Learning collective variables that respect permutational symmetry

In addition to translational and rotational symmetries, clusters of identical interacting particles possess permutational symmetry. Coarse-grained models for such systems are instrumental in identifying metastable states, providing an effective description of their dynamics, and estimating transition rates. We propose a numerical framework for learning collective variables that respect translational, rotational, and permutational symmetries and for estimating transition rates and residence times. It combines a sort-based featurization, residence manifold learning in the feature space, and learning of collective variables with autoencoders whose loss function utilizes the orthogonality relationship [F. Legoll and T. Lelievre, Nonlinearity 23, 2131-2163 (2010)]. The committor of the resulting reduced model is used as the reaction coordinate in the forward flux sampling and to design a control for sampling the transition path process. We offer two case studies, the Lennard-Jones-7 in 2D and the Lennard-Jones-8 in 3D. The transition rates and residence times computed with the aid of the reduced models agree with those obtained via brute-force methods.