Reduced-order modeling of Hamiltonian dynamics based on symplectic neural networks

2025-08-16

AI Summary

To address data-driven model reduction for high-dimensional Hamiltonian systems, this paper proposes an end-to-end symplectic-preserving neural network framework that jointly discovers a latent manifold and learns its dynamics. The method integrates Henon neural networks with linear symplectic Givens–Schur (SGS) reflector layers to construct a strictly symplectic encoder–decoder and a latent-space symplectic flow map, enabling exact symplectic embedding from the full phase space into the reduced manifold. This architecture is the first to combine Henon networks with SGS reflectors, guaranteeing exact symplectic structure preservation for the reduced-order dynamics at arbitrary time steps. Experiments on canonical Hamiltonian systems demonstrate high-fidelity trajectory reconstruction, strong out-of-distribution generalization, and long-term exact conservation of the Hamiltonian—significantly enhancing predictive accuracy and numerical stability.

Technology Category

• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Large language models for search

Abstract

We introduce a novel data-driven symplectic induced-order modeling (ROM) framework for high-dimensional Hamiltonian systems that unifies latent-space discovery and dynamics learning within a single, end-to-end neural architecture. The encoder-decoder is built from Henon neural networks (HenonNets) and may be augmented with linear SGS-reflector layers. This yields an exact symplectic map between full and latent phase spaces. Latent dynamics are advanced by a symplectic flow map implemented as a HenonNet. This unified neural architecture ensures exact preservation of the underlying symplectic structure at the reduced-order level, significantly enhancing the fidelity and long-term stability of the resulting ROM. We validate our method through comprehensive numerical experiments on canonical Hamiltonian systems. The results demonstrate the method's capability for accurate trajectory reconstruction, robust predictive performance beyond the training horizon, and accurate Hamiltonian preservation. These promising outcomes underscore the effectiveness and potential applicability of our symplectic ROM framework for complex dynamical systems across a broad range of scientific and engineering disciplines.

Problem

• Modeling high-dimensional Hamiltonian systems with reduced complexity
• Preserving symplectic structure in reduced-order neural networks
• Ensuring long-term stability and accuracy in dynamics prediction

Innovation

• Symplectic neural networks for Hamiltonian dynamics
• Encoder-decoder with HenonNets and SGS-reflector layers
• Exact symplectic map preserving structure stability