Time-adaptive SympNets for separable Hamiltonian systems
This work addresses the need for flexible machine learning methods in physics simulations, but it is incremental as it builds on existing SympNets.
The authors tackled the problem of learning time-adaptive symplectic integrators for Hamiltonian systems with irregularly sampled data, extending TSympNets to non-autonomous systems and providing a universal approximation theorem for separable cases while showing limitations for non-separable ones.
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.