Volume 42 Issue 6
Jun.  2021
Turn off MathJax
Article Contents
LAI Xuefang, WANG Xiaolong, NIE Yufeng. Nonlinear Model Reduction Based on the Mori-Zwanzig Scheme and Partial Least Squares[J]. Applied Mathematics and Mechanics, 2021, 42(6): 551-561. doi: 10.21656/1000-0887.410230
Citation: LAI Xuefang, WANG Xiaolong, NIE Yufeng. Nonlinear Model Reduction Based on the Mori-Zwanzig Scheme and Partial Least Squares[J]. Applied Mathematics and Mechanics, 2021, 42(6): 551-561. doi: 10.21656/1000-0887.410230

Nonlinear Model Reduction Based on the Mori-Zwanzig Scheme and Partial Least Squares

doi: 10.21656/1000-0887.410230

The National Natural Science Foundation of China(11871400;11971386)

  • Received Date: 2020-08-05
  • Rev Recd Date: 2021-01-06
  • The proper orthogonal decomposition and the Galerkin projection are widely used methods for solving the model reduction problems of complex nonlinear systems. However, only a part of basis function modes are extracted with these methods to construct the reduced systems, which usually makes the reduced systems inaccurate. For this issue a method was proposed to efficiently correct the errors of the reduced systems. First, the Mori-Zwanzig scheme was employed to analyze the errors of the reduced systems, with the theoretical form of the error model and the effective predictive variables obtained. Then, the error prediction model was built by means of the partial least square method to construct the multiple regression model between the predictive variables and the system errors. The constructed error prediction model was directly embedded into the original reduced system, to get a modified reduced system formally equivalent to the model obtained with the Petrov-Galerkin projection on the right side of the original model. The error estimation of the modified reduced system was given. Numerical results illustrate that, the proposed method can improve the stability and accuracy of the reduced systems effectively, and has high computation efficiency.
  • loading
  • [2]SCHILDERS W H A, VORST H A V D, ROMMES J. Model Order Reduction: Theory, Research Aspects and Applications[M]. Berlin, Heidelberg: Springer, 2008.
    蒋耀林. 模型降阶方法[M]. 北京:科学出版社, 2010.(JIANG Yaolin. Model Order Reduction Methods[M]. Beijing: Science Press, 2010.(in Chinese))
    [3]张珺, 李立州, 原梅妮. 径向基函数参数化翼型的气动力降阶模型优化[J]. 应用数学和力学, 2019,40(3): 250-258.(ZHANG Jun, LI Lizhou, YUAN Meini. Optimization of RBF parameterized airfoils with the aerodynamic ROM[J]. Applied Mathematics and Mechanics,2019,40(3): 250-258.(in Chinese))
    [4]SIROVICH L. Turbulence and the dynamics of coherent structures Ⅰ: coherent structures[J]. Quarterly of Applied Mathematics,1987,45(3): 561-571.
    [5]罗振东, 欧秋兰, 谢正辉. 非定常Stokes方程一种基于POD方法的简化有限差分格式[J]. 应用数学和力学, 2011,32(7): 795-806.(LUO Zhendong, OU Qiulan, XIE Zhenghui. A reduced finite difference scheme and error estimates based on POD method for the non-stationary Stokes equation[J]. Applied Mathematics and Mechanics,2011,32(7): 795-806.(in Chinese))
    [6]AMBILI M, SARKAR A, PADOUSSIS M P. Reduced-order models for nonlinear vibrations of cylindrical shells via the proper orthogonal decomposition method[J]. Journal of Fluids and Structures,2003,18(2): 227-250.
    [7]郭志文, 肖曼玉, 夏凉. 基于特征正交分解的材料微结构参数化表征模型及等效性能优化设计[J]. 应用数学和力学, 2017,38(7): 250-258.(GUO Zhiwen, XIAO Manyu, XIA Liang. A POD-based parameterization model for material microstructure representation and its application to optimal design of material effective mechanical properties[J]. Applied Mathematics and Mechanics,2017,38(7): 250-258.(in Chinese))
