Using conservation of energy - a fundamental property of closed classical and
quantum mechanical systems - we develop an efficient gradient-domain machine
learning (GDML) approach to construct accurate molecular force fields using a
restricted number of samples from ab initio molecular dynamics (AIMD)
trajectories. The GDML implementation is able to reproduce global potential
energy surfaces of intermediate-sized molecules with an accuracy of 0.3 kcal
$\text{mol}^{-1}$ for energies and 1 kcal $\text{mol}^{-1}$ $\text{\AA}^{-1}$
for atomic forces using only 1000 conformational geometries for training. We
demonstrate this accuracy for AIMD trajectories of molecules, including
benzene, toluene, naphthalene, ethanol, uracil, and aspirin. The challenge of
constructing conservative force fields is accomplished in our work by learning
in a Hilbert space of vector-valued functions that obey the law of energy
conservation. The GDML approach enables quantitative molecular dynamics
simulations for molecules at a fraction of cost of explicit AIMD calculations,
thereby allowing the construction of efficient force fields with the accuracy
and transferability of high-level ab initio methods.

利用能量守恒定律，我们开发了一种高效的梯度域机器学习方法（GDML），使用极少量的来自从头算分子动力学（AIMD）轨迹的样本构建准确的分子力场。这种方法可以在训练时仅使用 1000 个构象几何结构，以 0.3 kcal/mol 的能量准确度和 1 kcal/mol・Å^-1 的原子力准确度再现中等大小的分子的全局势能面。该方法使得分子动力学模拟可以像高级从头算法方法一样具有准确性和可转移性，同时减少了计算代价。