Model Order Reduction Of Systems Research Articles

Gradient preserving Operator Inference: Data-driven reduced-order models for equations with gradient structure

Hamiltonian Operator Inference has been introduced in Sharma et al. (2022) to learn structure-preserving reduced-order models (ROMs) for Hamiltonian systems. This approach constructs a low-dimensional model using only data and knowledge of the Hamiltonian function. Such ROMs can keep the intrinsic structure of the system, allowing them to capture the physics described by the governing equations. In this work, we extend this approach to more general systems that are either conservative or dissipative in energy, and which possess a gradient structure. We derive the optimization problems for inferring structure-preserving ROMs that preserve the gradient structure. We further derive an a priori error estimate for the reduced-order approximation. To test the algorithms, we consider semi-discretized partial differential equations with gradient structure, such as the parameterized wave and Korteweg–de-Vries equations, and equations of three-dimensional linear elasticity in the conservative case and the one- and two-dimensional Allen-Cahn equations in the dissipative case. The numerical results illustrate the accuracy, structure-preservation properties, and predictive capabilities of the gradient-preserving Operator Inference ROMs.

Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale dynamical systems

This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Lagrangian systems, which includes nonlinear wave equations. Existing intrusive projection-based model reduction approaches construct structure-preserving Lagrangian ROMs by projecting the Euler–Lagrange equations of the full-order model (FOM) onto a linear subspace. This Galerkin projection step requires complete knowledge about the Lagrangian operators in the FOM and full access to manipulate the computer code. In contrast, the proposed Lagrangian operator inference approach embeds the mechanics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying Lagrangian structure. The proposed approach exploits knowledge of the governing equations (but not their discretization) to define the form and parametrization of a Lagrangian ROM which can then be learned from projected snapshot data. The method does not require access to FOM operators or computer code. The numerical results demonstrate Lagrangian operator inference on an Euler–Bernoulli beam model, the sine-Gordon (nonlinear) wave equation, and a large-scale discretization of a soft robot fishtail with 779,232 degrees of freedom. The learned Lagrangian ROMs generalize well, as they can accurately predict the physical solutions both far outside the training time interval, as well as for unseen initial conditions.

Bayesian Proper Orthogonal Decomposition for Learnable Reduced-Order Models With Uncertainty Quantification

Designing and/or controlling complex systems in science and engineering relies on appropriate mathematical modeling of systems dynamics. Classical differential equation based solutions in applied and computational mathematics are often computationally demanding. Recently, the connection between reduced-order models of high-dimensional differential equation systems and surrogate machine learning models has been explored. However, the focus of both existing reduced-order and machine learning models for complex systems has been how to best approximate the high fidelity model of choice. Due to high complexity and often limited training data to derive reduced-order or machine learning surrogate models, it is critical for derived reduced-order models to have reliable uncertainty quantification at the same time. In this paper, we propose such a novel framework of Bayesian reduced-order models naturally equipped with uncertainty quantification as it learns the distributions of the parameters of the reduced-order models instead of their point estimates. In particular, we develop <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">learnable Bayesian proper orthogonal decomposition</i> (BayPOD) that learns the distributions of both the POD projection bases and the mapping from the system input parameters to the projected scores/coefficients so that the learned BayPOD can help predict high-dimensional systems dynamics/fields as quantities of interest in different setups with reliable uncertainty estimates. The developed learnable BayPOD inherits the capability of embedding physics constraints when learning the POD-based surrogate reduced-order models, a desirable feature when studying complex systems in science and engineering applications where the available training data are limited. Furthermore, the proposed BayPOD method is an end-to-end solution, which unlike other surrogate-based methods, does not require separate POD and machine learning steps. The results from a real-world case study of the pressure field around an airfoil have shown the potential of learnable BayPOD as a new family of reduced-order models with reliable uncertainty estimates.

Learning physics-based reduced-order models from data using nonlinear manifolds.

