Optimization of Hamiltonian Neural Networks using Liouville's Theorem

We study some optimization approaches on a specific type of artificial neural networks called Physics-Informed Neural Networks, PINNs for short. These types of networks make use of some of the groundwork already done in the scientific field, like Newtonian physical laws, and enforce this knowledge onto the network’s training to accurately predict data simulation of some systems, with as few data points as possible. The target system in this paper is the famous Hamiltonian system, which describes almost any classical/non-classical system, usually laid out in the form of partial differential equations. The main job here is then to find accurate approximations for the hidden solutions of these PDEs. In this work, examples expand to problems involving Hamiltonian systems such as the mass spring, pendulum, and 2 and 3-body systems. The work builds on the previous work of Greydanus paper on Hamiltonian Neural Networks, however we Integrate an additional theorem involved in Hamiltonian mechanics called Liouville’s theorem, which tells that the a conserved system’s dynamics propagation is in equilibrium with respect to its momentum and position combined. The results show that enforcing this equilibrium equation unto the network’s output and auto-differentiating it leads to better system integration and therefore, better energy outputs of the whole Hamiltonian system under study.