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

Kun Li; Ting-Zhu Huang; Liang Li; Ying Zhao; Stéphane Lanteri

doi:10.3934/dcdsb.2022084

Article Contents

2023, Volume 28, Issue 1: 449-473. Doi: 10.3934/dcdsb.2022084

This issue Previous Article Structure of non-autonomous attractors for a class of diffusively coupled ODE Next Article Free boundary problems for the local-nonlocal diffusive model with different moving parameters

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

1.
School of Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China
2.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
3.
Université Côte d'Azur, Inria, CNRS, LJAD, Sophia Antipolis Cedex, France

^*Corresponding author: Ting-Zhu Huang, Liang Li
^*Corresponding author: Ting-Zhu Huang, Liang Li

Received: September 30, 2021

Revised: February 28, 2022

Early access: May 2022

Published: January 2023

The first author is supported by the NSFC (Grant No. 12101511). The second author is supported by the NSFC (Grant No. 61772003) and the Key Projects of Applied Basic Research in Sichuan Province (Grant No. 2020YJ0216)

Abstract Full Text(HTML) Figure(12) / Table(8) Related Papers Cited by

Abstract

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 $ L $ ($ L\geq1 $) 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 $ L+1 $, which includes the sparse grids from approximation level $ L $. 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.

Keywords:

Mathematics Subject Classification: Primary: 78M34; Secondary: 65M22.

Citation:

Full Text(HTML)

Figure 1. 1-D Smolyak sparse grids with approximation levels 0, 1, 2, 3

Download: Full-size image PowerPoint slide

Figure 2. 2-D Smolyak sparse grid with approximation level 3 and full tensor product grid

Download: Full-size image PowerPoint slide

Figure 3. Scattering of plane wave by a dielectric disk: the convergence histories of $ \overline{e}_{ {\bf E}, {\rm{Pro}}^{(i)}} $, and $ \overline{e}_{ {\bf E}, {\rm{NIMOR}}^{(i)}} $ (a), $ \overline{e}_{ {\bf H}, {\rm{Pro}}^{(i)}} $, and $ \overline{e}_{ {\bf H}, {\rm{NIMOR}}^{(i)}} $ (b) ($ i = 1, 2 $) on the testing set $ \mathcal{T}_{te}\times\mathcal{P}_{te} $ with vary truncation tolerances $ \rho_\theta $, where $ \overline{e}_{ {\bf u}, {\rm{Pro}}^{(i)}} $ is the average projection error of $ {\rm{NIMOR}}^{(i)} $ method for $ {\bf u} $

Download: Full-size image PowerPoint slide

Figure 4. Scattering of plane wave by a dielectric disk: the $ 5 $-th, $ 10 $-th, $ 15 $-th, and $ 20 $-th exact and approximation reduced-order cofficients of $ E_z $ based on $ {\rm{NIMOR}}^{(1)} $

Download: Full-size image PowerPoint slide

Figure 5. Scattering of plane wave by a dielectric disk: the $ 5 $-th, $ 10 $-th, $ 15 $-th, and $ 20 $-th exact and approximation reduced-order cofficients of $ H_y $ based on $ {\rm{NIMOR}}^{(1)} $

Download: Full-size image PowerPoint slide

Figure 6. Scattering of plane wave by a dielectric disk: comparison of the 1-D x-wise distribution along $ y = 0 $ of the real part of $ H_y $ (left) and $ E_z $ (right) of four test points: $ \theta^{(1)} = 1.215 $ (1-th row), $ \theta^{(2)} = 2.215 $ (2-th row), $ \theta^{(3)} = 3.215 $ (3-th row) and $ \theta^{(4)} = 4.215 $ (4-th row)

Download: Full-size image PowerPoint slide

Figure 7. Scattering of plane wave by a dielectric disk: comparison of relative projection and NIMOR errors for $ {\bf E} $ (left) and $ {\bf H} $ (right) four the testing parameters

Download: Full-size image PowerPoint slide

Figure 8. Scattering of a plane wave by a multi-layer heterogeneous medium: geometry of the multi-layer medium

