Non-intrusive framework of reduced-order modeling based on proper orthogonal decomposition and polynomial chaos expansion

Sun, Xiaonan; Pan, Xiao‐Min; Choi, Jung Il

doi:10.1016/j.cam.2020.113372

Cited by 19 publications

(7 citation statements)

References 57 publications

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“…The high fidelity solver is only used to generate the snapshots and the training dataset, guaranteeing a complete decoupling between the online evaluation and the offline training. There is a lot of work on the non-intrusive method in the past, such as the regression-based non-intrusive RB methods including the tensor decomposition based regression (see, e.g., for parametrized time-dependent problem [2]), the artificial neural networks (ANNs) based regression (see, e.g., for steady-state problem [34], for combustion problem [18], and for transient flow [35]), and the Gaussian processes regression (GPR) based regression (see, e.g., for nonlinear structural analysis [36], and for compressible flow [31]), and the interpolation-based non-intrusive RB methods comprising the radial basis function (RBF) interpolations (see, e.g., for parametrized time-dependent PDE [37], for multiphase flows in porous media [38], and for shallow water equations [39]), the polynomial interpolations (see, e.g., for parametrized timedependent problems [40], for stochastic representations in UQ analysis [41]), and the cubic spline interpolations (CSI) (see, e.g., for parametric applications in transonic aerodynamics [42], and for non-linear parametrized physical problems [43]).…”

Section: Introductionmentioning

confidence: 99%

Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation

Huang

et al. 2021

J Sci Comput

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This paper presents a non-intrusive model order reduction (MOR) for the solution of parameterized electromagnetic scattering problems, which needs to prepare a database offline of full-order solution samples (snapshots) at some different parameter locations. The snapshot vectors are produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. Because the second dimension of snapshots matrix is large, a two-step or nested proper orthogonal decomposition (POD) method is employed to extract timeand parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the projection coefficient matrices (also referred to as the reduced coefficient matrices) of full-order solutions onto the RB subspace are extracted. A cubic spline interpolation-based (CSI) approach is proposed to approximate the dominating time-and parameter-modes of the reduced coefficient matrices without resorting to Galerkin projection. The generation of snapshot vectors, the construction of POD basis functions and the approximation of reduced coefficient matrices based on the CSI method are completed during the offline stage. The RB solutions for new time and parameter values can be rapidly recovered via outputs from the interpolation models in the online stage. In particular, the offline and online stages of the proposed RB method, termed as the POD-CSI method, are completely decoupled, which ensures the computational validity of the method. Moreover, a surrogate error model is constructed as an efficient error estimator for the POD-CSI method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of POD-CSI method.

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Section: Introductionmentioning

confidence: 99%

Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation

Huang

et al. 2021

J Sci Comput

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“…Different from the IMOR method, the NIMOR method only uses the approximation mappings obtained by some data-driven methods [6], such as interpolation, regression, and artificial neural networks (ANN) methods, to calculate the reducedorder coefficients for new time/parameter values. There have been a lot of works [22,13,46,53,49,1,2,7,10,19,42,11] on the NIMOR method in the very recent years. Audouze et al in [2] presented a NIMOR method for nonlinear parametric time dependent PDEs using POD and radial basis function (RBF) approximation.…”

mentioning

confidence: 99%

“…Hesthaven and Ubbiali in [15] built an ANN to compute the reduced-order coefficients of the ROM, where the nonlinear Poisson and steady-state incompressible Navier-Stokes equations are tested. In [42], the polynomial chaos expansion (PCE) method is used in the NI-MOR method for stochastic representations in UQ analysis. The GPR method is a machine learning regression method based on the Bayesian and statistical theories, which is suitable for dealing with complex problems such as high dimensions, small samples and nonlinearity.…”

