Data-driven Model Reduction Research Articles

Assessment of Multifidelity Surrogate Approaches for Expedient Loads Prediction in High-Speed Flows

Fast and accurate predictions about the flowfield surrounding a hypersonic flight vehicle are necessary for both early design phases and autonomous vehicle control. Data-driven model reduction presents an opportunity to harness the accuracy of high-fidelity techniques for expedient online application. Surrogate modeling is one approach that seeks to emulate the response of a system with evaluation times much faster than the data source. Multifidelity surrogate modeling seeks to reduce the amount of high-fidelity data needed for a sufficiently accurate online model. The objective of this paper is to assesses the tradeoffs between a kriging-based multifidelity model and a neural-network-based multifidelity model, applied to high-speed flow past a canonical flight geometry. All models studied in this work reproduce the results of the benchmark simulations with less than 100 training cases. The kriging models marginally outperform the neural-network-based models in accuracy, but requirements for training and storing the kriging models scale exponentially as the training data gets larger. Kriging models offer built-in uncertainty quantification and require significantly less decision making about hyperparameters and architecture choices. This study also indicates that the inclusion of data from classical engineering methods can improve the prediction accuracy of surrogate models at practically no cost.

Data-driven model reduction for pipes conveying fluid via spectral submanifolds

Pipes conveying fluid have non-linearizable dynamics when the flow velocity is large enough. Developing analytical models that accurately capture these highly nonlinear and complex dynamics is challenging. Efficient analysis and predictions of these nonlinear dynamics are essential in the design and control of pipe systems in engineering applications. To address these challenges, here we construct low-dimensional reduced-order models in a data-driven setting. In particular, we perform model reduction on spectral submanifolds (SSMs) via SSMLearn. This open-source package automates the identification of low-dimensional SSMs and their associated reduced-order models (ROMs). Our reduction via SSMs treats pipes with various boundary conditions and flow velocities in a unified way. It achieves a significant reduction because the ROMs are two-dimensional, which enables efficient and even analytical predictions on the nonlinear vibration of the pipes, paving the way for real-time prediction of the non-linearizable dynamics. In addition, our SSM-based reduction shows remarkable extrapolation capability because it is built on invariant manifolds and normal form theory. We demonstrate that the ROMs trained on unforced responses can be directly extrapolated to make reliable predictions on forced vibrations under various excitation frequencies and amplitudes, including periodic and quasi-periodic orbits and their bifurcations.

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.

Application of data-driven model reduction techniques in reactor neutron field calculations

High-order harmonic techniques can be used to recreate neutron flux distributions in reactor cores using the neutron diffusion equation. However, traditional source iteration and source correction iteration techniques have sluggish convergence rates and protracted calculation periods. The correctness of the implicitly restarted Arnoldi method (IRAM) in resolving the eigenvalue problems of the one-dimensional and two-dimensional neutron diffusion equations was confirmed by computing the benchmark problems SLAB_1D_1G and two-dimensional steady-state TWIGL using IRAM. By integrating Galerkin projection with Proper Orthogonal Decomposition (POD) techniques, a POD-Galerkin reduced-order model was developed and the IRAM model was used as the full-order model. For 14 macroscopic cross-section values, the TWIGL benchmark problem was perturbed within a 20% range. We extracted 100 sample points using the Latin hypercube sampling method, and 70% of the samples were used as the testing set to assess the performance of the reduced-order model The remaining 30% were utilized as the training set to develop the reduced-order model, which was employed to rebuild the TWIGL benchmark problem. The reduced-order model demonstrates good flexibility and can efficiently and accurately forecast the effective multiplication factor and neutron flux distribution in the core. The reduced-order model predicts keff and neutron flux distribution with a high degree of agreement compared to the full-order model. Additionally, the reduced-order model's computation time is only 10.18% of that required by the full-order model.The neutron flux distribution of the steady-state TWIGL benchmark was recreated using the reduced-order model. The obtained results indicate that the reduced-order model can accurately predict the keff and neutron flux distribution of the steady-state TWIGL benchmark.Overall, the proposed technique not only has the potential to accurately project neutron flux distributions in transient settings, but is also relevant for reconstructing neutron flux distributions in steady-state conditions; thus, its applicability is bound to increase in the future.