Download: Full-size image PowerPoint slide

Figure 9. Scattering of a plane wave by a multi-layer heterogeneous medium: the convergence histories of $ \overline{e}_{ {\bf E}, {\rm{Pro}}^{(i)}} $, and $ \overline{e}_{ {\bf E}, {\rm{NIMOR}}^{(i)}} $ (a), $ \overline{e}_{ {\bf H}, {\rm{Pro}}^{(i)}} $, and $ \overline{e}_{ {\bf H}, {\rm{NIMOR}}^{(i)}} $ (b) ($ i = 1, 2 $) on the testing set $ \mathcal{T}_{te}\times\mathcal{P}_{te} $ with vary truncation tolerances $ \rho_\theta $, where $ \overline{e}_{ {\bf u}, {\rm{Pro}}^{(i)}} $ is the average projection error of $ {\rm{NIMOR}}^{(i)} $ method for $ {\bf u} $

Download: Full-size image PowerPoint slide

Figure 10. Scattering of plane wave by a multi-layer heterogeneous medium: the 3th, 6th, 9th, and 12th exact ($ - $) and approximation ($ * $) time-modes for $ E_z $ (left) and $ H_y $ (right) (the 1th mode: black, the 3th mode: red, the 5th mode: brown; the 7th mode: blue; the 9th mode: green)

Download: Full-size image PowerPoint slide

Figure 11. Scattering of plane wave by a multi-layer heterogeneous medium: comparison of the 1-D x-wise distribution along $ y = 0 $ of the real part of $ H_y $ (left) and $ E_z $ (right) of four test points: $ \theta^{(1)} = \{(5.125, 3.375, 2.125, 1.375)\} $ (1-th row), $ \theta^{(2)} = \{(5.425, 3.625, 2.425, 1.625)\} $ (2-th row), $ \theta^{(3)} = \{(5.125, 3.625, 2.125, 1.625)\} $ (3-th row) and $ \theta^{(4)} = \{(5.425, 3.375, 2.425, 1.375)\} $ (4-th row)

Download: Full-size image PowerPoint slide

Figure 12. Scattering of plane wave by a multi-layer heterogeneous medium: comparison of relative projection and NIMOR errors for $ {\bf E} $ (left) and $ {\bf H} $ (right) four the testing parameters

Download: Full-size image PowerPoint slide

Table 1. Comparison of the number of sampling points using Smolyak sparse gird and full tensor product grid. Here $ M_1 $ is the number of sampling points using Smolyak sparse gird, and $ M_2 $ is the number of sampling points using full tensor product grid

Dimension size $ p $	Approximation level $ l $	$ M_1 $	$ M_2 $	$ \dfrac{M_2}{M_1} $
	2	13	25	1.923
2	3	29	81	2.793
	5	145	1089	7.510
	2	61	3125	51.230
5	3	241	59049	$ 2.450\times10^2 $
	5	2433	$ 3.914\times10^7 $	$ 1.609\times10^4 $
	2	221	$ 9.766\times10^6 $	$ 4.419\times10^4 $
10	3	1581	$ 3.487\times10^9 $	$ 2.206\times10^6 $
	5	41265	$ 1.532\times10^{15} $	$ 3.713\times10^{10} $

| Show Table

DownLoad: CSV

Table 2. Scattering of plane wave by a dielectric disk: settings for the training, and testing datasets

Data set	Training set	Testing set
Parameter sample points	65, uneven (Smolyak mehod)	40, random (LHS method)
Time sample points	263, uniform	263, uniform
Size	17095	10520

| Show Table

DownLoad: CSV

Table 3. Scattering of plane wave by a dielectric disk: the average projection and NIMOR errors on the testing set

Average relative errors	$ \overline{e}_{ {\bf E}, {\rm{Pro}}} $	$ \overline{e}_{ {\bf E}, {\rm{NIMOR}}} $	$ \overline{e}_{ {\bf H}, {\rm{Pro}}} $	$ \overline{e}_{ {\bf H}, {\rm{NIMOR}}} $
$ {\rm{NIMOR}}^{(1)} {\rm{method}} $	$ 1.170\times 10^{-2} $	$ 1.500\times 10^{-2} $	$ 1.065\times 10^{-2} $	$ 1.447\times 10^{-2} $
$ {\rm{NIMOR}}^{(2)} {\rm{method}} $	$ 1.173\times 10^{-2} $	$ 1.279\times 10^{-2} $	$ 1.069\times 10^{-2} $	$ 1.198\times 10^{-2} $

| Show Table

DownLoad: CSV

Table 4. Scattering of plane wave by a dielectric disk: computational times of $ {\rm{NIMOR}}^{(i)} $ ($ i = 1, 2 $) and DGTD methods in terms of CPU time. The unit of time cost is second

Method	Offlin stage (Snapshots, Nested POD, GRP training)	Online stage (one run for new paramter)
DGTD	-	$ 4.254\times10^{2} $
$ {\rm{NIMOR}}^{(1)} $	$ 1.444\times10^{4} $	3.8
$ {\rm{NIMOR}}^{(2)} $	$ 2.793\times10^{4} $	3.1

| Show Table

DownLoad: CSV

Table 5. Scattering of a plane wave by a multi-layer heterogeneous medium: the distribution and range of material parameters

Layer $ i $	$ \mathcal{P}^{(i)} $	$ \mu_{r, i} $	$ r_i $
1	$ \varepsilon_{r, 1}\in[5.0, 5.6] $	1	0.15
2	$ \varepsilon_{r, 2}\in[3.25, 3.75] $	1	0.3
3	$ \varepsilon_{r, 3}\in[2.0, 2.5] $	1	0.45
4	$ \varepsilon_{r, 4}\in[1.25, 1.75] $	1	0.6

| Show Table

DownLoad: CSV

Table 6. Scattering of plane wave by a multi-layer heterogeneous medium: settings for the training, and testing datasets

Data set	Training set	Testing set
Parameter sample points	137, uneven (Smolyak mehod)	40, random (LHS method)
Time sample points	253, uniform	253, uniform
Size	34661	10120

| Show Table

DownLoad: CSV

Table 7. Scattering of plane wave by a multi-layer heterogeneous medium: the average projection and NIMOR errors on the testing set

Average relative errors	$ \overline{e}_{ {\bf E}, {\rm{Pro}}} $	$ \overline{e}_{ {\bf E}, {\rm{NIMOR}}} $	$ \overline{e}_{ {\bf H}, {\rm{Pro}}} $	$ \overline{e}_{ {\bf H}, {\rm{NIMOR}}} $
$ {\rm{NIMOR}}^{(1)} {\rm{method}} $	$ 3.668\times 10^{-3} $	$ 6.011\times 10^{-3} $	$ 4.419\times 10^{-3} $	$ 6.695\times 10^{-3} $
$ {\rm{NIMOR}}^{(2)} {\rm{method}} $	$ 3.665\times 10^{-3} $	$ 4.744\times 10^{-3} $	$ 4.405\times 10^{-3} $	$ 5.229\times 10^{-3} $

| Show Table

DownLoad: CSV

Table 8. Scattering of plane wave by a multi-layer heterogeneous medium: computational times of $ {\rm{NIMOR}}^{(i)} $ ($ i = 1, 2 $) and DGTD methods in terms of CPU time. The unit of time cost is second

Method	Offlin stage (Snapshots, Nested POD, GRP training)	Online stage (one run for new paramter)
DGTD	-	$ 4.513\times10^{2} $
$ {\rm{NIMOR}}^{(1)} $	$ 1.912\times10^{4} $	3.9
$ {\rm{NIMOR}}^{(2)} $	$ 4.221\times10^{4} $	3.3

