2018, pp. 4087–4111) is further extended toward parametric systems with a focus on application to aerospace design. Specifically, the method is extended to address static systems with parametric geometry (that deforms the mesh) in addition to parametric freestream boundary conditions. The main idea is to first perform a transformation on the governing equations such that it is lifted to a higher-dimensional but linear underdetermined system. This enables one to extract the system matrices easily as compared to that of the original nonlinear system. The underdetermined system is closed with a set of model-dependent nonlinear constraints upon which the model reduction is finally performed. The methodology is validated on the subsonic and transonic inviscid flows past the NACA0012 and the RAE2822 airfoils with parametrized shapes. The utility of the approach is further demonstrated by applying it to two common problems in aerospace design, namely, derivative-free global optimization and parametric uncertainty quantification with Monte Carlo sampling. Overall, the methodology is shown to achieve accuracy up to 5% and computational speedup of two to three orders of magnitude relative to the full order model. Comparison against another nonintrusive model reduction method revealed that the proposed approach is more robust, accurate, and retains the consistency between the state variables.

On-the-fly model reduction for large-scale structural topology optimization using principal components analysis

Despite a solid theoretical foundation and straightforward application to structural design problems, 3D topology optimization still suffers from a prohibitively high computational effort that hinders its widespread use in industrial design. One major contributor to this problem is the cost of solving the finite element equations during each iteration of the optimization loop. To alleviate this cost in large-scale topology optimization, the authors propose a projection-based reduced-order modeling approach using proper orthogonal decomposition for the construction of a reduced basis for the FE solution during the optimization, using a small number of previously obtained and stored solutions. This basis is then adaptively enriched and updated on-the-fly according to an error residual, until convergence of the main optimization loop. The method of moving asymptotes is used for the optimization. The techniques are validated using established 3D benchmark problems. The numerical results demonstrate the advantages and the improved performance of our proposed approach.

Nonintrusive reduced order modeling framework for quasigeostrophic turbulence.

In this study, we present a nonintrusive reduced order modeling (ROM) framework for large-scale quasistationary systems. The framework proposed herein exploits the time series prediction capability of long short-term memory (LSTM) recurrent neural network architecture such that (1) in the training phase, the LSTM model is trained on the modal coefficients extracted from the high-resolution data snapshots using proper orthogonal decomposition (POD) transform, and (2) in the testing phase, the trained model predicts the modal coefficients for the total time recursively based on the initial time history. Hence, no prior information about the underlying governing equations is required to generate the ROM. To illustrate the predictive performance of the proposed framework, the mean flow fields and time series response of the field values are reconstructed from the predicted modal coefficients by using an inverse POD transform. As a representative benchmark test case, we consider a two-dimensional quasigeostrophic ocean circulation model which, in general, displays an enormous range of fluctuating spatial and temporal scales. We first demonstrate that the conventional Galerkin projection-based reduced order modeling of such systems requires a high number of POD modes to obtain a stable flow physics. In addition, ROM-Galerkin projection (ROM-GP) does not seem to capture the intermittent bursts appearing in the dynamics of the first few most energetic modes. However, the proposed nonintrusive ROM framework based on LSTM (ROM-LSTM) yields a stable solution even for a small number of POD modes. We also observe that the ROM-LSTM model is able to capture quasiperiodic intermittent bursts accurately, and yields a stable and accurate mean flow dynamics using the time history of a few previous time states, denoted as the lookback time window in this paper. We show several features of ROM-LSTM framework such as significantly higher accuracy than ROM-GP, and faster performance using larger time step size. Throughout the paper, we demonstrate our findings in terms of time series evolution of the field values and mean flow patterns, which suggest that the proposed fully nonintrusive ROM framework is robust and capable of predicting chaotic nonlinear fluid flows in an extremely efficient way compared to the conventional projection-based ROM framework.

