Time-adaptive SympNets for separable Hamiltonian systems

2025/09/19 by Konrad Janik, Peter Benner, Janik, Konrad +1 · 1 citation
Engineering · #Control and Stability of Dynamical Systems #FOS: Computer and information sciences #Machine Learning (cs.LG)

paper · pdf · doi:10.48550/arxiv.2509.16026

openalex publication_date 2025/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Measurement data is often sampled irregularly i.e. not on equidistant time grids. This is also true for Hamiltonian systems. However, existing machine learning methods, which learn symplectic integrators, such as SympNets [20] and HénonNets [4] still require training data generated by fixed step sizes. To learn time-adaptive symplectic integrators, an extension to SympNets, which we call TSympNets, was introduced in [20]. We adapt the architecture of TSympNets and extend them to non-autonomous Hamiltonian systems. So far the approximation qualities of TSympNets were unknown. We close this gap by providing a universal approximation theorem for separable Hamiltonian systems and show that it is not possible to extend it to non-separable Hamiltonian systems. To investigate these theoretical approximation capabilities, we perform different numerical experiments. Furthermore we fix a mistake in a proof of a substantial theorem [25, Theorem 2] for the approximation of symplectic maps in general, but specifically for symplectic machine learning methods.

Citations

• Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftings
• Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps
• Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks
• Nonseparable Symplectic Neural Networks
• Fast neural Poincaré maps for toroidal magnetic fields
• Lagrangian Neural Networks
• SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems
• Hamiltonian Neural Networks
• Neural Ordinary Differential Equations
• A Multivariate Faa di Bruno Formula with Applications

Cited by

• Time-adaptive HénonNets for separable Hamiltonian systems
• Symplectic convolutional neural networks

Related

• Time-adaptive HénonNets for separable Hamiltonian systems
• SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems
• Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps
• Learning Hamiltonian Systems with Pseudo-symplectic Neural Network
• Hamiltonian Matching for Symplectic Neural Integrators