通过变量提升将非线性能量转为二次型,实现高维保守系统的保能降维。
Nonlinear energy-preserving model reduction with lifting transformations that quadratize the energy
- 引入升维变量使能量表达式变为二次型,构建保结构的降维模型。
- 在4个典型保守偏微分方程上验证,精度与现有最优方法相当,线上效率更优。
- 适合需精确保能且处理复杂非线性系统的科学计算场景。
针对高维保守偏微分方程(PDEs)中非多项式非线性导致的传统模型降维方法计算瓶颈,本文提出一种基于变量提升的非线性降维方法。通过能量二次化策略,将能量表达式中的非线性项转化为辅助变量形式,构造等价的二次提升系统,使其能量保持二次结构。结合本征正交分解(POD),可得到在高维下精确保持二次化提升能量的二次型降维模型。该方法在四个非线性保守PDE上进行了验证:一维指数非线性波动方程、二维Sine-Gordon方程、含参数依赖的二维Klein-Gordon方程,以及二维Klein-Gordon-Zakharov方程。数值结果表明,该方法在线阶段的精度和计算效率与当前最优的保结构超还原方法相当,且在离线阶段具有显著计算优势。
原文摘要 · Abstract (English)
Existing model reduction techniques for high-dimensional models of conservative partial differential equations (PDEs) encounter computational bottlenecks when dealing with systems featuring non-polynomial nonlinearities. This work presents a nonlinear model reduction method that employs lifting variable transformations to derive structure-preserving quadratic reduced-order models for conservative PDEs with general nonlinearities. We present an energy-quadratization strategy that defines the auxiliary variable in terms of the nonlinear term in the energy expression to derive an equivalent quadratic lifted system with quadratic system energy. The proposed strategy combined with proper orthogonal decomposition model reduction yields quadratic reduced-order models that conserve the quadratized lifted energy exactly in high dimensions. We demonstrate the proposed model reduction approach on four nonlinear conservative PDEs: the one-dimensional wave equation with exponential nonlinearity, the two-dimensional sine-Gordon equation, the two-dimensional Klein-Gordon equation with parametric dependence, and the two-dimensional Klein-Gordon-Zakharov equations. The numerical results show that the proposed lifting approach is competitive with the state-of-the-art structure-preserving hyper-reduction method in terms of both accuracy and computational efficiency in the online stage while providing significant computational gains in the offline stage.
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