  For nearly the past 30 years, Centroid Molecular Dynamics (CMD) has proven to
be a viable classical-like phase space formulation for the calculation of
quantum dynamical properties. However, calculation of the centroid effective
force remains a significant computational cost and limits the ability of CMD to
be an efficient approach to study condensed phase quantum dynamics. In this
paper we introduce a neural network-based methodology for first learning the
centroid effective force from path integral molecular dynamics data, which is
subsequently used as an effective force field to evolve the centroids directly
with the CMD algorithm. This method, called Machine-Learned Centroid Molecular
Dynamics (ML-CMD) is faster and far less costly than both standard on the fly
CMD and ring polymer molecular dynamics (RPMD). The training aspect of ML-CMD
is also straightforwardly implemented utilizing the DeepMD software kit. ML-CMD
is then applied to two model systems to illustrate the approach: liquid
para-hydrogen and water. The results show comparable accuracy to both CMD and
RPMD in the estimation of quantum dynamical properties, including the
self-diffusion constant and velocity time correlation function, but for
significantly reduced overall computational cost.
