  This article discusses applications of Bayesian machine learning for quantum
molecular dynamics. One particular formulation of quantum dynamics advocated
here is in the form of a machine learning simulator of the Schr\"{o}dinger
equation. If combined with the Bayesian statistics, such a simulator allows one
to obtain not only the quantum predictions but also the error bars of the
dynamical results associated with uncertainties of inputs (such as the
potential energy surface or non-adiabatic couplings) into the nuclear
Schr\"{o}dinger equation. Instead of viewing atoms as undergoing dynamics on a
given potential energy surface, Bayesian machine learning allows one to
formulate the problem as the Schr\"{o}dinger equation with a non-parametric
distribution of potential energy surfaces that becomes conditioned by the
desired dynamical properties (such as the experimental measurements). Machine
learning models of the Schr\"{o}dinger equation solutions can identify the
sensitivity of the dynamical properties to different parts of the potential
surface, the collision energy, angular momentum, external field parameters and
basis sets used for the calculations. This can be used to inform the design of
efficient quantum dynamics calculations. Machine learning models can also be
used to correlate rigorous results with approximate calculations, providing
accurate interpolation of exact results. Finally, there is evidence that it is
possible to build Bayesian machine learning models capable of physically
extrapolating the solutions of the Schr\"{o}dinger equation. This is
particularly valuable as such models could complement common discovery tools to
explore physical properties at Hamiltonian parameters not accessible by
rigorous quantum calculations or experiments, and potentially be used to
accelerate the numerical integration of the nuclear Schr\"{o}dinger equation.
