Hamiltonian Neural Networks for solving differential equations

There has been a wave of interest in applying machine learning to study dynamical systems. In particular, neural networks have been applied to solve the equations of motion, and therefore, track the evolution of a system. In contrast to other applications of neural networks and machine learning, dynamical systems -- depending on their underlying symmetries -- possess invariants such as energy, momentum, and angular momentum. Traditional numerical iteration methods usually violate these conservation laws, propagating errors in time, and reducing the predictability of the method. We present a Hamiltonian neural network that solves the differential equations that govern dynamical systems. This unsupervised model is learning solutions that satisfy identically, up to an arbitrarily small error, Hamilton's equations and, therefore, conserve the Hamiltonian invariants. Once it is optimized, the proposed architecture is considered a symplectic unit due to the introduction of an efficient parametric form of solutions. In addition, by sharing the network parameters and the choice of an appropriate activation function drastically improve the predictability of the network. An error analysis is derived and states that the numerical errors depend on the overall network performance. The symplectic architecture is then employed to solve the equations for the nonlinear oscillator and the chaotic Henon-Heiles dynamical system. In both systems, the symplectic Euler integrator requires two orders more evaluation points than the Hamiltonian network in order to achieve the same order of the numerical error in the predicted phase space trajectories.