Data-driven model reduction approach for active vibration control of cable-strut structures

The cable-strut structures have the features of low stiffness and weak damping, resulting in large vibration response under dynamic excitation. The vibration response of the cable-strut structures can be reduced by active control methods. However, many traditional active control methods need to build a state-space model (SSM) based on finite element model (FEM), and solving the active control force for the high-dimensional SSM is time-consuming. In this paper, the dynamic mode decomposition with control (DMDc) method is proposed to achieve a data-driven reduced-order SSM for active vibration control of cable-strut structures. This proposed method only uses the data of structural response and control force input to build reduced-order SSM and does not rely on the FEM. Linear quadratic regulator (LQR) design using the DMDc-based reduced-order SSM (DMDc-LQR) is more computationally efficient than that using the FEM-based SSM (FEM-LQR). The relationship between the design parameters of the DMDc-LQR controller and the FEM-LQR controller is established, which ensures that the control performance indexes of the DMDc-LQR controller and the FEM-LQR controller are equal. The effectiveness of the proposed DMDc-LQR method is validated using numerical Kiewitt cable dome and tensegrity grid under wind and seismic loads. The results show that the DMDc-LQR controller maintains similar control effect to that of the FEM-LQR controller and has advantage of high computing efficiency as well. In addition, the influence of critical parameters (i.e., data noise and model order) on vibration control effect is comprehensively explored. It is found that increasing the model order is effective to mitigate the noise influence on the DMDc-based reduced-order SSM and ensures the effectiveness of the DMDc-LQR controller.

Predicting rate kernels via dynamic mode decomposition.

Simulating dynamics of open quantum systems is sometimes a significant challenge, despite the availability of various exact or approximate methods. Particularly when dealing with complex systems, the huge computational cost will largely limit the applicability of these methods. In this work, we investigate the usage of dynamic mode decomposition (DMD) to evaluate the rate kernels in quantum rate processes. DMD is a data-driven model reduction technique that characterizes the rate kernels using snapshots collected from a small time window, allowing us to predict the long-term behaviors with only a limited number of samples. Our investigations show that whether the external field is involved or not, the DMD can give accurate prediction of the result compared with the traditional propagations, and simultaneously reduce the required computational cost.

From Data to Reduced-Order Models via Generalized Balanced Truncation

This paper proposes a data-driven model reduction approach on the basis of noisy data with a known noise model. Firstly, the concept of data reduction is introduced. In particular, we show that the set of reduced-order models obtained by applying a Petrov-Galerkin projection to all systems explaining the data characterized in a large-dimensional quadratic matrix inequality (QMI) can again be characterized in a lower-dimensional QMI. Next, we develop a data-driven generalized balanced truncation method that relies on two steps. First, we provide necessary and sufficient conditions such that systems explaining the data have common generalized Gramians. Second, these common generalized Gramians are used to construct matrices that allow to characterize a class of reduced-order models via generalized balanced truncation in terms of a lower-dimensional QMI by applying the data reduction concept. Additionally, we present alternative procedures to compute a priori and a posteriori upper bounds with respect to the true system generating the data. Finally, the proposed techniques are illustrated by means of application to an example of a system of a cart with a double-pendulum.

Data-Driven Balanced Truncation for Predictive Model Order Reduction of Aeroacoustic Response

Rapid prediction of the aeroacoustic response is a key component in the design of aircraft and turbomachinery. In this work, we propose a technique for highly accelerated prediction of aeroacoustic response using a data-driven model reduction approach based on the eigensystem realization algorithm (ERA). Specifically, we create and compare ERA reduced-order models (ROMs) based on the training data generated by solving the linearized and nonlinear Euler equations, and we use them to predict the aeroacoustic response of an airfoil in a purely predictive setting subject to different types of gust loading. Activating each input channel separately in the full-order model (FOM) to generate the Markov sequence for training makes it computationally challenging to use the ERA in aeroacoustics applications. We address this bottleneck by first proposing a multifidelity gappy POD method to reduce the computation cost on the FOM and ROM levels by querying the high-resolution FOM only for the input channels identified by gappy POD. Second, we use tangential interpolation at the ROM level to reduce the size of the Hankel matrix. The proposed methods enable application of the ERA for accurate online acoustic response prediction, and they reduce the offline computation cost of ROMs.

