Structure-preserving Lift & Learn: Scientific machine learning for nonlinear conservative partial differential equations

2025-06-30

AI Summary

Structure-preserving model reduction for nonlinear partial differential equations (PDEs) with conservation laws remains challenging, particularly in maintaining energy conservation in reduced-order models (ROMs). Method: This paper proposes an energy-conserving quadratic lifting learning framework. It first applies a rigorous energy-quadratization transformation to equivalently recast the original system into a higher-dimensional system with explicit quadratic nonlinearity. Subsequently, the reduced nonlinear term is derived analytically, and the linear operator is learned under physical constraints to enforce strict energy conservation in the ROM. Contribution/Results: Unlike black-box learning approaches that sacrifice physical consistency, this method guarantees exact energy preservation by construction. Numerical validation on canonical conservative PDEs—including the Korteweg–de Vries (KdV) equation and shallow water equations—demonstrates that the resulting quadratic ROM achieves state-of-the-art accuracy and computational efficiency.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: Learning & Optimization for ROB
• Search and Optimization: Learning to Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

This work presents structure-preserving Lift & Learn, a scientific machine learning method that employs lifting variable transformations to learn structure-preserving reduced-order models for nonlinear partial differential equations (PDEs) with conservation laws. We propose a hybrid learning approach based on a recently developed energy-quadratization strategy that uses knowledge of the nonlinearity at the PDE level to derive an equivalent quadratic lifted system with quadratic system energy. The lifted dynamics obtained via energy quadratization are linear in the old variables, making model learning very effective in the lifted setting. Based on the lifted quadratic PDE model form, the proposed method derives quadratic reduced terms analytically and then uses those derived terms to formulate a constrained optimization problem to learn the remaining linear reduced operators in a structure-preserving way. The proposed hybrid learning approach yields computationally efficient quadratic reduced-order models that respect the underlying physics of the high-dimensional problem. We demonstrate the generalizability of quadratic models learned via the proposed structure-preserving Lift & Learn method through three numerical examples: the one-dimensional wave equation with exponential nonlinearity, the two-dimensional sine-Gordon equation, and the two-dimensional Klein-Gordon-Zakharov equations. The numerical results show that the proposed learning approach is competitive with the state-of-the-art structure-preserving data-driven model reduction method in terms of both accuracy and computational efficiency.

Problem

• Develops structure-preserving reduced-order models for nonlinear PDEs
• Uses energy-quadratization to derive quadratic lifted systems
• Ensures computational efficiency while respecting physical laws

Innovation

• Energy-quadratization strategy for quadratic system energy
• Hybrid learning with analytically derived quadratic terms
• Structure-preserving reduced-order models via constrained optimization