| 基于Mori-Zwanzig格式和偏最小二乘的非线性模型降阶 |
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| 引用本文： | 赖学方,王晓龙,聂玉峰.基于Mori-Zwanzig格式和偏最小二乘的非线性模型降阶[J].应用数学和力学,2021,42(6):551-561. |
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| 作者姓名： | 赖学方 王晓龙 聂玉峰 |
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| 作者单位： | 西北工业大学 数学与统计学院， 西安 710129 |
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| 基金项目： | 国家自然科学基金(11871400；11971386) |
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| 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.
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| 关 键 词： | 模型降阶 Mori-Zwanzig格式 偏最小二乘 误差校正 Petrov-Galerkin投影 |
| 收稿时间： | 2020-08-05 |
Nonlinear Model Reduction Based on the Mori-Zwanzig Scheme and Partial Least Squares |
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| Affiliation: | School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129, P.R.China |
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| 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. |
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