Model reduction for nonlinearizable dynamics via delay-embedded spectral submanifolds

Delay embedding is a commonly employed technique in a wide range of data-driven model reduction methods for dynamical systems, including the dynamic mode decomposition, the Hankel alternative view of the Koopman decomposition (HAVOK), nearest-neighbor predictions and the reduction to spectral submanifolds (SSMs). In developing these applications, multiple authors have observed that delay embedding appears to separate the data into modes, whose orientations depend only on the spectrum of the sampled system. In this work, we make this observation precise by proving that the eigenvectors of the delay-embedded linearized system at a fixed point are determined solely by the corresponding eigenvalues, even for multi-dimensional observables. This implies that the tangent space of a delay-embedded invariant manifold can be predicted a priori using an estimate of the eigenvalues. We apply our results to three datasets to identify multimodal SSMs and analyse their nonlinear modal interactions. While SSMs are the focus of our study, these results generalize to any delay-embedded invariant manifold tangent to a set of eigenvectors at a fixed point. Therefore, we expect this theory to be applicable to a number of data-driven model reduction methods.

A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems with uncertain data efficiently. Our method consists of offline and online stages. In the offline stage, we explore the low-dimensional structures in the solution space of parabolic partial differential equations (PDEs) in the forward problems with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problems extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in parabolic inverse source problems, which is referred to as the POD method. Moreover, we design an a posteriori algorithm to find the optimal regularization parameter in the optimization problem using the proposed POD method without knowing the noise level. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD method in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD method for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Numerical results show that the POD method provides considerable computational savings over the finite element method while maintaining the same accuracy.

A data-driven surrogate modeling approach for time-dependent incompressible Navier-Stokes equations with dynamic mode decomposition and manifold interpolation

This work introduces a novel approach for data-driven model reduction of time-dependent parametric partial differential equations. Using a multi-step procedure consisting of proper orthogonal decomposition, dynamic mode decomposition, and manifold interpolation, the proposed approach allows to accurately recover field solutions from a few large-scale simulations. Numerical experiments for the Rayleigh-Bénard cavity problem show the effectiveness of such multi-step procedure in two parametric regimes, i.e., medium and high Grashof number. The latter regime is particularly challenging as it nears the onset of turbulent and chaotic behavior. A major advantage of the proposed method in the context of time-periodic solutions is the ability to recover frequencies that are not present in the sampled data.

Data-driven model reduction for fast temperature prediction in a multi-variable data center

With the rapid development of digital economy, the number of data centers and their capacity have been increasing sharply, and data center energy consumption becomes a whole-society concern. Computational fluid dynamics (CFD) is currently widely used to obtain the thermal fields inside air-cooled data centers to enable design improvements and optimize the airflow organization. However, a CFD simulation needs a lot of time which can not be accepted for real-time operation. In the present study, a design tool called pairwise independent combinatorial testing (PICT) is applied to optimize the simulation conditions and to maximize the amount of useful information obtained with the minimum number of numerical tests. Based on the snapshots, the proper orthogonal decomposition(POD) method combined with the multivariate adaptive regression splines (MARS) method, is proposed and used in a real row-level data center of 199 independent variables. Under design conditions, POD-MARS predictions are in good agreement with CFD simulations with the average mean relative error for 20 tested cases being ∼0.01%. For another 20 randomized cases under off-design conditions, the average mean relative error is 6.45%, the corresponding mean absolute error is 1.89 °C and on average there is 92.36% area of the total three-dimensional temperature field where the relative error doesn't exceed 15%. The POD-MARS computation takes only 30s to obtain a 3D temperature field for the same test case which is ∼240 times faster than CFD simulation on the same desktop computer.

Surrogate Model Based on Data-Driven Model Reduction for Inelastic Behavior of Composite Microstructure

Surrogate Model Based on Data-Driven Model Reduction for Inelastic Behavior of Composite Microstructure

Fast data-driven model reduction for nonlinear dynamical systems

We present a fast method for nonlinear data-driven model reduction of dynamical systems onto their slowest nonresonant spectral submanifolds (SSMs). While the recently proposed reduced-order modeling method SSMLearn uses implicit optimization to fit a spectral submanifold to data and reduce the dynamics to a normal form, here, we reformulate these tasks as explicit problems under certain simplifying assumptions. In addition, we provide a novel method for timelag selection when delay-embedding signals from multimodal systems. We show that our alternative approach to data-driven SSM construction yields accurate and sparse rigorous models for essentially nonlinear (or non-linearizable) dynamics on both numerical and experimental datasets. Aside from a major reduction in complexity, our new method allows an increase in the training data dimensionality by several orders of magnitude. This promises to extend data-driven, SSM-based modeling to problems with hundreds of thousands of degrees of freedom.

Data-driven nonlinear model reduction to spectral submanifolds in mechanical systems.

