Title: Learning Symplectic Model Reduction Based on an Approximation Theorem for Symplectic Embeddings

High-dimensional Hamiltonian systems are fundamental to numerous fields within science and engineering, characterized by dynamics that evolve along symplectic manifolds. While deep learning offers robust mechanisms for generating low-dimensional surrogates from data, the process of model reduction often inadvertently disrupts the intrinsic symplectic structure. Consequently, standard autoencoders may yield latent coordinates incapable of supporting Hamiltonian flow, which typically results in unstable predictions over extended time periods.

To address these challenges, this study first derives a universal approximation theorem specifically for symplectic embeddings. Building upon this theoretical foundation, we introduce symplecticity-preserving autoencoders (SpAE). In this architecture, the decoder is defined as a symplectic embedding, while the encoder is formulated as the corresponding symplectic projection. This design possesses sufficient expressive power to approximate both nonlinear symplectic embeddings and their associated projections. Crucially, it maintains the symplectic structure exactly by design and remains compatible with standard unconstrained optimization techniques. These features collectively enhance both reconstruction fidelity and predictive accuracy. The efficacy of the proposed approach is validated through comprehensive experiments involving high-dimensional lattice and particle systems.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC