Learning symplectic model reduction based on a approximation theorem of symplectic embeddings

2026-06-03

AI Summary

This work addresses the instability in long-term predictions of high-dimensional Hamiltonian systems caused by the loss of symplectic structure during dimensionality reduction, which prevents the latent space from supporting Hamiltonian flow. To resolve this, the authors propose the Symplecticity-preserving Autoencoder (SpAE), whose decoder acts as a symplectic embedding and encoder as its corresponding symplectic projection, thereby rigorously preserving the symplectic structure. Built upon the first-ever universal approximation theorem for symplectic embeddings, this architecture enables unconstrained optimization during training without requiring additional constraints to enforce symplecticity. Experiments on high-dimensional lattice and particle systems demonstrate that SpAE significantly improves reconstruction accuracy and long-term dynamical stability, confirming the method’s effectiveness and superiority.

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.

Problem

• symplectic structure
• model reduction
• Hamiltonian systems
• autoencoder
• long-time prediction

Innovation

• symplectic autoencoder
• symplectic embedding
• Hamiltonian system
• model reduction
• structure-preserving learning