# Learning symplectic model reduction based on a approximation theorem of symplectic embeddings Source: [Learning symplectic model reduction based on a approximation theorem of symplectic embeddings]() Collection index: [llms.txt]() Data updated: 2026-06-05T05:00:04.418Z Authors: Liyi Feng, Yifa Tang, Yulin Xie, Ruili Zhang, Aiqing Zhu Published: 2026-06-03T08:58:44.000Z ## Abstract High-dimensional Hamiltonian systems play a central role in many scientific and engineering disciplines, with dynamics evolving on symplectic manifolds. Although deep learning provides powerful tools for constructing low-dimensional surrogates from data, the intrinsic symplectic structure is easily destroyed during model reduction. As a result, a standard autoencoder may produce latent coordinates that do not support a Hamiltonian flow, leading to unstable long-time prediction. In this paper, we first establish a universal approximation theorem for symplectic embeddings. Based on this theory, we propose symplecticity-preserving autoencoders (SpAE), in which the decoder is parameterized as a symplectic embedding and the encoder is constructed as the corresponding symplectic projection. This architecture is expressive enough to approximate nonlinear symplectic embeddings and the associated symplectic projections, preserves the symplectic structure exactly by construction, and can be trained by standard unconstrained optimization, thereby improving both reconstruction and prediction accuracy. Extensive experiments on high-dimensional lattice and particle systems demonstrate the effectiveness of the proposed method. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Reconstruction on Two-Stream System](): Reconstruction; Two-Stream System; Absolute Error: 1.3656. - [Full Prediction on Two-Stream System](): Full Prediction; Two-Stream System; Absolute Error: 5.3817. - [Reconstruction on Crystal Lattice System](): Reconstruction; Crystal Lattice System; Absolute Error: 2.6579. - [Trajectory Prediction on Crystal Lattice System](): Trajectory Prediction; Crystal Lattice System; Absolute Error: 8.1953. - [Full Prediction on Tokamak System](): Full Prediction; Tokamak System; Absolute Error: 3.4109. - [Reconstruction on Tokamak System](): Reconstruction; Tokamak System; Absolute Error: 9.8653. ## Original sources and implementation [Original paper]() [PDF]()