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	F. Alsayyari, Z. Perkó, M. Tiberga, J. L. Kloosterman and D. Lathouwers, A fully adaptive nonintrusive reduced-order modelling approach for parametrized time-dependent problems, Comput. Methods Appl. Mech. Engrg., 373 (2021), Paper No. 113483, 21 pp. doi: 10.1016/j.cma.2020.113483.
[2]	C. Audouze, F. D. Vuyst and P. B. Nair, Nonintrusive reduced-order modeling of parametrized time-dependent partial differential equations, Numer. Methods Partial Differential Equations, 29 (2013), 1587-1628. doi: 10.1002/num.21768.
[3]	M. Barrault, Y. Maday, N. C. Nguyen and A. T. Patera, An 'empirical interpolation' method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Math., 339 (2004), 667-672. doi: 10.1016/j.crma.2004.08.006.
[4]	P. Benner, S. Gugercin and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57 (2015), 483-531. doi: 10.1137/130932715.
[5]	M. Bernacki, L. Fezoui, S. Lanteri and S. Piperno, Parallel discontinuous Galerkin unstructured mesh solvers for the calculation of three-dimensional wave propagation problems, Appl. Math. Model., 30 (2006), 744-763.
[6]	F. Casenave, A. Ern and T. Leliévre, A nonintrusive reduced basis method applied to aeroacoustic simulations, Adv. Comput. Math., 41 (2015), 961-986. doi: 10.1007/s10444-014-9365-0.
[7]	R. Chakir, Y. Maday and P. Parnaudeau, A non-intrusive reduced basis approach for parametrized heat transfer problems, J. Comput. Phys., 376 (2019), 617-633. doi: 10.1016/j.jcp.2018.10.001.
[8]	S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32 (2010), 2737-2764. doi: 10.1137/090766498.
[9]	W. Chen, Q. Wang, J. S. Hesthaven and C. Zhang, Physics-informed machine learning for reduced-order modeling of nonlinear problems, J. Comput. Phys., 446 (2021), Paper No. 110666, 28 pp. doi: 10.1016/j.jcp.2021.110666.
[10]	N. Dal Santo, S. Deparis and L. Pegolotti, Data driven approximation of parametrized pdes by reduced basis and neural networks, J. Comput. Phys., 416 (2020), 109550. doi: 10.1016/j.jcp.2020.109550.
[11]	S. Fresca, L. Dede and A. Manzoni, A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized pdes, J. Sci. Comput., 87 (2021). doi: 10.1007/s10915-021-01462-7.
[12]	T. Gerstner and M. Griebel, Numerical integration using sparse grids, Numer. Algorithms, 18 (1998). doi: 10.1023/A:1019129717644.
[13]	M. Guo and J. S. Hesthaven, Data-driven reduced order modeling for time-dependent problems, Comput. Methods Appl. Mech. Engrg., 345 (2019), 75-99. doi: 10.1016/j.cma.2018.10.029.
[14]	J. S. Hesthaven, G. Rozza and B. Stamm, et al., Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2016. doi: 10.1007/978-3-319-22470-1.
[15]	J. S. Hesthaven and U. Stefano, Non-intrusive reduced order modeling of nonlinear problems using neural networks, J. Comput. Phys., 363 (2018), 55-78. doi: 10.1016/j.jcp.2018.02.037.
[16]	J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, New York, 2008. doi: 10.1007/978-0-387-72067-8.
[17]	K. L. Judd, L. Maliar, S. Maliar and R. Valero, Smolyak method for solving dynamic economic models: Lagrange interpolation, anisotropic grid and adaptive domain, J. Econom. Dynam. Control, 44 (2014), 92-123. doi: 10.1016/j.jedc.2014.03.003.
[18]	E. N. Karatzas, F. Ballarin and G. Rozza, Projection-based reduced order models for a cut finite element method in parametrized domains, Comput. Math. Appl., 79 (2020), 833-851. doi: 10.1016/j.camwa.2019.08.003.
[19]	M. Kast, M. Guo and J. S. Hesthaven, A non-intrusive multifidelity method for the reduced order modeling of nonlinear problems, Comput. Methods Appl. Mech. Engrg., 364 (2020), 112947, 28 pp. doi: 10.1016/j.cma.2020.112947.
[20]	K. Kunisch and S. Volkwein, Optimal snapshot location for computing POD basis functions, M2AN Math. Model. Numer. Anal., 44 (2010), 509-529. doi: 10.1051/m2an/2010011.