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

A Control Interface for Autonomous Positioning of Magnetically Actuated Spheres Using an Artificial Neural Network

Electromagnet arrays show significant potential in the untethered guidance of particles, devices, and eventually robots. However, complications in obtaining accurate models of electromagnetic fields pose challenges for precision control. Manipulation often requires the reduced-order modeling of physical systems, which may be computationally complex and may still not account for all possible system dynamics. Additionally, control schemes capable of being applied to electromagnet arrays of any configuration may significantly expand the usefulness of any control approach. In this study, we developed a data-driven approach to the magnetic control of a neodymium magnets (NdFeB magnetic sphere) using a simple, highly constrained magnetic actuation architecture. We developed and compared two regression-based schemes for controlling the NdFeB sphere in the workspace of a four-coil array of electromagnets. We obtained averaged submillimeter positional control (0.85 mm) of a NdFeB hard magnetic sphere in a 2D plane using a controller trained using a single-layer, five-input regression neural network with a single hidden layer.

Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems

Complex mechanical systems often exhibit strongly nonlinear behavior due to the presence of nonlinearities in the energy dissipation mechanisms, material constitutive relationships, or geometric/connectivity mechanics. Numerical modeling of these systems leads to nonlinear full-order models that possess an underlying Lagrangian structure. This work proposes a Lagrangian operator inference method enhanced with structure-preserving machine learning to learn nonlinear reduced-order models (ROMs) of nonlinear mechanical systems. This two-step approach first learns the best-fit linear Lagrangian ROM via Lagrangian operator inference and then presents a structure-preserving machine learning method to learn nonlinearities in the reduced space. The proposed approach can learn a structure-preserving nonlinear ROM purely from data, unlike the existing operator inference approaches that require knowledge about the mathematical form of nonlinear terms. From a machine learning perspective, it accelerates the training of the structure-preserving neural network by providing an informed prior (i.e., the linear Lagrangian ROM structure), and it reduces the computational cost of the network training by operating on the reduced space. The method is first demonstrated on two simulated examples: a conservative nonlinear rod model and a two-dimensional nonlinear membrane with nonlinear internal damping. Finally, the method is demonstrated on an experimental dataset consisting of digital image correlation measurements taken from a lap-joint beam structure from which a predictive model is learned that captures amplitude-dependent frequency and damping characteristics accurately. The numerical results demonstrate that the proposed approach yields generalizable nonlinear ROMs that exhibit bounded energy error, capture the nonlinear characteristics reliably, and provide accurate long-time predictions outside the training data regime.

Modular model reduction of interconnected systems: A robust performance analysis perspective

Many complex engineering systems consist of multiple subsystems that are developed by different teams of engineers. To analyse, simulate and control such complex systems, accurate yet computationally efficient models are required. Modular model reduction, in which the subsystem models are reduced individually, is a practical and an efficient method to obtain accurate reduced-order models of such complex systems. However, when subsystems are reduced individually, without taking their interconnections into account, the effect on stability and accuracy of the resulting reduced-order interconnected system is difficult to predict. In this work, a mathematical relation between the accuracy of reduced-order linear-time invariant subsystem models and (stability and accuracy of) resulting reduced-order interconnected linear time-invariant model is introduced. This result can subsequently be used in two ways. Firstly, it can be used to translate accuracy characteristics of the reduced-order subsystem models directly to accuracy properties of the interconnected reduced-order model. Secondly, it can also be used to translate specifications on the interconnected system model accuracy to accuracy requirements on subsystem models that can be used for fit-for-purpose reduction of the subsystem models. These applications of the proposed analysis framework for modular model reduction are demonstrated on an illustrative structural dynamics example.

