Time-adaptive HénonNets for separable Hamiltonian systems
This work addresses the need for time-adaptive methods in Hamiltonian systems for applications like physics simulations, but it is incremental as it extends existing HénonNets.
The authors tackled the problem of learning symplectic integrators for Hamiltonian systems with irregularly sampled data by proposing T-HénonNets, a neural network architecture that is symplectic by design and handles adaptive time steps, achieving results demonstrated through numerical experiments.
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 [1] and HénonNets [2] still require training data generated by fixed step sizes. To learn time-adaptive symplectic integrators, an extension to SympNets called TSympNets is introduced in [3]. The aim of this work is to do a similar extension for HénonNets. We propose a novel neural network architecture called T-HénonNets, which is symplectic by design and can handle adaptive time steps. We also extend the T-HénonNet architecture to non-autonomous Hamiltonian systems. Additionally, we provide universal approximation theorems for both new architectures for separable Hamiltonian systems and discuss why it is difficult to handle non-separable Hamiltonian systems with the proposed methods. To investigate these theoretical approximation capabilities, we perform different numerical experiments.