  We develop a combined machine learning (ML) and quantum mechanics approach
that enables data-efficient reconstruction of flexible molecular force fields
from high-level ab initio calculations, through the consideration of
fundamental physical constraints. We discuss how such constraints are recovered
and incorporated into ML models. Specifically, we use conservation of energy -
a fundamental property of closed classical and quantum mechanical systems -- to
derive an efficient gradient-domain machine learning (GDML) model. The
challenge of constructing conservative force fields is accomplished by learning
in a Hilbert space of vector-valued functions that obey the law of energy
conservation. We proceed with the development of a multi-partite matching
algorithm that enables a fully automated recovery of physically relevant
point-group and fluxional symmetries from the training dataset into a symmetric
variant of our model. The developed symmetric GDML (sGDML) approach is able to
faithfully reproduce global force fields at the accuracy high-level ab initio
methods, thus enabling sample intensive tasks like molecular dynamics
simulations at that level of accuracy.
