  The typically rugged nature of molecular free energy landscapes can frustrate
efficient sampling of the thermodynamically relevant phase space due to the
presence of high free energy barriers. Enhanced sampling techniques can improve
phase space exploration by accelerating sampling along particular collective
variables (CVs). A number of techniques exist for data-driven discovery of CVs
parameterizing the important large scale motions of the system. A challenge to
CV discovery is learning CVs invariant to symmetries of the molecular system,
frequently rigid translation, rigid rotation, and permutational relabeling of
identical particles. Of these, permutational invariance have proved a
persistent challenge in frustrating the the data-driven discovery of
multi-molecular CVs in systems of self-assembling particles and
solvent-inclusive CVs for solvated systems. In this work, we integrate
Permutation Invariant Vector (PIV) featurizations with autoencoding neural
networks to learn nonlinear CVs invariant to translation, rotation, and
permutation, and perform interleaved rounds of CV discovery and enhanced
sampling to iteratively expand sampling of configurational phase space and
obtain converged CVs and free energy landscapes. We demonstrate the
Permutationally Invariant Network for Enhanced Sampling (PINES) approach in
applications to the self-assembly of a 13-atom Argon cluster,
association/dissociation of a NaCl ion pair in water, and hydrophobic collapse
of a C45H92 n-pentatetracontane polymer chain. We make the approach freely
available as a new module within the PLUMED2 enhanced sampling libraries.
