Symmetry Control Neural Networks

This paper continues the quest for designing the optimal physics bias for neural networks predicting the dynamics of systems when the underlying dynamics shall be inferred from the data directly. The description of physical systems is greatly simplified when the underlying symmetries of the system are taken into account. In classical systems described via Hamiltonian dynamics this is achieved by using appropriate coordinates, so-called cyclic coordinates, which reveal conserved quantities directly. Without changing the Hamiltonian, these coordinates can be obtained via canonical transformations. We show that such coordinates can be searched for automatically with appropriate loss functions which naturally arise from Hamiltonian dynamics. As a proof of principle, we test our method on standard classical physics systems using synthetic and experimental data where our network identifies the conserved quantities in an unsupervised way and find improved performance on predicting the dynamics of the system compared to networks biasing just to the Hamiltonian. Effectively, these new coordinates guarantee that motion takes place on symmetry orbits in phase space, i.e.~appropriate lower dimensional sub-spaces of phase space. By fitting analytic formulae we recover that our networks are utilising conserved quantities such as (angular) momentum.