Hamiltonian neural networks for solving equations of motion
This provides a more efficient method for simulating dynamical systems in physics and engineering, though it is incremental as it builds on existing neural network approaches.
The paper tackles solving differential equations for dynamical systems by introducing a Hamiltonian neural network that learns solutions without ground truth data, achieving the same numerical error in phase space trajectories with two orders fewer evaluation points than a symplectic Euler integrator for nonlinear oscillator and chaotic Henon-Heiles systems.
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 equation-driven 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 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.