Stabilized principal interval decomposition method for model reduction of nonlinear convective systems with moving shocks

An efficient and reliable model order reduction of nonlinear systems poses a challenge for nonstationary problems with convective, non-periodic, and non-equilibrium dynamics. To that end, we put forth a localized basis selection strategy based on the proper orthogonal decomposition (POD) and principal interval decomposition (PID) to construct a stable reduced-order modeling framework to capture the unsteady dynamics of nonlinear systems effectively. The implementation of an eddy viscosity (EV) based closure model in POD–PID approach yields the proposed POD–PID–EV projection-based reduced-order modeling approach for nonlinear partial differential equations. Solving the nonlinear Burgers’ equation with various spatio-temporal dynamical complexities, it is shown that the present approach yields significant improvements in accuracy over the standard POD–Galerkin model with a negligibly small computational overhead. Furthermore, we show that strong moving discontinuities can be effectively captured in the low-dimensional space with the proposed approach.

Koopman-Based Approach to Nonintrusive Projection-Based Reduced-Order Modeling with Black-Box High-Fidelity Models

This paper presents a methodology that enables projection-based model reduction for black-box high-fidelity models governing nonlinear static parametric systems. The methodology specifically addresses the situation in which the high-fidelity model may be a black box, but there is knowledge of the governing equations in continuous partial differential equation (PDE) form. Drawing from the Koopman theory, it first obtains an underdetermined linear representation of the governing equations in terms of a set of observables. Then, the linear operator is extracted by a direct discretization of the linear differential terms using a suitable method, such as the finite volume method. By applying the snapshots of the observables to the discrete linear operator, a right-hand-side vector is obtained, providing the necessary system matrices for the projection step, which is done via proper orthogonal decomposition. The underdetermined linear system is closed with a set of nonlinear algebraic equations that act as constraints, whose computation is efficiently handled via the discrete empirical interpolation method. An offline database of reduced-order models (ROMs) corresponding to each parameter snapshot is generated, which are then interpolated online to predict the state for new parameter instances. The resulting ROM is posed and solved as a nonlinear constrained optimization problem. The method is tested on a canonical PDE, followed by the 2-D compressible Euler equations governing the flow past an airfoil, under both the subsonic and transonic regimes. It is demonstrated that the ROM predicts both the state and outputs within 5% of the full-order model given adequate snapshots and the computational speedup is up to two to three orders of magnitude.

Neural network closures for nonlinear model order reduction

Many reduced-order models are neither robust with respect to parameter changes nor cost-effective enough for handling the nonlinear dependence of complex dynamical systems. In this study, we put forth a robust machine learning framework for projection-based reduced-order modeling of such nonlinear and nonstationary systems. As a demonstration, we focus on a nonlinear advection-diffusion system given by the viscous Burgers equation, which is a prototypical setting of more realistic fluid dynamics applications due to its quadratic nonlinearity. In our proposed methodology the effects of truncated modes are modeled using a single layer feed-forward neural network architecture. The neural network architecture is trained by utilizing both the Bayesian regularization and extreme learning machine approaches, where the latter one is found to be more computationally efficient. A significant emphasis is laid on the selection of basis functions through the use of both Fourier bases and proper orthogonal decomposition. It is shown that the proposed model yields significant improvements in accuracy over the standard Galerkin projection methodology with a negligibly small computational overhead and provide reliable predictions with respect to parameter changes.

Non-intrusive Sparse Subspace Learning for Parametrized Problems

We discuss the use of hierarchical collocation to approximate the numerical solution of parametric models. With respect to traditional projection-based reduced order modeling, the use of a collocation enables non-intrusive approach based on sparse adaptive sampling of the parametric space. This allows to recover the low-dimensional structure of the parametric solution subspace while also learning the functional dependency from the parameters in explicit form. A sparse low-rank approximate tensor representation of the parametric solution can be built through an incremental strategy that only needs to have access to the output of a deterministic solver. Non-intrusiveness makes this approach straightforwardly applicable to challenging problems characterized by nonlinearity or non affine weak forms. As we show in the various examples presented in the paper, the method can be interfaced with no particular effort to existing third party simulation software making the proposed approach particularly appealing and adapted to practical engineering problems of industrial interest.