[21]	O. Lass and S. Volkwein, POD-Galerkin schemes for nonlinear elliptic-parabolic systems, SIAM J. Sci. Comput., 35 (2013), 1271-1298. doi: 10.1137/110848414.
[22]	K. Li, T.-Z. Huang, L. Li and S. Lanteri, Non-intrusive reduced-order modeling of parameterized electromagnetic scattering problems using cubic spline interpolation, J. Sci. Comput., 87 (2021), Paper No. 52, 29 pp. doi: 10.1007/s10915-021-01467-2.
[23]	K. Li, T.-Z. Huang, L. Li and S. Lanteri, POD-based model order reduction with an adaptive snapshot selection for a discontinuous Galerkin approximation of the time-domain Maxwell's equations, J. Comput. Phys., 396 (2019), 106-128. doi: 10.1016/j.jcp.2019.05.051.
[24]	K. Li, T.-Z. Huang, L. Li, S. Lanteri, L. Xu and B. Li, A reduced-order discontinuous Galerkin method based on POD for electromagnetic simulation, IEEE Trans. Antennas and Propagation, 66 (2018), 242-254.
[25]	S. L. Lohr, Sampling: Design and Analysis, 2$^{nd}$ edition, Brooks/Cole, Cengage Learning, Boston, MA, 2010.
[26]	S. Lorenzi, A. Cammi, L. Luzzi and G. Rozza, POD-Galerkin method for finite volume approximation of Navier–Stokes and RANS equations, Comput. Methods Appl. Mech. Engrg., 311 (2016), 151-179. doi: 10.1016/j.cma.2016.08.006.
[27]	D. Loukrezis, U. Römer and H. D. Gersem, Assessing the performance of Leja and Clenshaw-Curtis collocation for computational electromagnetics with random input data, Int. J. Uncertain. Quantif., 9 (2019), 33-57. doi: 10.1615/Int.J.UncertaintyQuantification.2018025234.
[28]	Z. Luo, Proper orthogonal decomposition-based reduced-order stabilized mixed finite volume element extrapolating model for the nonstationary incompressible Boussinesq equations, J. Math. Anal. Appl., 425 (2015), 259-280. doi: 10.1016/j.jmaa.2014.12.011.
[29]	Z. Luo and J. Gao, A POD reduced-order finite difference time-domain extrapolating scheme for the 2D Maxwell equations in a lossy medium, J. Math. Anal. Appl., 444 (2016), 433-451. doi: 10.1016/j.jmaa.2016.06.036.
[30]	Z. Luo and W. Jiang, A reduced-order extrapolated technique about the unknown coefficient vectors of solutions in the finite element method for hyperbolic type equation, Appl. Numer. Math., 158 (2020), 123-133. doi: 10.1016/j.apnum.2020.07.025.
[31]	Z. Luo, H. Li, Y. Zhou and X. Huang, A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem, J. Math. Anal. Appl., 385 (2012), 310-321. doi: 10.1016/j.jmaa.2011.06.057.
[32]	Z. Luo, Q. Ou and Z. Xie, Reduced finite difference scheme and error estimates based on POD method for non-stationary Stokes equation, Appl. Math. Mech., 32 (2011), 847-858. doi: 10.1007/s10483-011-1464-9.
[33]	Z. Luo and F. Teng, A reduced-order extrapolated finite difference iterative scheme based on POD method for 2D sobolev equation, Appl. Math. Comput., 329 (2018), 374-383. doi: 10.1016/j.amc.2018.02.022.
[34]	Z. Luo, F. Teng and H. Xia, A reduced-order extrapolated Crank-Nicolson finite spectral element method based on POD for the 2D non-stationary boussinesq equations, J. Math. Anal. Appl., 471 (2019), 564-583. doi: 10.1016/j.jmaa.2018.10.092.
[35]	Z. Luo and H. Ren, A reduced-order extrapolated finite difference iterative method for the riemann-liouville tempered fractional derivative equation, Appl. Numer. Math., 157 (2020), 307-314. doi: 10.1016/j.apnum.2020.05.028.
[36]	Z. Luo and J. Shiju, A reduced-order extrapolated Crank-Nicolson collocation spectral method based on proper orthogonal decomposition for the two-dimensional viscoelastic wave equations, Numer. Methods Partial Differential Equations, 36 (2020), 49-65. doi: 10.1002/num.22397.
[37]	B. Peherstorfer, K. Willcox and M. Gunzburger, Survey of multifidelity methods in uncertainty propagation, inference, and optimization, SIAM Rev., 60 (2018), 550-591. doi: 10.1137/16M1082469.