    [8]GUO M W, HESTHAVEN J S. Data-driven reduced order modeling for time-dependent problems[J]. Computer Methods in Applied Mechanics and Engineering,2019,345(1): 75-99.
    [9]KUNISCH K, VOLKWEIN S. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics[J]. SIAM Journal on Numerical Analysis,2019,40(2): 492-515.
    [10]CARLBERG K, BOU-MOSLEH C, CHARBEL F. Efficient nonlinear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations[J]. International Journal for Numerical Methods in Engineering,2011,86(2): 155-181.
    [11]CARLBERG K, BARONE M, ANTIL H. Galerkin v. least-squares Petrov-Galerkin projection in nonlinear model reduction[J]. Journal of Computational Physics,2017,330: 693-734.
    [12]SAGAUT P. Large Eddy Simulation for Incompressible Flows: an Introduction[M]. New York: Springer Science and Business Media, 2006.
    [13]SAN O, MAULIK R. Neural network closures for nonlinear model order reduction[J]. Advances in Computational Mathematics,2018,44(6): 1717-1750.
    [14]XIE X, MOHEBUJJAMAN M, REBHOLZ L, et al. Data-driven filtered reduced order modeling of fluid flows[J]. SIAM Journal on Scientific Computing,2018,40(3): 834-857.
    [15]WANG Z, AKHTAR I, BORGGAARD J, et al. Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison[J].Computer Methods in Applied Mechanics and Engineering,2012,237: 10-26.
    [16]JING L, PANOS S. Mori-Zwanzig reduced models for uncertainty quantification[J]. Journal of Computational Dynamics,2019,6(1): 39-68.
    [17]MA C, WANG J C, EE W N. Model reduction with memory and the machine learning of dynamical systems[J]. Communications in Computational Physics,2019,25(4): 947-962.
    [18]PARISH E J, WENTLAND C R, DURAISAMY K. The adjoint Petrov-Galerkin method for non-linear model reduction[J]. Computer Methods in Applied Mechanics and Engineering,2020,365: 112991.
    [19]PAN S, DURAISAMY K. Data-driven discovery of closure models[J]. SIAM Journal on Applied Dynamical Systems,2018,17(4): 2381-2413.
    [20]WANG Q, RIPAMONTI N, HESTHAVEN J S. Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori-Zwanzig formalism[J]. Journal of Computational Physics,2020,410: 109402.
    [21]WOLD S, RUHE A, WOLD H, et al. The collinearity problem in linear regression. the partial least squares (PLS) approach to generalized inverses[J]. SIAM Journal on Scientific and Statistical Computing,1984,5(1): 735-743.
    [22]ABDI H. Partial least squares regression and projection on latent structure regression (PLS regression)[J]. Wiley Interdisciplinary Reviews: Computational Statistics,2010,2(1): 97-106.
    [23]CHATURANTABUT S, SORENSEN D C. Nonlinear model reduction via discrete empirical interpolation[J]. SIAM Journal on Scientific Computing,2010,32(5): 2737-2764.
    [24]ALLA A, KUTZ J N. Nonlinear model order reduction via dynamic mode decomposition[J]. SIAM Journal on Scientific Computing,2017,39(5): 778-796.
    [25]PARISH E J, DURAISAMY K. A dynamic subgrid scale model for large eddy simulations based on the Mori-Zwanzig formalism[J]. Journal of Computational Physics,2017,349: 154-175.
    [26]PARISH E J, DURAISAMY K. A unified framework for multiscale modeling using Mori-Zwanzig and the variational multiscale method[Z/OL]. arXiv Preprint, 2018. [2020-11-29]. https://arxiv.org/pdf/1712.09669.pdf.
    [27]CHAFEE N, INFANTE E F. A bifurcation problem for a nonlinear partial differential equation of parabolic type[J]. Applicable Analysis,1974,4(1): 17-37.
    [28]BENNER P, BREITEN T. Two-sided projection methods for nonlinear model order reduction[J]. SIAM Journal on Scientific Computing,2015,37(2): 239-260.
  • 加载中


    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (704) PDF downloads(100) Cited by()
    Proportional views


    DownLoad:  Full-Size Img  PowerPoint