基于数据驱动的流场控制方程的稀疏识别

江昊; 王伯福; 卢志明

doi:10.6052/0459-1879-21-052

基于数据驱动的流场控制方程的稀疏识别

doi: 10.6052/0459-1879-21-052

上海大学, 力学与工程科学学院, 上海市应用数学和力学研究所, 上海 200072

基金项目: 1)国家自然科学基金资助项目(12072185);国家自然科学基金资助项目(11732010);国家自然科学基金资助项目(11972220)

详细信息

作者简介:
2)卢志明, 教授, 主要研究方向: 湍流, 环境流体力学, 计算流体力学. E-mail: zmlu@shu.edu.cn

通讯作者:
卢志明

中图分类号: O357
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- 文章访问数: 219
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- 被引次数: 0
出版历程
- 收稿日期: 2021-01-29
- 录用日期: 2021-06-18

DATA-DRIVEN SPARSE IDENTIFICATION OF GOVERNING EQUATIONS FOR FLUID DYNAMICS

Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, China

摘要

摘要: 利用有限数据建立系统的非线性动力学模型是具有挑战性的重要课题. 数据驱动的稀疏识别方法是近年来发展的从数据识别动力系统控制方程的有效方法. 本文基于数据驱动稀疏识别方法对不同流场的控制方程进行了识别. 采用非线性动力学偏微分方程函数识别(partial differential equations functional identification of nonlinear dynamics, PDE-FIND)方法和最小绝对收缩和选择算子(least absolute shrinkage and selection operator, LASSO)方法对二维圆柱绕流、顶盖驱动方腔流、Rayleigh-Bénard (RB)对流和三维槽道湍流的控制方程进行了识别. 在稀疏识别过程中, 采用直接数值模拟得到的流场数据来计算过完备候选库中的每一项, 候选库中变量最高保留到二次, 变量导数最高保留到二阶, 非线性项最高保留到四阶. 结果发现PDE-FIND方法和LASSO方法对于不含有非线性项的控制方程, 如涡量输运方程、热输运方程和连续性方程, 都能准确识别. 对于含有强非线性项的控制方程, 如Navier-Stokes方程的识别, PDE-FIND方法正确地识别出了控制方程及流场的Rayleigh数和Reynolds数, 而LASSO方法识别结果不正确, 这是因为候选库中的项之间存在分组效应, LASSO方法通常只取分组中的一项. 本文还发现选择流动结构丰富的区域的数据进行控制方程的稀疏识别可以提高识别的准确性.
- 数据驱动 /
- 纳维斯托克斯方程 /
- 稀疏识别 /
- 偏微分方程识别方法 /
- 套索方法
Abstract: It is a challenging and important issue to establish a nonlinear dynamic model of system by use of limited data. The data-driven sparse identification method is an effective method developed recently to identify the governing equations of the dynamic system from data developed in recent years. In this paper, governing equations for different flows are identified by data-driven sparse identification methods. Partial differential equation functional identification of nonlinear dynamics (PDE-FIND) scheme and least absolute shrinkage and selection operator (LASSO) scheme are used to identify the governing equations of two-dimensional flow past a circular cylinder, liddriven cavity flow, Rayleigh-Bénard convection and three-dimensional turbulent channel flow. An over-complete candidate library is constructed by direct numerical simulation flow field data in the process of identification. Variables in the library are retained up to second order, variable derivatives are retained up to second order, and nonlinear terms are retained up to fourth order. By comparing the results from the two methods, we find both methods show good performance in identifying governing equation with no nonlinear terms, i.e., vorticity transport equation, heat transport equation and continuity equation. PDE-FIND scheme correctly identified the governing equations and Rayleigh number and Reynolds number for the flow field. But LASSO scheme failed to identify the governing equations which contain strong nonlinear terms, i.e., Navier-Stokes equations. This is because grouping effect may occur among the items in the candidate library and only one item in the group is chosen in such case in LASSO scheme. So PDE-FIND scheme is more effective than LASSO scheme in sparse identification of strongly nonlinear partial differential equation. It is also found that selecting data from regions with abundant flow structures can improve the accuracy of data-driven sparse identification results.
- data-driven /
- Navier-Stokes equation /
- sparse identification /
- PDE-FIND /
- LASSO

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参考文献(33)

张慧洁, 郝瑞林. 电网中动态电压恢复器的非线性控制研究. 电子测试, 2019(Z1): 14-16

(Zhang Huijie, Hao Ruilin. Research on dynamic voltage restorer based on nonlinear control theory in the power grid. Electronic Test, 2019(Z1): 14-16 (in Chinese))

Yang CH, Tsai CY. Nonlinear dynamic analysis of finance system and its analog/digital circuit implementation. Journal of Computational and Theoretical Nanoscience, 2015, 12(10): 3538-3546

Navarro LG, Jurado FL, Castillo MJ, et al. Assessment of autonomous nerve system through non-linear heart rate variability outcomes in sedentary healthy adults. Peer J, 2020, 8(11): e10178

张伟伟, 寇家庆, 刘溢浪. 智能赋能流体力学展望. 航空学报, 2021, 42(4): 1-51

(Zhang Weiwei, Kou Jiaqing, Liu Yilang. Prospect of artificial intelligence empowering fluid mechanics. Acta Aeronautica et Astronautica Sinica, 2021, 42(4): 1-51 (in Chinese))

Crutchfield JP, McNamara BS. Equations of motion from a data series. Complex Systems, 1987, 1(121): 417-452