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confidence: 99%

A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations

Huang

et al. 2023

DCDS-B

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<p style='text-indent:20px;'>We present a non-intrusive model order reduction (NIMOR) approach with an offline-online decoupling for the solution of parameterized time-domain Maxwell's equations. During the offline stage, the training parameters are chosen by using Smolyak sparse grid method with an approximation level <inline-formula><tex-math id="M1">\begin{document}$ L $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ L\geq1 $\end{document}</tex-math></inline-formula>) over a target parameterized space. For each selected parameter, the snapshot vectors are first produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. In order to minimize the overall computational cost in the offline stage and to improve the accuracy of the NIMOR method, a radial basis function (RBF) interpolation method is then used to construct more snapshot vectors at the sparse grid with approximation level <inline-formula><tex-math id="M3">\begin{document}$ L+1 $\end{document}</tex-math></inline-formula>, which includes the sparse grids from approximation level <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula>. A nested proper orthogonal decomposition (POD) method is employed to extract time- and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the reduced coefficient matrices of the high-fidelity solutions onto the reduced-order subspace spaned by the POD basis functions are extracted. Moreover, a Gaussian process regression (GPR) method is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices. During the online stage, the reduced-order solutions for new time and parameter values can be rapidly recovered via outputs from the regression models without using the DGTD method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of the NIMOR method.</p>

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“…As a first test case, we introduce a stochastic version of the Ackley function, a highly irregular baseline with multiple extrema presented inSun et al (2019), which takes P = 3 parameters. Being real-valued (D = 1) and two-dimensional in space (n = 2), it is defined asu : IR 2+P →IR(38) (x, y; s) → − 20 (1 + 0.1s 3 ) exp −0.2(1 + 0.1s 2 ) 0.5(x 2 + y 2 ) − exp (0.5(cos(2π(1 + 0.1s 1 )x) + cos(2π(1 + 0.1s 1 )y))) + 20 + exp(0), with the non-spatial parameters vector s of size P = 3, and each element s i randomly sampled over Ω = [−1, 1], as in Sun et al (2019).…”

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confidence: 99%

Non-Intrusive Reduced-Order Modeling Using Uncertainty-Aware Deep Neural Networks and Proper Orthogonal Decomposition: Application to Flood Modeling

Jacquier,

Abdedou,

Delmas

et al. 2020

Preprint

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Deep Learning research is advancing at a fantastic rate, and there is much to gain from transferring this knowledge to older fields like Computational Fluid Dynamics in practical engineering contexts. This work compares state-of-the-art methods that address uncertainty quantification in Deep Neural Networks, pushing forward the reduced-order modeling approach of Proper Orthogonal Decomposition-Neural Networks (POD-NN) with Deep Ensembles and Variational Inference-based Bayesian Neural Networks on two-dimensional problems in space. These are first tested on benchmark problems, and then applied to a real-life application: flooding predictions in the Mille Îles river in the Montreal, Quebec, Canada metropolitan area. Our setup involves a set of input parameters, with a potentially noisy distribution, and accumulates the simulation data resulting from these parameters. The goal is to build a non-intrusive surrogate model that is able to know when it doesn't know, which is still an open research area in Neural Networks (and in AI in general). With the help of this model, probabilistic flooding maps are generated, aware of the model uncertainty. These insights on the unknown are also utilized for an uncertainty propagation task, allowing for flooded area predictions that are broader and safer than those made with a regular uncertainty-uninformed surrogate model. Our study of the time-dependent and highly nonlinear case of a dam break is also presented. Both the ensembles and the Bayesian approach lead to reliable results for multiple smooth physical solutions, providing the correct warning when going out-of-distribution. However, the former, referred to as POD-EnsNN, proved much easier to implement and showed greater flexibility than the latter in the case of discontinuities, where standard algorithms may oscillate or fail to converge.

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Non-intrusive framework of reduced-order modeling based on proper orthogonal decomposition and polynomial chaos expansion

Cited by 19 publications

References 57 publications

Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation

Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation

A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations

Non-Intrusive Reduced-Order Modeling Using Uncertainty-Aware Deep Neural Networks and Proper Orthogonal Decomposition: Application to Flood Modeling

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