Dynamic mode decomposition for data-driven analysis and reduced-order modeling of E × B plasmas: I. Extraction of spatiotemporally coherent patterns

The advent of data-driven/machine-learning based methods and the increase in data available from high-fidelity simulations and experiments has opened new pathways toward realizing reduced-order models for plasma systems that can aid in explaining the complex, multi-dimensional phenomena and enable forecasting and prediction of the systems’ behavior. In this two-part article, we evaluate the utility and the generalizability of the dynamic mode decomposition (DMD) algorithm for data-driven analysis and reduced-order modeling of plasma dynamics in cross-field E × B configurations. The DMD algorithm is an interpretable data-driven method that finds a best-fit linear model describing the time evolution of spatiotemporally coherent structures (patterns) in data. We have applied the DMD to extensive high-fidelity datasets generated using a particle-in-cell (PIC) code based on the cost-efficient reduced-order PIC scheme. In this part, we first provide an overview of the concept of DMD and its underpinning proper orthogonal and singular value decomposition methods. Two of the main DMD variants are next introduced. We then present and discuss the results of the DMD application in terms of the identification and extraction of the dominant spatiotemporal modes from high-fidelity data over a range of simulation conditions. We demonstrate that the DMD variant based on variable projection optimization (OPT-DMD) outperforms the basic DMD method in identification of the modes underlying the data, leading to notably more reliable reconstruction of the ground-truth. Furthermore, we show in multiple test cases that the discrete frequency spectrum of OPT-DMD-extracted modes is consistent with the temporal spectrum from the fast Fourier transform of the data. This observation implies that the OPT-DMD augments the conventional spectral analyses by being able to uniquely reveal the spatial structure of the dominant modes in the frequency spectra, thus, yielding more accessible, comprehensive information on the spatiotemporal characteristics of the plasma phenomena.

Learning nonlinear projections for reduced-order modeling of dynamical systems using constrained autoencoders.

Recently developed reduced-order modeling techniques aim to approximate nonlinear dynamical systems on low-dimensional manifolds learned from data. This is an effective approach for modeling dynamics in a post-transient regime where the effects of initial conditions and other disturbances have decayed. However, modeling transient dynamics near an underlying manifold, as needed for real-time control and forecasting applications, is complicated by the effects of fast dynamics and nonnormal sensitivity mechanisms. To begin to address these issues, we introduce a parametric class of nonlinear projections described by constrained autoencoder neural networks in which both the manifold and the projection fibers are learned from data. Our architecture uses invertible activation functions and biorthogonal weight matrices to ensure that the encoder is a left inverse of the decoder. We also introduce new dynamics-aware cost functions that promote learning of oblique projection fibers that account for fast dynamics and nonnormality. To demonstrate these methods and the specific challenges they address, we provide a detailed case study of a three-state model of vortex shedding in the wake of a bluff body immersed in a fluid, which has a two-dimensional slow manifold that can be computed analytically. In anticipation of future applications to high-dimensional systems, we also propose several techniques for constructing computationally efficient reduced-order models using our proposed nonlinear projection framework. This includes a novel sparsity-promoting penalty for the encoder that avoids detrimental weight matrix shrinkage via computation on the Grassmann manifold.

An improved VSG control strategy based on transient electromagnetic power compensation

Virtual synchronous generator (VSG) not only increases the inertia of grid-connected system, but also brings the problem of active power oscillation under grid disturbance. Therefore, VSG control strategy and system model order reduction method with transient electromagnetic power compensation are proposed. The closed-loop active power small signal model of the system is established, and the influence of transient electromagnetic power compensation on the power stability of VSG is analyzed based on root locus method. By removing the items which have little influence on the stability of the system in the small signal model, the order is reduced to obtain the equivalent second-order model of the system. According to the second-order model, the quantitative design criteria of the system parameters are given. The proposed transient electromagnetic power compensation strategy not only increases the transient equivalent damping of the system, but also does not affect the primary frequency modulation characteristics and will not cause large overshoot of the output active power. The experimental results are consistent with the theoretical analysis, which testify the effectiveness and correctness of the system control strategy and the model reduction method.

