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

2025-03-04

AI Summary

Existing model reduction techniques for conservative partial differential equations (PDEs) with non-polynomial nonlinearities suffer from prohibitively expensive offline computations and difficulty in preserving energy conservation. To address this, we propose a structure-preserving nonlinear model reduction method based on lifting variable transformations. Our approach introduces a novel energy quadratization strategy that exactly recasts the original nonlinear energy functional into a quadratic form; energy conservation is then rigorously enforced in the lifted state space. The reduced-order model is constructed via proper orthogonal decomposition (POD) applied to the quadratized system, yielding a quadratic reduced model. We validate the method on four canonical classes of nonlinear conservative PDEs. Numerical results demonstrate that the online phase achieves accuracy and efficiency comparable to state-of-the-art hyper-reduction methods, while significantly reducing offline computational cost. This work establishes a new paradigm for efficient, structure-preserving simulation of high-dimensional nonlinear conservative systems.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Addresses computational bottlenecks in model reduction for conservative PDEs.
• Proposes a method to preserve energy in reduced-order models with nonlinearities.
• Demonstrates efficiency and accuracy in handling high-dimensional nonlinear PDEs.

Innovation

• Lifting transformations quadratize energy for model reduction.
• Energy-quadratization strategy preserves system structure.
• Combines with proper orthogonal decomposition for efficiency.