[Paper Review] Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds

Patrick Buchfink, Silke Glas+1 more|arXiv (Cornell University)|Dec 20, 2021

Model Reduction and Neural Networks4 citations

TL;DR

This paper introduces symplectic manifold Galerkin (SMG) projection for model order reduction of Hamiltonian systems on nonlinear manifolds, preserving symplecticity, energy, and stability. The method uses a weakly symplectic deep convolutional autoencoder to construct a nonlinear symplectic trial manifold, achieving superior accuracy with lower reduced dimensions than classical linear-subspace or non-symplectic manifold methods.

ABSTRACT

Classical model reduction techniques project the governing equations onto linear subspaces of the high-dimensional state-space. For problems with slowly decaying Kolmogorov-n-widths such as certain transport-dominated problems, however, classical linear-subspace reduced-order models (ROMs) of low dimension might yield inaccurate results. Thus, the concept of classical linear-subspace ROMs has to be extended to more general concepts, like Model Order Reduction (MOR) on manifolds. Moreover, as we are dealing with Hamiltonian systems, it is crucial that the underlying symplectic structure is preserved in the reduced model, as otherwise it could become unphysical in the sense that the energy is not conserved or stability properties are lost. To the best of our knowledge, existing literature addresses either MOR on manifolds or symplectic model reduction for Hamiltonian systems, but not their combination. In this work, we bridge the two aforementioned approaches by providing a novel projection technique called symplectic manifold Galerkin (SMG), which projects the Hamiltonian system onto a nonlinear symplectic trial manifold such that the reduced model is again a Hamiltonian system. We derive analytical results such as stability, energy-preservation and a rigorous a-posteriori error bound. Moreover, we construct a weakly symplectic deep convolutional autoencoder as a computationally practical approach to approximate a nonlinear symplectic trial manifold. Finally, we numerically demonstrate the ability of the method to outperform (non-)structure-preserving linear-subspace ROMs and non-structure-preserving MOR on manifold techniques.

Motivation & Objective

• Address the limitation of classical linear-subspace reduced-order models (ROMs) in capturing transport-dominated problems with slow Kolmogorov-n-width decay.
• Bridge the gap between symplectic model reduction and manifold-based model order reduction (MOR), which have been studied separately.
• Develop a projection technique that preserves the Hamiltonian structure—specifically symplecticity, energy conservation, and Lyapunov stability—on nonlinear manifolds.
• Construct a computationally practical, weakly symplectic deep convolutional autoencoder (DCA) to approximate the nonlinear symplectic trial manifold.
• Demonstrate that the proposed SMG-ROM outperforms both non-structure-preserving ROMs and non-symplectic manifold-based MOR in accuracy and long-term stability.

Proposed method

• Propose the symplectic manifold Galerkin (SMG) projection technique, which projects the full-order Hamiltonian system onto a nonlinear symplectic trial manifold.
• Ensure the reduced system remains Hamiltonian by enforcing symplecticity through the manifold's geometric structure and projection constraints.
• Derive a rigorous a-posteriori error bound based on a time-discrete formulation to enable error control in the reduced model.
• Design a weakly symplectic deep convolutional autoencoder (DCA) that enforces symplecticity via architectural constraints rather than penalty terms.
• Use the same high-fidelity snapshots for training the DCA as in classical symplectic ROM, enabling fair comparison.
• Integrate the DCA with the SMG projection to generate a reduced model that preserves energy and stability under mild assumptions.

Experimental results

Research questions

• RQ1 Can symplectic model reduction be effectively extended to nonlinear manifolds to improve accuracy for problems with slow Kolmogorov-n-width decay?
• RQ2 Does preserving symplecticity in the reduced model lead to better energy conservation and long-term stability compared to non-symplectic approaches on manifolds?
• RQ3 Can a weakly symplectic deep convolutional autoencoder effectively approximate a nonlinear symplectic trial manifold while maintaining computational feasibility?
• RQ4 How does the SMG-ROM compare in accuracy and stability to classical linear-subspace ROMs and non-symplectic manifold-based MOR for Hamiltonian systems?
• RQ5 What is the impact of symplecticity enforcement via architecture versus loss-based penalty in autoencoder-based manifold learning for ROM?

Key findings

• The SMG-ROM preserves symplecticity, energy, and Lyapunov stability under mild assumptions, ensuring physical consistency of the reduced model.
• The weakly symplectic DCA ensures that the Hamiltonian error remains below 5.5×10⁻⁴ over time, significantly outperforming non-symplectic DCAs, which exhibit Hamiltonian errors up to 2 at t=1.
• For a moving pulse governed by a wave equation, the SMG-ROM achieves low reduction error with a very low-dimensional manifold, outperforming classical linear-subspace ROMs.
• Non-symplectic manifold-based MOR methods yield poor solutions despite manifold learning, indicating that symplecticity is essential for accuracy in Hamiltonian systems.
• The method inherits challenges from autoencoder-based approaches, including increased hyperparameters and lack of full offline-online separation, motivating future work on hyper-reduction and symplectic architecture design.
• Numerical results show that the SMG-ROM with weakly symplectic DCA maintains low Hamiltonian error and symplecticity error, while non-symplectic variants diverge significantly in energy preservation.