基于Mori-Zwanzig格式和偏最小二乘的非线性模型降阶

赖学方; 王晓龙; 聂玉峰

doi:10.21656/1000-0887.410230

基于Mori-Zwanzig格式和偏最小二乘的非线性模型降阶

doi: 10.21656/1000-0887.410230

西北工业大学数学与统计学院，西安 710129

基金项目:

国家自然科学基金(11871400；11971386)

详细信息

作者简介:
赖学方(1988—)，男，博士生(E-mail: xfanglai@mail.nwpu.edu.cn);聂玉峰(1968—)，男，教授，博士，博士生导师(通讯作者. E-mail: yfnie@nwpu.edu.cn).

通讯作者:
聂玉峰(1968—)，男，教授，博士，博士生导师(通讯作者. E-mail: yfnie@nwpu.edu.cn).

中图分类号: O242.2
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- 文章访问数: 860
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- PDF下载量: 110
- 被引次数: 0
出版历程
- 收稿日期: 2020-08-05
- 修回日期: 2021-01-06

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

School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129, P.R.China

Funds:

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

摘要

摘要: 本征正交分解及Galerkin投影是解决复杂非线性系统模型降阶问题常用的方法.然而，该方法在构造降阶系统过程中只截取基函数的部分模态，这通常会使得降阶系统不准确.针对该问题，提出了对降阶系统误差进行快速校正的方法.首先应用Mori-Zwanzig格式对降阶系统的误差进行分析，理论上得到误差模型的形式和有效预测变量.再通过偏最小二乘方法构造预测变量和系统误差的多元回归模型，建立误差预测模型.将所构造的误差预测模型直接嵌入到原降阶系统，得到新的降阶系统在形式上等价于对原模型的右端采用Petrov-Galerkin投影.最后给出了新的降阶系统的误差估计.数值结果进一步说明了所提方法能有效地提高降阶系统的稳定性和准确性，且具有较高计算效率.
- 模型降阶 /
- Mori-Zwanzig格式 /
- 偏最小二乘 /
- 误差校正 /
- Petrov-Galerkin投影
Abstract: 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.
- model reduction /
- Mori-Zwanzig scheme /
- partial least square /
- error correction /
- Petrov-Galerkin projection

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参考文献(28)

[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.

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基于Mori-Zwanzig格式和偏最小二乘的非线性模型降阶

doi: 10.21656/1000-0887.410230

作者简介:
赖学方(1988—)，男，博士生(E-mail: xfanglai@mail.nwpu.edu.cn);聂玉峰(1968—)，男，教授，博士，博士生导师(通讯作者. E-mail: yfnie@nwpu.edu.cn).

通讯作者:
聂玉峰(1968—)，男，教授，博士，博士生导师(通讯作者. E-mail: yfnie@nwpu.edu.cn).

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

计量

目录

留言板

基于Mori-Zwanzig格式和偏最小二乘的非线性模型降阶

doi: 10.21656/1000-0887.410230

作者简介: 赖学方(1988—)，男，博士生(E-mail: xfanglai@mail.nwpu.edu.cn);聂玉峰(1968—)，男，教授，博士，博士生导师(通讯作者. E-mail: yfnie@nwpu.edu.cn).

通讯作者: 聂玉峰(1968—)，男，教授，博士，博士生导师(通讯作者. E-mail: yfnie@nwpu.edu.cn).

计量

出版历程

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

计量

出版历程

目录

作者简介:
赖学方(1988—)，男，博士生(E-mail: xfanglai@mail.nwpu.edu.cn);聂玉峰(1968—)，男，教授，博士，博士生导师(通讯作者. E-mail: yfnie@nwpu.edu.cn).

通讯作者:
聂玉峰(1968—)，男，教授，博士，博士生导师(通讯作者. E-mail: yfnie@nwpu.edu.cn).