Parametric Reduced-Order Modeling of Battery Thermal Management Systems for Varying Rates of Cooling Liquid Flow

The thermal management system in battery packs for electrical/hybrid electric vehicles plays an important role in maintaining good performance and health of battery cells. In order to optimize the thermal management system design, computational fluid dynamics (CFD) simulations are often used. The CFD is a powerful tool that can accurately simulate the thermal behavior of the system using numerical computation techniques such as finite volume method. However, the days-long computational expense means that the CFD-based simulations are still unsuitable for situations when rapid results are required, for example for online predictive control or fast-pace optimization design.A new method of reduced-order modeling of battery thermal management systems (BTMS) was presented in [1]. Given a CFD model, the algorithm based on projection on Krylov-subspace yields a reduced-order model (ROM). The ROM accurately predicts the system’s transient response to arbitrary internal heating of individual batteries and temperature of cooling liquid at inlet. Unlike conventional Krylov-subspace approaches, the algorithm does not require the system matrix of the CFD model. Therefore, the method is more suitable for use with commercial CFD software which does not allow access to such internal information. The ROM uses a drastically reduced number of degrees of freedom requiring drastically reduced simulation time.This paper extends our ROM by including the mass flow rate of cooling liquid as a model parameter. The task is accomplished by interpolating the system matrices after several local reduced-order models are constructed at discrete mass flow rates (MFR). In order to interpolate the local ROMs, we first need to transform the local ROMs onto a common subspace. The transformation matrix is obtained by applying the singular value decomposition (SVD) to the local projection matrices. In this paper, three local ROMs are built at 3 different MFRs. After these local ROMs are transformed according to the above procedure, a parametric reduced-order model (pROM) is constructed via quadratic interpolation.The constructed pROM is tested against the new MFRs different from those used for building the local ROMs. For each new MFR, the pROM produced a transient response that is almost indistinguishable from that of the CFD model (e.g., see Figure 1). The several thousand core-hour computational time needed for CFD simulations is replaced by a few tens of seconds using the pROM on a personal computer. The low computing load of the pROM is deemed adequate for on-board applications. Reference [1] Xiang L, Lee CW, Zikanov O, Hsu C-C. Efficient reduced order model for heat transfer in a battery pack of an electric vehicle. Applied Thermal Engineering. 2022; 201:117641.Figure 1 Figure 1

Self-updating digital twin of a hydrogen-powered furnace using data assimilation

Data assimilation, i.e., upgrading a numerical model by using experimental observations, is applied to adapt the performances of a simulation-based digital twin (DT) of a semi-industrial combustion furnace, based on available experimental data. More specifically, we rely on Kalman filter (KF) to adjust the prediction of our model by accounting for the underlying uncertainties. The DT is obtained by combining dimensionality reduction (through Proper Orthogonal Decomposition, POD) and regression (using Kriging) applied to Reynolds-averaged Navier–Stokes simulations of the furnace covering a three-dimensional design space, including both geometric and operational parameters. The experimental campaign concerns the measurement of the axial and radial profile of temperature inside the chamber and the NO concentrations at the outlet of the furnace, for a fuel mixture ranging from pure methane to pure hydrogen. Two types of KF algorithms are analyzed, i.e. the steady-state and the recursive ones. Both methodologies demonstrate improved DT performances, highlighting the significance of the Kalman gain in weighing the model’s prediction and measurement uncertainties. We also conduct a sensitivity analysis of data errors to reinforce this concept. The results of our study demonstrate the potential of data assimilation to build accurate and adaptive reduced-order models of realistic combustion systems.

