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

TL;DR

Conservative nonlinear PDEs require reduced-order models that respect underlying invariants. The authors introduce structure-preserving Lift & Learn, which uses energy-quadratization lifting to transform nonlinear dynamics into a quadratic lifted form and learns a linear reduced operator under energy-conservation constraints, ensuring a perturbed lifted energy is exactly conserved. The approach is demonstrated on three problems (1D nonlinear wave with exponential nonlinearity, 2D sine-Gordon, and 2D Klein-Gordon–Zakharov), showing accuracy and efficiency comparable to or better than nonintrusive Hamiltonian Operator Inference with spDEIM, while avoiding additional hyper-reduction. This yields scalable, physics-respecting ROMs for conservative PDEs and extends applicability to coupled systems lacking a canonical Hamiltonian structure.

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.

Figures (11)

• Figure 1: The proposed structure-preserving Lift & Learn approach (right path) learns computationally efficient quadratic ROMs without any additional hyper-reduction step whereas the Hamiltonian Operator Inference approach (left path) requires additional Jacobian snapshot data for structure-preserving hyper-reduction. The dashed line style for the arrow from the nonlinear conservative PDE to the lifted quadratic FOM indicates that we only derive the symbolic form of the lifted quadratic FOM, and constructing the lifted FOM operators is not required.
• Figure 2: One-dimensional sine-Gordon PDE. Plot (a) shows that the standard Lift & Learn approach yields quadratic ROMs that achieve relative state error below $10^{-2}$ for ROMs of size $2r>6$ in the training data regime. However, the energy error comparison in plot (b) demonstrates that the standard learning approach yields unstable ROMs that exhibit unbounded energy error growth outside the training data regime. The solid black line in plot (b) indicates end of the training time interval.
• Figure 3: Comparison of structure-preserving Lift & Learn and HOpInf with spDEIM. Plots (a) and (b) show that both approaches learn accurate and stable ROMs with bounded FOM energy error.
• Figure 4: Nonlinear wave equation with exponential nonlinearity. Quadratic ROMs obtained via structure-preserving Lift & Learn achieve similar state error to nonlinear ROMs obtained via intrusive lifting and HOpInf with spDEIM in both training and test data regimes.
• Figure 5: Nonlinear wave equation with exponential nonlinearity. The efficacy comparison in plot (a) shows that the structure-preserving Lift & Learn approach achieves higher efficacy than the HOpInf with spDEIM approach. Plot (b) shows that all ROMs demonstrate bounded energy error. The solid black line in plot (b) indicates the end of the training data regime.

Theorems & Definitions (4)

• Proposition 1
• proof
• Remark 1
• Remark 2