论文标题
用于求解运动方程的哈密顿神经网络
Hamiltonian neural networks for solving equations of motion
论文作者
论文摘要
将机器学习应用于研究动态系统一直存在兴趣。我们提出了一个哈密顿神经网络，该神经网络解决了控制动态系统的微分方程。这是一种方程式驱动的机器学习方法，其中网络的优化过程仅取决于预测功能而无需使用任何基础真实数据。该模型学习了满足汉密尔顿方程式的解决方案，因此可以保存汉密尔顿不变的解决方案。适当的激活函数的选择大大提高了网络的可预测性。此外，得出了错误分析，并指出数值错误取决于整体网络性能。然后，使用哈密顿网络来求解非线性振荡器和混乱的亨逊 - 赫尔斯动力学系统的方程。在这两个系统中，与哈密顿网络相比，符合性的Euler积分器需要两个订单的评估点，以便在预测的相空间轨迹中达到数值误差相同的顺序。
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.
