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.

翻译：针对保守偏微分方程高维模型的现有模型降阶技术，在处理具有非多项式非线性项的系统时会遇到计算瓶颈。本文提出一种非线性模型降阶方法，该方法采用提升变量变换，为具有一般非线性项的保守偏微分方程推导出结构保持的二次型降阶模型。我们提出一种能量二次化策略，该策略根据能量表达式中的非线性项定义辅助变量，从而推导出具有二次系统能量的等价二次提升系统。所提策略与适当正交分解模型降阶相结合，可得到在高维空间中精确保持二次化提升能量的二次型降阶模型。我们在四个非线性保守偏微分方程上验证了所提出的模型降阶方法：具有指数非线性项的一维波动方程、二维正弦-戈尔登方程、具有参数依赖性的二维克莱因-戈尔登方程，以及二维克莱因-戈尔登-扎哈罗夫方程组。数值结果表明，所提出的提升方法在在线阶段的精度和计算效率方面与最先进的保结构超减缩方法相当，同时在离线阶段提供了显著的计算增益。