  Free energy perturbation (FEP) was proposed by Zwanzig more than six decades
ago as a method to estimate free energy differences, and has since inspired a
huge body of related methods that use it as an integral building block. Being
an importance sampling based estimator, however, FEP suffers from a severe
limitation: the requirement of sufficient overlap between distributions. One
strategy to mitigate this problem, called Targeted Free Energy Perturbation,
uses a high-dimensional mapping in configuration space to increase overlap of
the underlying distributions. Despite its potential, this method has attracted
only limited attention due to the formidable challenge of formulating a
tractable mapping. Here, we cast Targeted FEP as a machine learning problem in
which the mapping is parameterized as a neural network that is optimized so as
to increase overlap. We develop a new model architecture that respects
permutational and periodic symmetries often encountered in atomistic
simulations and test our method on a fully-periodic solvation system. We
demonstrate that our method leads to a substantial variance reduction in free
energy estimates when compared against baselines, without requiring any
additional data.
