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Chin. Phys. B, 2021, Vol. 30(3): 030204    DOI: 10.1088/1674-1056/abd92e
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Constructing reduced model for complex physical systems via interpolation and neural networks

Xuefang Lai(赖学方), Xiaolong Wang(王晓龙)†, and Yufeng Nie(聂玉峰)
1 Research Center for Computational Science, School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129, China
Abstract  The work studies model reduction method for nonlinear systems based on proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). Instead of using the classical DEIM to directly approximate the nonlinear term of a system, our approach extracts the main part of the nonlinear term with a linear approximation before approximating the residual with the DEIM. We construct the linear term by Taylor series expansion and dynamic mode decomposition (DMD), respectively, so as to obtain a more accurate reconstruction of the nonlinear term. In addition, a novel error prediction model is devised for the POD-DEIM reduced systems by employing neural networks with the aid of error data. The error model is cheaply computable and can be adopted as a remedy model to enhance the reduction accuracy. Finally, numerical experiments are performed on two nonlinear problems to show the performance of the proposed method.
Keywords:  model reduction      discrete empirical interpolation method      dynamic mode decomposition      neural networks  
Received:  12 November 2020      Revised:  17 December 2020      Accepted manuscript online:  07 January 2021
PACS:  02.30.Jr (Partial differential equations)  
  02.60.Cb (Numerical simulation; solution of equations)  
  02.60.Gf (Algorithms for functional approximation)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11871400 and 11971386) and the Natural Science Foundation of Shaanxi Province, China (Grant No. 2017JM1019).
Corresponding Authors:  Corresponding author. E-mail: xlwang@nwpu.edu.cn   

Cite this article: 

Xuefang Lai(赖学方), Xiaolong Wang(王晓龙, and Yufeng Nie(聂玉峰) Constructing reduced model for complex physical systems via interpolation and neural networks 2021 Chin. Phys. B 30 030204

1 Wang X L and Jiang Y L 2013 Math. Comp. Model. Dyn. 19 575
2 Liu X Z, Yu J, Lou Z M and Qian X M 2019 Chin. Phys. B 28 010201
3 Qu Y H, Wang A N and Lin S 2018 Chin. Phys. B 27 010203
4 Chen Y and White J Proc. Int. Conf. Modeling and Simulation of Microsystems 2000 477
5 Rewienski M and White J IEEE. T. Comput. Aid. D 22 155
6 Rewie\'nski M A Trajectory Piecewise-Linear Approach to Model Order Reduction of Nonlinear Dynamical Systems (Ph.D. Dissertation) (Boston: Massachusetts Institute of Technology)
7 Ning D and Roychowdhury J IEEE Trans. Comput-Aided Des. Integr. Circuits Syst. 27 249
8 Holmes P, Lumley J L, Berkooz G and Rowley C W 2012 Turbulence, coherent structures, dynamical systems and symmetry, 2nd edition (England: Cambridge University Press) pp. 68-105
9 Li S, Jiang N, Yang S Q, Huang Y X and Wu Y H 2018 Chin. Phys. B 27 104701
10 Xu W, Li S and Li R H 2007 Chin. Phys. B 16 1591
11 Hesthaven J S and Ubbiali S 2018 J. Comput. Phys. 363 55
12 Guo M W and Hesthaven J S 2019 Comput. Method Appl. M. 345 75
13 Kunisch K and Volkwein S 2002 SIAM J. Numer. Anal. 40 492
14 Everson R and Sirovich L 1995 J. Opt. Soc. Am. A 12 1657
15 Barrault M, Maday Y and Nguyen N C 2004 C. R. Math. 339 667
16 Nguyen N C, Patera A T and Peraire J 2008 Int. J. Numer. Meth. Eng. 73 521
17 Chaturantabut S and Sorensen D C 2011 SIAM J. Sci. Comput. 32 2737
18 Willcox K 2006 Comput. Fluids 35 208
19 Kutz J N, Brunton S L, Brunton B W and Proctor J L 2016 Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems(Philadelphia: SIAM)
20 Xiao D, Fang F, Buchan A G and Ionel M N 2014 J. Comput. Phys. 8 1
21 Bistrian D A and Navon I M 2017 Int. J. Numer. Meth. Eng. 112 3
22 Alla A and Kutz J N 2017 SIAM J. Sci. Comput. 39 B778
23 Chaturantabut S and Sorensen D C 2012 SIAM J. Numer. Anal. 50 46
24 Wirtz D, Sorensen D C and Haasdonk B 2014 SIAM J. Sci. Comput. 36 A311
25 Zhao Z Q and Huang D S 2007 Appl. Math. Model. 31 1271
26 Dimitriu G, Stefanescu R and Navon I M 2017 J. Comput. Appl. Math.310 32
27 Drohmann M and Carlberg K 2015 SIAM/ASA J. Uncertain. 3 116
28 Xiao D 2019 Comput. Method. Appl. M. 355 513
29 Lipponen A, Kolehmainen V, Romakkaniemi S and Kokkola H 2013 Geosci. Model Dev. 6 2087
30 Lipponen A, Huttunen J M J, Romakkaniemi S, Kokkola H and Kolehmainen V 2018 SIAM J. Sci. Comput. 40 B305
31 Freno B A and Carlberg K T 2019 Comput. Method. Appl. M. 348 250
32 Zhang W B, Guo X C, Wang C Y and Wu C G 2007 Adaptive and Natural Computing Algorithms, Lecture Notes in Computer Science(Berlin: Springer) pp. 189-198
33 Liberty E, Woolfe F, Martinsson P G, Rokhlin V and Tygert M 2007 Proc. Natl. Acad. Sci. USA 51 104
34 Gu\acuteeniat F, Mathelin L and Pastur L R 2015 Phys. Fluids 27 025113
35 Lai X F, Wang X L, Nie Y F and Li Q 2020 Int. J. Numer. Meth. Fluids 92 587
36 Zhang W W, Kou J Q and Wang Z Y 2016 AIAA J. 54 3302
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