Hamiltonian neural networks for solving equations of motion
Abstract
There has been a wave of interest in applying machine learning to study dynamical systems. We present a Hamiltonian neural network that solves the differential equations that govern dynamical systems. This is an equationdriven machine learning method where the optimization process of the network depends solely on the predicted functions without using any ground truth data. The model learns solutions that satisfy, up to an arbitrarily small error, Hamilton's equations and, therefore, conserve the Hamiltonian invariants. The choice of an appropriate activation function drastically improves the predictability of the network. Moreover, an error analysis is derived and states that the numerical errors depend on the overall network performance. The Hamiltonian network is then employed to solve the equations for the nonlinear oscillator and the chaotic HénonHeiles dynamical system. In both systems, a symplectic Euler integrator requires two orders more evaluation points than the Hamiltonian network to achieve the same order of the numerical error in the predicted phase space trajectories.
 Publication:

Physical Review E
 Pub Date:
 June 2022
 DOI:
 10.1103/PhysRevE.105.065305
 arXiv:
 arXiv:2001.11107
 Bibcode:
 2022PhRvE.105f5305M
 Keywords:

 Physics  Computational Physics;
 Computer Science  Machine Learning
 EPrint:
 Phys. Rev. E 105, 065305 (2022)