Learning symplectic model reduction based on a approximation theorem of symplectic embeddings
For scientists and engineers modeling Hamiltonian systems, this work provides a method to maintain long-term stability in reduced-order models, which is a known bottleneck in structure-preserving model reduction.
The paper addresses the problem of preserving symplectic structure in model reduction of high-dimensional Hamiltonian systems. It proposes symplecticity-preserving autoencoders (SpAE) that exactly preserve symplectic structure, improving reconstruction and prediction accuracy, as demonstrated on lattice and particle systems.
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.