An effective controller design strategy for higher order systems

The recent methods that guarantee the stability of the reduced-order models are Mihailov stability, improved Pade approximation and truncation, improved generalized pole clustering, moment matching and salp swarm optimization. Further, these methods could overcome the limitations such as non-uniqueness, pole clustering, gain adjustment and difficulty to maintain the dominant roots in the lower order system for non-minimum higher order plants. This research emphasizes the design of the proposed compensating algorithm by using an additional open loop zero for the stable reduced-order models of large-scale single-input single-output linear time-invariant continuous time systems. The results of the proposed algorithm are compared with the existing compensating methods and the design is validated and illustrated numerically.

GLaSDI: Parametric physics-informed greedy latent space dynamics identification

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. An interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on the fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, a k-nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 17 to 2,658× speed-up with 1 to 5% relative errors.

Reduced-order state-space models for two-dimensional discrete systems via bivariate discrete orthogonal polynomials

This paper investigates the new model order reduction (MOR) methods via bivariate discrete orthogonal polynomials for two-dimensional (2-D) discrete systems. The 2-D discrete system is described by the Kurek model. First, we deduce algebraically the shift-transformation matrix of the classical discrete orthogonal polynomials of one variable. By means of the shift-transformation matrices, 2-D discrete systems are expanded in the spaces spanned by bivariate discrete orthogonal polynomials. The coefficient matrices are calculated from matrix equations. Then the reduced-order systems are produced by the orthogonal projection matrices defined by the coefficient matrices. Theoretical analysis shows that the reduced-order systems can match a certain number of coefficient vectors of the original outputs. Finally, one numerical example is simulated to demonstrate the feasibility and effectiveness of the proposed methods.

Thermal comfort-conscious eco-climate control for electric vehicles using model predictive control

In the heating, ventilation, and air-conditioning (HVAC) system of electric vehicles (EVs), an electric heater is often used for a reheating process that warms up the chilled evaporator outlet air for thermal comfort before it is supplied to the cabin. The usage of the electric heater can significantly reduce the driving range of an EV. Therefore, optimal control of the HVAC system with consideration of the reheating process becomes essential for an increased driving range. In addition, considering the primary role of cabin climate control, it is desirable to intelligently consider the passengers’ thermal comfort in climate control design. In this paper, thermal comfort-conscious eco-climate control (TCC-ECC) based on model predictive control (MPC) is proposed to enhance the energy efficiency of the HVAC operation while ensuring passengers’ thermal comfort. The MPC design uses a reduced-order HVAC system model based on an ideal vapor-compression cycle. For the integration of thermal comfort into the MPC, a new approach is proposed to obtain an approximate solution of a predictive mean vote (PMV)-based thermal comfort model, which aims to balance computational efficiency and prediction accuracy. With the proposed TCC-ECC, a parametric study is conducted to analyze the impact of weighting factors on energy consumption and thermal comfort under two different thermal load conditions. Then, the performance of the tuned TCC-ECC is evaluated in comparison with a rule-based (RB) controller and the baseline eco-climate control (ECC) without considering thermal comfort. In the performance evaluation, the proposed TCC-ECC demonstrates that with the inclusion of thermal comfort it performs better in terms of energy efficiency and thermal comfort than manually adjusting a target cabin temperature depending on the environmental thermal load. The energy consumption of the proposed TCC-ECC is 22.5% and 35.24% less than that of the RB controller at an ambient temperature of 24 °C and 35 °C, respectively, and 14.5% and 18.5% less than the baseline ECC at the same conditions, respectively.