洪泽华, 章佳君, 周金鹏等. 复杂目标红外辐射特性大数据知识表示及建模方法. 上海航天, 2020, 37(6): 52-57

(Hong Zehua, Zhang Jiajun, Zhou Jinpeng, et al. Big Data knowledge representation and modeling method for infrared radiation characteristics of comp. Aerospace Shanghai, 2020, 37(6): 52-57 (in Chinese))

曾贲, 房霄, 孔德帅等. 一种数据驱动的对抗博弈智能体建模方法. 系统仿真学报, 2020: 1-7

(Zeng Bi, Fang Xiao, Kong Deshuai, et al. A data-driven modeling method for game adversity agent. Journal of System Simulation, 2020: 1-7 (in Chinese))

李星. 数据驱动建模中精度与泛化平衡问题研究. [硕士论文]. 西安: 西安理工大学, 2020

(Li Xing. Research on the balance between accuracy and generalization in data-driven modeling. [Master Thesis]. Xi'an: Xi'an University of Technology, 2020 (in Chinese))

赵丹. 非线性系统即时学习建模方法研究. [硕士论文]. 镇江: 江苏大学, 2017

(Zhao Dan. Research on just-in-time learning modeling for nonlinear systems. [Master Thesis]. Zhenjiang: Jiangsu University, 2017 (in Chinese))

González-García R, Rico-Martínez R, Kevrekidis IG. Identification of distributed parameter systems: A neural net based approach. Computers & Chemical Engineering, 1998, 22: S965-S968

Kevrekidis IG, Gear CW, Hyman JM, et al. Equation-free, coarse-grained multiscale computation: Enabling mocroscopic simulators to perform system-level analysis. Communications in Mathematical Sciences, 2003, 1(4): 715-762

Daniels BC, Nemenman I. Automated adaptive inference of phenomenological dynamical models. Nature Communications, 2015, 6(1): 1-8

Bongard J, Lipson H. Automated reverse engineering of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 2007, 104(24): 9943-9948

Tibshirani R. Regression shrinkage and selection via the lasso. Stat. Soc. B, 1996, 58(1): 267-288

Tibshirani R. The lasso method for variable selection in the Cox model. Statistics in Medicine, 1997, 16(4): 385-395

Schmidt M, Lipson H. Distilling free-form natural laws from experimental data. Science, 2009, 324(5923): 81-85

Brunton SL, Kutz JN. Data-driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. England: Cambridge University Press, 2019

Brunton SL, Proctor JL, Kutz JN. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences of the United States of America, 2016, 113(15): 3932-3937

Rudy SH, Brunton SL, Proctor JL, et al. Data-driven discovery of partial differential equations. Science Advances, 2017, 3(4): e1602614

胡军, 刘全, 倪国喜. 时变偏微分方程的贝叶斯稀疏识别方法. 计算物理, 2021, 38(1): 25-34

(Hu Jun, Liu Quan, Ni Guoxi. Bayesian sparse identification of time-varying partial differential equations. Chinese Journal of Computational Physics, 2021, 38(1): 25-34 (in Chinese))

Berkooz G, Holmes P, Lumley JL. The proper orthogonal decomposition in the analysis of turbulent flows. Annual Review of Fluid Mechanics, 1993, 25(1): 539-575

Brunton SL, Noack BR, Koumoutsakos P. Machine learning for fluid mechanics. Annual Review of Fluid Mechanics, 2020, 52: 477-508

Chang H, Zhang D. Machine learning subsurface flow equations from data. Computational Geosciences, 2019, 23(5): 895-910

Raissi M, Yazdani A, Karniadakis GE. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Science, 2020, 367(6481): 1026-1030

Zhang J, Ma W. Data-driven discovery of governing equations for fluid dynamics based on molecular simulation. Journal of Fluid Mechanics, 2020, 892: A5

Schmelzer M, Dwight RP, Cinnella P. Discovery of algebraic Reynolds-stress models using sparse symbolic regression. Flow Turbulence and Combustion, 2020, 104(2-3): 579-603

张亦知, 程诚, 范钇彤等. 基于物理知识约束的数据驱动式湍流模型修正及槽道湍流计算验证. 航空学报, 2020, 41(3): 119-128

(Zhang Yizhi, Cheng Cheng, Fan Yitong, et al. Data-driven correction of turbulence model with physics knowledge constrains in channel flow. Acta Aeronautica et Astronautica Ainica, 2020, 41(3): 119-128 (in Chinese))

Hoerl AE, Kennard RW. Ridge regression: Applications to nonorthogonal problems. Technometrics, 1970, 12(1): 69-82

Benjamin AT, Ericksen L, Jayawant P, et al. Combinatorial trigonometry with Chebyshev polynomials. Journal of Statistical Planning and Inference, 2010, 140(8): 2157-2160

Zhang YZ, Xia SN, Dong YH, et al. An efficient parallel algorithm for DNS of buoyancy-driven turbulent flows. Journal of Hydrodynamics, Ser. B, 2019, 31(6): 1159-1169

Zou H, Hastie T. Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2005, 67(2): 301-320

Maddu S, Cheeseman BL, Sbalzarini IF, et al. Stability selection enables robust learning of partial differential equations from limited noisy data. arXiv preprint, 2019, arXiv:1907.07810

Bo Y, Wang P, Guo Z, et al. DUGKS simulations of three-dimensional Taylor-Green vortex flow and turbulent channel flow. Computers & Fluids, 2017, 155: 9-21

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