Mattheakis Marios, Sondak David, Dogra Akshunna S, Protopapas Pavlos
John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA.
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom.
Phys Rev E. 2022 Jun;105(6-2):065305. doi: 10.1103/PhysRevE.105.065305.
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 Hénon-Heiles 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.
将机器学习应用于研究动力系统已引发一股热潮。我们提出了一种哈密顿神经网络，它能求解支配动力系统的微分方程。这是一种方程驱动的机器学习方法，其中网络的优化过程仅取决于预测函数，而不使用任何真实数据。该模型学习的解在任意小的误差范围内满足哈密顿方程，因此能守恒哈密顿不变量。选择合适的激活函数可显著提高网络的可预测性。此外，还进行了误差分析，结果表明数值误差取决于整体网络性能。然后，利用哈密顿网络求解非线性振荡器和混沌的亨农 - 海尔斯动力系统的方程。在这两个系统中，对于预测的相空间轨迹，辛欧拉积分器要达到与哈密顿网络相同的数值误差阶数，所需的评估点数要多两个阶数。
