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 possess invariants such as energy, momentum, and angular momentum, depending on their underlying symmetries. Traditional numerical integration methods sometimes violate these conservation laws, propagating errors in time, ultimately reducing the predictability of the method. We present a data-free Hamiltonian neural network that solves the differential equations that govern dynamical systems. This is an equation-driven unsupervised learning method where the optimization process of the network depends solely on the predicted functions without using any ground truth data. This unsupervised model learns solutions that satisfy identically, up to an arbitrarily small error, Hamiltons equations and, therefore, conserve the Hamiltonian invariants. Once the network is optimized, the proposed architecture is considered a symplectic unit due to the introduction of an efficient parametric form of solutions. In addition, the choice of an appropriate activation function drastically improves 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, a 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.
Download