[38]	P. Phalippou, S. Bouabdallah, P. Breitkopf, P. Villon and M. Zarroug, 'On-the-fly' snapshots selection for proper orthogonal decomposition with application to nonlinear dynamics, Comput. Methods Appl. Mech. Engrg., 367 (2020), 113120. doi: 10.1016/j.cma.2020.113120.
[39]	C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning, MIT press, 2006.
[40]	N. Schmitt, C. Scheid, J. Viquerat and S. Lanteri, Simulation of three-dimensional nanoscale light interaction with spatially dispersive metals using a high order curvilinear DGTD method, J. Comput. Phys., 373 (2018), 210-229. doi: 10.1016/j.jcp.2018.06.033.
[41]	G. Stabile and G. Rozza, Finite POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations, Comput. & Fluids, 173 (2018), 273-284. doi: 10.1016/j.compfluid.2018.01.035.
[42]	X. Sun, X. Pan and J.-I. Choi, Non-intrusive framework of reduced-order modeling based on proper orthogonal decomposition and polynomial chaos expansion, J. Comput. Appl. Math., 390 (2021), 113372. doi: 10.1016/j.cam.2020.113372.
[43]	S. Ullmann, M. Rotkvic and J. Lang, POD-Galerkin reduced-order modeling with adaptive finite element snapshots, J. Comput. Phys., 325 (2016), 244-258. doi: 10.1016/j.jcp.2016.08.018.
[44]	F. Vidal-Codina, N. C. Nguyen and J. Peraire, Computing parametrized solutions for plasmonic nanogap structures, J. Comput. Phys., 366 (2018), 89-106. doi: 10.1016/j.jcp.2018.04.009.
[45]	J. Viquerat and S. Lanteri, Simulation of near-field plasmonic interactions with a local approximation order discontinuous Galerkin time-domain method, Photonics Nanostruct., 18 (2016), 43-58.
[46]	Q. Wang, J. S. Hesthaven and D. Ray, Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem, J. Comput. Phys., 384 (2019), 289-307. doi: 10.1016/j.jcp.2019.01.031.
[47]	Q. Wang, N. Ripamonti and J. S. Hesthaven, Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori-Zwanzig formalism, J. Comput. Phys., 410 (2020), 109402, 32 pp. doi: 10.1016/j.jcp.2020.109402.
[48]	C. K. I. Williams and C. E. Rasmussen, Gaussian processes for regression, In Proceedings of the 8th International Conference on Neural Information Processing Systems, MIT, (1995), 514–520.
[49]	D. Xiao, F. Fang, C. Pain and I. Navon, A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications, Comput. Methods Appl. Mech. Engrg., 317 (2017), 868-889. doi: 10.1016/j.cma.2016.12.033.
[50]	D. Xiao, Z. Lin, F. Fang, C. C. Pain, L. M. Navon, P. Salinas and A. Muggeridge, Non-intrusive reduced-order modeling for multiphase porous media flows using smolyak sparse grids, Internat. J. Numer. Methods Fluids, 83 (2017), 205-219. doi: 10.1002/fld.4263.
[51]	B. Xu and X. Zhang, An efficient high-order compact finite difference scheme based on proper orthogonal decomposition for the multi-dimensional parabolic equation, Adv. Differ. Equ., 2019 (2019), 341. doi: 10.1186/s13662-019-2273-3.
[52]	R. Yondo, E. Andrés and E. Valero, A review on design of experiments and surrogate models in aircraft real-time and many-query aerodynamic analyses, Prog. Aerosp. Sci., 96 (2018), 23-61.
[53]	J. Yu, C. Yan, Z. Jiang, W. Yuan and S. Chen, Adaptive non-intrusive reduced order modeling for compressible flows, J. Comput. Phys., 397 (2019), 108855. doi: 10.1016/j.jcp.2019.07.053.
[54]	X. Zhang and P. Zhang, A reduced high-order compact finite difference scheme based on proper orthogonal decomposition technique for KdV equation, Appl. Math. Comput., 339 (2018), 535-545. doi: 10.1016/j.amc.2018.07.017.
[55]	Y. Zhou, Y. Zhang, Y. Liang and Z. Luo, A reduced-order extrapolated model based on splitting implicit finite difference scheme and proper orthogonal decomposition for the fourth-order nonlinear rosenau equation, Appl. Numer. Math., 162 (2021), 192-200. doi: 10.1016/j.apnum.2020.12.020.