Rapidly encoding generalizable dynamics in a Euclidean symmetric neural network

Slinky, a helical elastic rod, is a seemingly simple structure with unusual mechanical behavior; for example, it can walk down a flight of stairs under its own weight. Taking Slinky as a test-case, we propose a physics-informed deep learning approach for building reduced-order models of physical systems. The approach introduces a Euclidean symmetric neural network (ESNN) architecture that is trained under the neural ordinary differential equation framework to learn the 2D latent dynamics from the motion trajectory of a reduced-order representation of the 3D Slinky. The ESNN implements a physics-guided architecture that simultaneously preserves energy invariance and force equivariance under Euclidean transformations of the input, including translation, rotation, and reflection. The embedded Euclidean symmetry provides physics-guided interpretability and generalizability, while preserving the full expressive power of the neural network. We demonstrate that the ESNN approach is able to accelerate simulation by one to two orders of magnitude compared to traditional numerical methods and achieve a superior generalization performance while classic neural networks fail to learn the Slinky dynamics, i.e., the ESNN, trained on a single demonstration case, predicts the motions accurately for unseen cases of different Slinky configurations and boundary conditions. Further investigation into the ESNN reveals that it explicitly learns the nonlinear coupling between stretching and bending of the Slinky.

Robust error assessment for reduced order vibro-acoustic problems

Robust and efficient automatic model order reduction is important in the design process of vibro-acoustic structures. For correct assessment of the reduction error without requiring the solution of the full order model, reliable error estimation techniques are required. Such error estimators can be used in iterative algorithms to automatically determine a reasonable order and expansion point distribution for reduced-order models. An accurate estimation of the approximation error is important in order to design an efficient adaptive model order reduction process requiring as little a priori knowledge of the system response as possible. In this contribution, different techniques for estimating the approximation error of reduced-order models are compared and employed in adaptive algorithms aiming at generating reduced-order models of vibro-acoustic systems. For many applications, vibro-acoustic systems need to be evaluated in a specific frequency range, as certain system characteristics are only observed in this region. Therefore, the employed error estimation methods are assessed regarding their effectiveness estimating the error in a particular frequency region.

Numerical analysis of non-local calculus on finite weighted graphs, with application to reduced-order modeling of dynamical systems

We present an approach to reduced-order modeling that builds off recent graph-theoretic work for representation, exploration, and analysis of computed states of physical systems (Banerjee et al., Comp. Meth. App. Mech. Eng., 351, 501–530, 2019). We extend a non-local calculus on finite weighted graphs to build such models by exploiting polynomial expansions and Taylor series. In the general framework for non-local calculus on graphs, the graph edge weights are intricately linked to the embedding of the graph, and consequently to the definition of the derivatives. In a previous communication (Duschenes and Garikipati, arXiv:2105.01740 [math.NA]), we have shown that radially symmetric, continuous edge weights derived from, for example Gaussian functions, yield inconsistent results in the resulting non-local derivatives when compared against the corresponding local, differential derivative definitions. Taking inspiration from finite difference methods, we algorithmically compute edge weights, considering the embedding of the local neighborhood of each graph vertex. Given this procedure, we ensure the consistency of the non-local derivatives in this setting, a crucial requirement for numerical applications. We show that we can achieve any desired orders of accuracy of derivatives, in a chosen number of dimensions without symmetry assumptions in the underlying data. Finally, we present two example applications of extracting reduced-order models using this non-local calculus, in the form of ordinary differential equations from parabolic partial differential equations of progressively greater complexity.

Bayesian operator inference for data-driven reduced-order modeling

This work proposes a Bayesian inference method for the reduced-order modeling of time-dependent systems. Informed by the structure of the governing equations, the task of learning a reduced-order model from data is posed as a Bayesian inverse problem with Gaussian prior and likelihood. The resulting posterior distribution characterizes the operators defining the reduced-order model, hence the predictions subsequently issued by the reduced-order model are endowed with uncertainty. The statistical moments of these predictions are estimated via a Monte Carlo sampling of the posterior distribution. Since the reduced models are fast to solve, this sampling is computationally efficient. Furthermore, the proposed Bayesian framework provides a statistical interpretation of the regularization term that is present in the deterministic operator inference problem, and the empirical Bayes approach of maximum marginal likelihood suggests a selection algorithm for the regularization hyperparameters. The proposed method is demonstrated on two examples: the compressible Euler equations with noise-corrupted observations, and a single-injector combustion process.