Nonlinear energy-preserving model reduction with lifting transformations that quadratize the energy

TL;DR

The paper tackles the challenge of reducing high-dimensional conservative PDEs with non-polynomial nonlinearities while preserving energy. It introduces an energy-quadratization lifting strategy that augments the state with auxiliary variables so that the lifted energy becomes quadratic; after POD projection, the resulting ROMs are quadratic and energy-conserving in the lifted sense. The authors prove that, under mild assumptions, the proposed lifting yields ROMs that conserve the lifted FOM energy exactly, and they demonstrate this on four nonlinear PDEs, including the Klein-Gordon–Zakharov system with 960,000 DOF. Numerically, the structure-preserving lifting approach achieves accuracy comparable to state-of-the-art structure-preserving hyper-reduction (spDEIM) methods while reducing offline cost and maintaining energy conservation online, with strong performance in time- and parameter-extrapolation scenarios. The work broadens the class of nonlinear conservative PDEs amenable to energy-preserving, efficient reduced-order modeling and provides a practical framework for future data-driven extensions and noncanonical Hamiltonian systems.

Abstract

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.

Figures (12)

• Figure 1: Comparative study for the one-dimensional sine-Gordon equation. Plot (a) shows that the proposed energy-quadratization strategy yields quadratic ROMs that achieve lower relative state error than the quadratic ROMs obtained using the standard lifting approach. The energy error comparison in plot (b) demonstrates that the quadratic ROMs derived using proposed structure-preserving lifting approach conserve the lifted FOM energy exactly.
• Figure 2: Nonlinear wave equation with exponential nonlinearity. The relative state error comparison for $q$ in plots (a) and (b) show that quadratic ROMs obtained via the proposed structure-preserving lifting approach achieve similar state error to nonlinear ROMs obtained via PSD and spDEIM with $r_{\text{spDEIM}}=2r$ in both training and test data regimes. The relative state error comparison for $p$ in plots (a) and (b) indicates that, while all ROMs perform poorly with relative error in $p$ above $10^{-1}$, the nonlinear ROMs obtained via PSD and spDEIM achieve higher accuracy than the quadratic ROMs in both training and test regimes.
• Figure 3: Nonlinear wave equation with exponential nonlinearity. The efficacy comparison in plot (a) shows that the proposed structure-preserving lifting approach achieves similar accuracy at a substantially lower computational cost in the online stage than the spDEIM approach. Plot (b) shows that all ROMs demonstrate bounded energy error, with the PSD ROM achieving the lowest energy error. The solid black line in plot (b) indicates the end of the training data regime.
• Figure 4: Two-dimensional sine-Gordon equation. The relative state error comparison in plot (a) shows that proposed structure-preserving lifting approach achieves higher accuracy than the spDEIM approach in the training regime. Plot (b) shows that both approaches yield similar accuracy in the test data regime with the structure-preserving lifting approach performing marginally better.
• Figure 5: Two-dimensional sine-Gordon wave equation. Plot (a) shows that both spDEIM with $r_{\text{spDEIM}}=2r$ and structure-preserving lifting approaches provide similar efficacy whereas spDEIM ROMs with $r_{\text{spDEIM}}=r$ yield substantially lower efficacy. The energy error comparison in plot (b) shows that the structure-preserving lifting approach achieves lower energy error than both spDEIM ROMs whereas the PSD ROM achieves the lowest energy error. The solid black line in plot (b) indicates the end of the training time interval.

Theorems & Definitions (6)

• Remark 1
• Definition 1: Polynomialization and Quadratization bychkov2024exact
• Remark 2
• Theorem 1
• proof
• Remark 3