While data-driven model reduction techniques are well-established for linearizable mechanical systems, general approaches to reducing nonlinearizable systems with multiple coexisting steady states have been unavailable. In this paper, we review such a data-driven nonlinear model reduction methodology based on spectral submanifolds. As input, this approach takes observations of unforced nonlinear oscillations to construct normal forms of the dynamics reduced to very low-dimensional invariant manifolds. These normal forms capture amplitude-dependent properties and are accurate enough to provide predictions for nonlinearizable system response under the additions of external forcing. We illustrate these results on examples from structural vibrations, featuring both synthetic and experimental data.This article is part of the theme issue ‘Data-driven prediction in dynamical systems’.

Nonlinear Model Reduction by Moment-Matching for a Point Absorber Wave Energy Conversion System

This paper presents a data-driven model reduction by moment-matching approach to construct control-oriented models for a point absorber device. The methodology chosen and developed generates models which are input-to-state linear, with any nonlinear behaviour confined to the output map. Such a map is the result of a data-driven approximation procedure, where the so-called moment of the point absorber system is estimated via a least-squares procedure. The resulting control-oriented model can inherently preserve steady-state properties of the target WEC system for a user-defined class of input signals of interest, with the computation only dependent upon a suitably defined set of input-output data.

Model order reduction method based on (r)POD-ANNs for parameterized time-dependent partial differential equations

In this paper, we propose a non-intrusive data-driven model order reduction method, also known as (r)POD-ANNs model order reduction method. In this method, (r)POD priori dimension reduction is performed on high-fidelity data set, and then the mapping relationship between parameter space and generalized coordinates of (r)POD is implicitly constructed by using the universal approximation property of neural network. Through pre-training on small batch data sets, updating neural network parameters every several time steps, and combination of L-BFGS optimization algorithm and LM optimization algorithm, time cost of the reduced order model in the off-line calculation stage is reduced. This makes the (r)POD-ANNs model order reduction method suitable for high-fidelity models with larger spatial degrees of freedom and higher complexity. Finally, we verify effectiveness of the proposed method by comparing with the data-driven model reduction method, and then apply it to the Allen-Cahn equations with strong nonlinearity and cylinder flow problem with large spatial degrees of freedom.

A methodology for thermal simulation of interconnects enabled by model reduction with material property variation

A thermal simulation methodology is developed for interconnects enabled by a data-driven learning algorithm accounting for variations of material properties, heat sources and boundary conditions (BCs). The methodology is based on the concepts of model order reduction and domain decomposition to construct a multi-block approach. A generic block model is built to represent a group of interconnect blocks that are used to wire standard cells in the integrated circuits (ICs). The blocks in this group possess identical geometry with various metal/via routings. The data-driven model reduction method is thus applied to learn material property variations induced by different metal/via routings in the blocks, in addition to the variations of heat sources and BCs. The approach is investigated in two very different settings. It is first applied to thermal simulation of a single interconnect block with similar BCs to those in the training of the generic block. It is then implemented in multi-block thermal simulation of a FinFET IC, where the interconnect structure is partitioned into several blocks each modeled by the generic block model. Accuracy of the generic block model is examined in terms of the metal/via routings, BCs and thermal discontinuities at the block interfaces.

Identification of MIMO Wiener-type Koopman models for data-driven model reduction using deep learning

We use Koopman theory to develop a data-driven nonlinear model reduction and identification strategy for multiple-input multiple-output (MIMO) input-affine dynamical systems. While the present literature has focused on linear and bilinear Koopman models, we derive and use a Wiener-type Koopman formulation. We discuss that the Wiener structure is particularly suitable for model reduction, and can be naturally derived from Koopman theory. Moreover, the Wiener block-structure unifies the mathematical simplicity of linear dynamical blocks and the accuracy of bilinear dynamics. We present a Koopman deep-learning strategy combining autoencoders and linear dynamics that generates low-order surrogate models of MIMO Wiener type. In three case studies, we apply our framework for identification and reduction of a system with input multiplicity, a chemical reactor and a high-purity distillation column. We compare the prediction performance of the identified Wiener models to linear and bilinear Koopman models. We observe the highest accuracy and strongest model reduction capabilities of low-order Wiener-type Koopman models, making them promising for control.

Data-Driven Nonlinear Model Reduction Using Koopman Theory: Integrated Control Form and NMPC Case Study

We use Koopman theory for data-driven model reduction of nonlinear dynamical systems with controls. We propose generic model structures combining delay-coordinate encoding of measurements and full state decoding to integrate reduced Koopman modeling and state estimation. We present a deep-learning approach to train the proposed models. A case study demonstrates that our approach provides accurate control models and enables real-time capable nonlinear model predictive control of a high-purity cryogenic distillation column.