  We present an approach that extends the theory of targeted free energy
perturbation (TFEP) to calculate free energy differences and free energy
surfaces at an accurate quantum mechanical level of theory from a cheaper
reference potential. The convergence is accelerated by a mapping function that
increases the overlap between the target and the reference distributions.
Building on recent work, we show that this map can be learned with a
normalizing flow neural network, without requiring simulations with the
expensive target potential but only a small number of single-point
calculations, and, crucially, avoiding the systematic error that was found
previously. We validate the method by numerically evaluating the free energy
difference in a system with a double-well potential and by describing the free
energy landscape of a simple chemical reaction in the gas phase.
